Beyond the Standard Model

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As of 2006, in particle physics, the predictions coming from the Standard Model—a model discovered in the early 1970s—agree with most of the experiments that have been conducted so far within current experimental bounds to a phenomenal accuracy despite decades of determined efforts by experimentalists to find all sorts of deviations from the Standard Model. The exceptions are neutrino oscillations and the 3.4 σ deviation of the anomalous magnetic dipole moment of the muon^[1]. Nevertheless, the vast majority of the high energy physicists feel that the Standard Model is not a complete model, but rather, an effective field theory, and has to be extended by some new physics at some higher energy scale. The most convincing reasons given are

Neutrino oscillations (not predicted by the Standard Model).
The Standard Model does not include gravity and gravitational corrections to the Renormalization group equation (RGE) will become significant near the Planck scale. Any UV fixed point for quantum gravity is likely to be very sensitive to the nongravitational sector.
New physics beyond the Standard Model may explain (fix) the various parameters of the Standard Model, such as mass parameters, which are completely free in the Standard Model, and the relations between them. The parameters of the Standard Model do not appear to be completely random. Many patterns can be found.
The Standard Model has a hierarchy problem, which requires fine-tuning.
Dark matter from cosmology.
The Standard Model has a Landau pole (this is larger than the Planck scale, though).

Of all of these reasons, only one (the gauge hierarchy problem) suggests new physics at the TeV scale.

1 Introduction
2 The Higgs sector
3 Operators
4 Symmetry violating operators
- 4.1 Global spacetime symmetries
  - 4.1.1 Lorentz/CPT violating operators
  - 4.1.2 CP violating operators
- 4.2 Global internal symmetries
  - 4.2.1 B/L/B−L violating operators
  - 4.2.2 Flavor changing operators
5 Various constraints
6 Why does the Standard Model have the field content it does?
7 New conserved charges
8 Decoupled light hidden sector
9 Existence/nonexistence of the Higgs boson
10 Elementary/composite
11 Supersymmetric/not supersymmetric
12 Four dimensions or more?
13 The anthropic principle
14 Various models
15 References
16 External links

[edit] Introduction

Renormalization group theory tells us that the description of a field theory depends upon the scale at which one examines the theory and that the descriptions at various scales are related by the renormalization group flow. It has also given us the concept of universality classes and effective field theories. If there are particles with masses of the order of Λ_new, then as we run the renormalization group scale Λ below Λ_new, these massive particles are effectively integrated out and their fields do not appear in the RG any longer. As these particles appear at scales higher than we have measured so far, their dynamics are called new physics. According to the logic of the renormalization group, the Standard Model happens to lie on a universality class of models. At the scale at which the new physics begins to get integrated out, we would often expect that the irrelevant coupling strengths of the unintegrated fields will be of order one due to contributions from the new physics (there are exceptions which will be mentioned later in this article). Being irrelevant, as we run further down the RG, these couplings will get smaller and smaller and we will get closer and closer to the universality class of the Standard Model. At the electroweak scale and for many orders of magnitude above it, the Standard Model happens to be a weakly coupled theory, which means that it lies fairly close to a Gaussian fixed point and can be well described by perturbation theory. This means that the RG parameters run fairly close to what one would expect for a classical field theory with only logarithmic corrections from the classical (tree level) predictions. Putting all of this together, if the classical scaling dimension of an operator happens to be Δ, then we will expect that its coefficient in the scale dependent action will be of order $\mathcal{O}(1/\Lambda_{new}^{\Delta-4})$ . The logarithmic corrections do not affect the order of magnitude prediction.

One of the goals of precision tests of the Standard Model is to measure the coupling strengths of various irrelevant operators and so far, with the exception of neutrino oscillations, none of them have measured any deviations from zero within current experimental bounds. This puts somewhat stringent constraints upon any extension of the Standard Model.

[edit] The Higgs sector

The Standard Model assumes that the Higgs sector only consists of one electroweak Higgs doublet. Many extensions to the Standard Model, like for instance MSSM, have a larger Higgs sector and moreover, these Higgs fields all appear at around the 100GeV scale (order of magnitude) and so, we can't integrate over any of them, and there are even some models without a Higgs sector (e.g. Higgsless models) which aren't technically extensions of the Standard Model.

As the Higgs sector is the least known part of the Standard Model (we haven't even detected the Higgs boson), the operators listed below which involve the Higgs field might not be appropriate because there might be more or even no Higgs fields.

[edit] Operators

Here, we will list all the Lorentz invariant operators which respect the Standard Model gauge symmetries with classical scaling dimensions less than 6 and some operators with classical scaling dimensions of 6.

H stands for the Standard Model Higgs field, B, W and G for the hypercharge, weak and chromodynamic gauge fields respectively and q,d^c,u^c,l,e^c for the fermions (each letter is understood to stand for all the three generations). Sometimes, f is used to represent all the fermions together for brevity and A for all the gauge fields together and F for the curvature forms.

There is some freedom in the choice of the flavor basis. Some fixing is assumed to have been made.

[edit] Dimension-0 operators ( $\sim \mathcal{O}(\Lambda^4)$ )

The cosmological constant

[edit] Dimension-1 operators ( $\sim \mathcal{O}(\Lambda^3)$ )

None (no gauge singlet scalar fields)

[edit] Dimension-2 operators ( $\sim \mathcal{O}(\Lambda^2)$ )

H*H (tachyonic mass) (the only dimensionful parameter in the Standard Model; dimensional transmutation aside)

[edit] Dimension-3 operators ( $\sim \mathcal{O}(\Lambda)$ )

None (no Majorana/Dirac fermions prior to electroweak breaking)

[edit] Dimension-4 operators ( $\sim \mathcal{O}(1)$ )

$DH^* \cdot DH$ (kinetic term) (this is normalized to 1 after wave function renormalization)
$\frac{1}{2g_1^2}dB\cdot dB$ (kinetic Yang-Mills term)
$\frac{1}{2g_2^2}Tr\left[DW \cdot DW\right]$ (kinetic Yang-Mills term)
$\frac{1}{2g_3^2}Tr\left[DG \cdot DG\right]$ (kinetic Yang-Mills term)
$\theta_W Tr\left[DW\cdot *DW\right]$ (theta term; this is a topological surface term which only comes into play nonperturbatively)
$\theta_C Tr\left[DG\cdot *DG\right]$ (theta term; topological surface term; only affects nonperturbative physics)
$\bar{f}\sigma\cdot Df$ (kinetic term) (normalized to 1_3×3; wavefunction renormalization)
$λ(H * H) 2$ (self-interaction)
$H q u c$ +c.c. (Yukawa)
$H * q d c$ +c.c. (Yukawa)
$H * l e c$ +c.c. (Yukawa)

[edit] Dimension-5 operators ( $\sim \mathcal{O}(1/\Lambda)$ )

$(H l) 2$ +c.c.

[edit] Some Dimension 6 operators ( $\sim \mathcal{O}(1/\Lambda^2)$ )

$H * l σ μ σ ν F μν e c$ +c.c.
$\partial^\rho F^{\mu\nu} \partial_\rho F_{\mu\nu}$
$f f f f$ +c.c.
$\bar{f}\bar{f}ff$ +c.c.
$| H * D H | 2$
$H^*F_{B\mu\nu}F_W^{\mu\nu}H$

[edit] Symmetry violating operators

Symmetry violating operators are often the most studied because a slight symmetry violation in a model which mostly symmetric (approximate symmetry) is much easier to detect. These symmetries might be explicitly broken by the new physics or they might be spontaneously broken.

[edit] Global spacetime symmetries

The only exact global spacetime symmetries (ignoring general relativity) of the Standard Model are the Poincaré symmetry and the CPT symmetry. Both the C-symmetry and the P-symmetry are maximally violated by the chiral fermions. The CP-symmetry is only slightly violated.

[edit] Lorentz/CPT violating operators

While there has been no evidence of Lorentz/CPT violation so far, the possibility for such violations is still there. CPT violations are only possible if there are also Lorentz violations.

[edit] CP violating operators

The Standard Model is mostly CP preserving. The imaginary components of the CKM entries are fairly small and the theta angle is exceedingly small. Tests include anomalous electric dipole moments and neutral kaon-antikaon oscillations.

[edit] Global internal symmetries

In the absence of the Yukawa couplings, the Standard Model has a $U(3)_q \times U(3)_{u^c} \times U(3)_{d^c} \times U(3)_{l} \times U(3)_{e^c}$ global internal symmetry. Each U(3) factor can be split as $U(1) \times SU(3)$ where the SU(3) part represents the flavor symmetries. Most linear combinations of the U(1)'s (as Lie algebras) are broken by anomalies and one particular combination turns out to be none other than U(1)_Y, the hypercharge, which is really a gauge symmetry. Apart from hypercharge, only B−L isn't anomalously broken. However, the anomalous breaking of these U(1) symmetries are pretty much unobservable in colliders and so, they still function pretty well as approximate symmetries.

The Yukawa coupling $λ l H * l e c$ where λ_l is a 3×3 matrix explicitly breaks $U(3)_l \times U(3)_{e^c}$ into $U (1) 3$ , corresponding to the electron lepton number, the muon lepton number and the tauon lepton number (the lepton flavor symmetries). Majorana neutrino masses, unlike Dirac neutrino masses, would break lepton number and in either case neutrino mixings break the lepton flavor symmetries although their effects are tiny.

$λ u H q u c$ by itself will break $U(3)_q \times U(3)_{u^c} \times U(3)_{d^c}$ into $U(1)^3 \times U(3)_{d^c}$ and $λ d H * q d c$ by itself will break $U(3)_q \times U(3)_{u^c} \times U(3)_{d^c}$ into $U(1)^3 \times U(3)_{u^c}$ . If both λ_u and λ_d are simultaneously diagonal, the combination of both Yukawa couplings will break the symmetry to $U (1) 3$ which will give both the baryon number and the quark flavor symmetries. It turns out that both matrices are nearly simultaneously diagonalizable, but not quite, leaving the flavor symmetries as approximate. In reality, the only unbroken symmetry (ignoring anomalies) is the baryon number.

[edit] B/L/B−L violating operators

Tests for baryon number violation, lepton number violations, proton decay and neutron-antineutron oscillations.

[edit] Flavor changing operators

Flavor is approximately conserved in the Standard Model. Only the weak interactions (or from another point of view, the coupling to the Higgs sector) violate it and even there, the violations are fairly small (weak processes are fairly slow). There are really two different kinds of flavor changing to consider: lepton flavor changing and quark flavor changing.

[edit] Various constraints

[edit] Cosmological constant problem

Main article: Cosmological constant problem

The cosmological constant is far smaller than what one would expect from radiative corrections. Supersymmetry protects the cosmological constant, but only up to the supersymmetry breaking scale, which is still too high. However, most particle physicists ignore this problem and leave it to cosmologists and quantum gravity researchers. Proposed solutions to this problem include the anthropic principle and inflation mechanisms.

[edit] Hierarchy problem

Main article: Hierarchy problem

The smallness of the Higgs mass. There are basically two parts to this problem:

Stabilizing the square of the Higgs mass from quadratic radiative corrections (the hierarchy problem)
Explaining why the Higgs mass is orders of magnitude smaller than the Planck scale (the mu problem)

supersymmetry, mu problem, doublet-triplet splitting problem, NMSSM, sliding VEV mechanism, composite Higgs, Randall-Sundrum models, pseudo Nambu-Goldstone mechanisms, anthropic principle, etc.

[edit] Gauge coupling unification

Main article: Gauge coupling unification

The three gauge couplings appear to unify in the renormalization group flow at high energies if we normalize hypercharge to agree with SU(5). This has been used as an argument for Grand Unified Theories.

[edit] Strong CP problem

Main article: Strong CP problem

The Standard Model has a few CP violating terms, namely the theta term and the complex parameters in the CKM matrix. The strong CP problem is the smallness of the theta term. Proposed solutions include the Peccei-Quinn theory involving axions.

[edit] Hierarchy of the Yukawa couplings between the three generations

The coefficients for the Yukawa couplings for the first generation are about 10^-5 times smaller than for the third generation. This is considered as something which needs to be explained by many physicists.

Froggatt-Nielsen mechanism, textures, Randall-Sundrum models

[edit] Fermion mass relations

The Yukawa couplings of the leptons and quarks for each generation have the same order of magnitude. Some Grand Unified Theories like SU(5) and SO(10) make predictions on the ratio of these Yukawa couplings for each generation.

[edit] Smallness of the offdiagonal entries of the CKM matrix

The offdiagonal entries of the CKM matrix, are small but not zero. Some (simple) GUTs predict zero offdiagonal entries.

[edit] Anomalous magnetic dipole moment

Main article: anomalous magnetic dipole moment

The dimension 6 operator $H * l σ μ σ ν F μν e c / Λ 2$ +c.c. (and more general functions of momentum in the vertex function) has a CP-even part and a CP-odd part. Once the Higgs field acquires a VEV, the CP-even part contributes to the anomalous magnetic dipole moment of the charged leptons and the CP-odd part contributes to the anomalous electric dipole moment. Tests have focused primarily on the muon as the electron is too much lighter than the electroweak scale to be sensitive and the tauon is too shortlived to be measured.

[edit] Anomalous electric dipole moment

This is a measurement of CP violations.

[edit] μ^-→e^-γ

The decay of a muon to an electron and a photon has not been observed experimentally even though it is predicted by quite a number of extensions beyond the standard model. This constrains the possible models we can construct.

[edit] Neutrino masses and mixings (MNS matrix)

Main articles: neutrino mass, neutrino oscillations, seesaw mechanism

The masses of the neutrinos have not actually been measured yet. However, neutrino oscillations have been detected, and they are only possible if the neutrinos are massive. The (left-handed) neutrinos can be given a tiny Majorana mass by the dimension 5 operator

(H l) 2 / Λ

after the Higgs field acquires a VEV. This operator can be generated, for example by the seesaw mechanism. Basically, a very massive Majorana neutrino (called a sterile neutrino or a right handed neutrino) couples to the ordinary neutrino via a Yukawa coupling with the electroweak Higgs doublet in the simplest seesaw mechanism. Once the massive neutrino is integrated out of the renormalization group, the dimension 5 operator is generated. In fact, the masses of the neutrinos somewhat seems to agree with Λ being the GUT scale, which has been claimed as an evidence in favor of GUTs.

Other proposed explanations also exist (e.g. electroweak triplet Higgs, the generation of the dimension 5 operator without right-handed/sterile neutrinos, etc.).

Unlike the CKM matrix, the offdiagonal entries of the MNS matrix are large.

This dimension 5 operator is lepton number violating and generic lepton number violating processes at the TeV scale are expected to induce a neutrino Majorana mass which is too large.

Of course, it is also possible that the neutrino really has a tiny Dirac mass instead of a tiny Majorana mass. In that case, we need to have a right-handed neutrino which cross couples to it and lepton number isn't violated. This Dirac mass must have a different origin from the dimension 5 operator, which can only give a Majorana mass.

[edit] Neutrinoless double beta decay

Main article: Neutrinoless double beta decay

If the neutrino has a nonzero mass, it is possible to have neutrinoless double beta decay.

[edit] Proton decay

Main article: Proton decay

R-parity, GUTs, doublet-triplet splitting problem

[edit] Neutron-antineutron oscillations

[edit] Nonuniversal couplings

Main article: nonoblique correction

The weak charged current and weak neutral current couplings of the fermions might show some slight generation dependence.

[edit] Fourth generation

Electroweak precision experiments strongly constrain a yet undiscovered fourth generation of Standard Model fermions. Stable heavy 4-th generation quarks have been recently proposed as dark matter candidates ^[2]^[3].

The Standard Model extension includes all possible Lorentz violating operators of low dimensions to the Standard Model.

[edit] Solitons

Many models beyond the standard model, like grand unified theories and string theory among others, predict the existence of topological solitons like monopoles, cosmic strings and domain walls. An overabundance of monopoles and domain walls will overclose the universe and as such, are ruled out cosmologically. This constrains quite a number of models, depending upon how much we trust current cosmologolical calculations. The constraints coming from cosmic strings are less stringent.

[edit] Why does the Standard Model have the field content it does?

This is a question which some models try to explain.

[edit] New conserved charges

Many extensions of the Standard Model have additional charges besides the Standard Model charges with all the Standard Model fields being neutral under it. Some examples include the magnetic charge in GUTs, R-parity in MSSM, the momentum/parity in the extra dimensions in some Kaluza-Klein theories and T parity in some little Higgs models. The lightest particle which happens to be charged under this symmetry (e.g. magnetic monopoles and the lightest supersymmetric particle and lightest T particle) will be stable and will persist despite having masses higher than what we can currently produce in particle accelerators. This means that we won't find these particles in current particle accelerators but if any of these particles happened to have been produced in the early universe, they will still persist today and we might be able to detect them in the cosmic rays or they might function as dark matter (e.g. WIMPs). Cosmic ray experiments and the inferred dark matter density can place constraints on these models.

[edit] Decoupled light hidden sector

It is not necessary for all particles other than the Standard Model particles to have masses above the electroweak scale provided that they (the hidden sector) happen to be decoupled from the Standard Model phyics. Examples of this could be mirror matter models. Such models might be able to account for dark matter.

[edit] Existence/nonexistence of the Higgs boson

Here, by the Higgs boson, we do not refer to the scalar field responsible for the electroweak symmetry breaking which acquires a VEV or its Goldstone bosons which become the longitutinal components of the W and the Z bosons. Instead, we mean its quanta in the components of the field transverse to the Goldstone directions. The Higgs boson is present in most models like the Standard Model, MSSM and the little Higgs but not present in technicolor or Higgless models.

[edit] Elementary/composite

Models beyond the Standard Model can be classified according to whether the Standard Model fields are elementary or composite. All the SM fields are elementary in MSSM and GUTs (or at least, their compositeness scale is higher than the GUT scale). In composite Higgs models like top quark condensate and some UV completions of little Higgs models, only the Higgs boson is composite. Technicolor is an example of a model without a Higgs boson but with a composite Higgs field. In preon models, the SM fermions are composite. Some models have composite electroweak gauge bosons.

A problem of many composite models is explaining why the mass of the composite particle is so much smaller than the compositeness scale. Pseudo Nambu-Goldstone mechanisms have been invoked to solve this problem. It certainly isn't easy to get composite chiral fermions.

[edit] Supersymmetric/not supersymmetric

SUSY stabilizes the hierarchy problem and makes the gauge coupling unification more accurate. Models may be supersymmetric or not. Supersymmetric models may have low scale SUSY breaking or high scale SUSY breaking or even a mixture of both as in split supersymmetry.

[edit] Four dimensions or more?

If there are extra dimensions, what is the number of extra dimensions? Are they large or small? Or are some small and others large? Are they compact or noncompact? Are they flat or warped or are they curved spheres or Calabi-Yau manifolds or whatnot? Are the extra dimensions continuous or deconstructed (i.e. "discrete")? Are the extra dimensions orbifolded or not? If they are, what are the orbifold projections? Are there branes in the extra dimensions or not? If yes, what are their properties? As can be seen, introducing extra dimensions gives a lot of freedom to model builders.

A generic prediction of extra dimensional models is the existence of a tower of Kaluza-Klein excitations, that is, a tower of particles with all the same quantum numbers except for mass.

Most interacting models with extra dimensions are nonrenormalizable. However, this need not be a problem as there might be nontrivial UV fixed points or there might be some UV completion like string theory or dimensional deconstruction. However, the strength of the bulk couplings (e.g. the bulk Planck mass, the bulk gauge coupling strengths) gives an upper limit to the scale Λ for the UV completion in the bulk.

[edit] The anthropic principle

Apparently, the vast majority of the values for the parameters of the Standard Model are incompatible with the existence of life (see fine-tuned universe for more details). The anthropic principle tells us that the Standard Model has the field content it does and the parameters it has because these are the values that are likely to give rise to the existence of lifeforms intelligent enough to be self-aware. If we know the landscape of possible theories and prior distribution of these theories and also know the probability that any given theory will give rise to life, we will be able to make a statistical prediction of the parameters of the Standard Model.

[edit] Various models

A nonexhaustive list of various models

The Standard Model fermions as S dual solitons

antiGUT

[edit] References

^ Hagiwara, K.; Martin, A. D. and Nomura, Daisuke and Teubner, T. (2006). "Improved predictions for g-2 of the muon and alpha(QED)(M(Z)**2)".
^ S. L. Glashow, A Sinister Extension of the Standard Model to SU(3)XSU(2)XSU(2)XU(1), talk at XI Workshop on Neutrino Telescopes, Venice, preprint.
^ M. Yu. Khlopov, Composite dark matter from 4th generation, Pisma Zh. Eksp. Teor. Fiz. 83, 3-6 (2006) preprint.

[edit] External links

Articles from arXiv.org

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