LOG#059. Standard Model (I).

What is the Standard Model? We are going to study this theory in a long series of blog posts. I don’t pretend to be completely exhaustive but a starting point even for the layman. I do want to review what we know about the SM. That is, this one and the next blog-posts will be an introduction to the main ideas, concepts and issues about the SM. What it is, what it does, what it is not and what it does not should be clear at the end…Enjoy it!

The Standard Model (SM) of elementary ( or fundamental) particles and interactions is, from the mathematical viewpoint, a local, relativistic, gauge and quantum theory. Therefore, the SM is a theory made of 4 magic keywords

$\boxed{\mbox{SM=LOCAL+RELATIVISTIC+GAUGE+QUANTUM THEORY}}$

Local means that the gauge fields are functions depending on the local place you are. Gauge fields and matter fields are local functions of spacetime points! Gauge fields transform in certain form preserving covariance of equations in every frame, and similarly with fermionic fields. Relativistic means that the SM is a theory containing the requirements of Special Relativity. Gauge means that there are some kind of transformations called gauge transformations acting on gauge fields and spacetime, in a consistent way with locality and relativity. Finally, quantum means that objects described by the SM are quantum, and they manifest wave-particle duality. Furthermore, being a quantum theory means that there is some class of discreteness in the spectrum, although the most intricated issues of quantum theories (like the interpretation of wave functions) are also present in the SM. In fact, quantum theory means here that the SM is a theory satisfying the Planck’s minimum energy condition, i.e., $E=n\hbar \omega$ and similar relationships, like de Broglie’s wavelength. We can also use the Feynman path integral in the SM with suitable cautions. The SM describes 3 fundamental interactions:

i) Electromagnetic interactions.

ii) Weak nuclear interactions.

iii) Strong nuclear interactions.

In fact, the counting of “fundamental” interactions varies as energy increases, and when energy is about 100 GeV, the electromagnetic and the weak nuclear force are unified into the electroweak (EW) interaction. EW theory treats electromagnetism and weak nuclear forces on equal footing. Therefore, from this viewpoint at energies about 100 GeV ( in fact, the EW bosons, the W and Z bosons, arise at pole masses about 81 and 91 GeV, respectively) or higher, electromagnetism and weak nuclear forces are “unified”. This phenomenon says that at those energies and higher, at quantum level, we work with only “two” SM interactions:

i) Electroweak interactions.

ii) Strong nuclear interactions.

As the SM is a quantum theory, it contains quantum fields and it satisfies the main quantum principles found since the QM birth. In short, the SM is some kind of Quantum Field Theory (QFT) of Yang-Mills ( gauge) type. Indeed, it is defined by its own “gauge group” of local transformations $G_{SM}\equiv G_{local}$ , where

$G_{SM}=SU(3)_c\times SU(2)_L\times U(1)_Y$

Remark: sometimes $SU(2)_L$ is written as $SU(2)_w$

Remark (II): $\times$ is sometimes written $\otimes$ , and it means direct product of group elements.

The groups $SU(3), SU(2), U(1)$ are Lie groups. The infinitesimal generators of the corresponding Lie algebras

$su(3), su(2), u(1)$

are associated to each one of the interacting bosons which appear in Nature, i.e., to the bosons $Z, W^{\pm}, \gamma$ . Moreover, the indexes $c, L, Y$ distinguish these groups from appproximate symmetries which sometimes are considered in different frameworks and theories.

Yang-Mills (YM) theories like the SM have certain gauge theories as fundamental input and the representations of that gauge group consist of two important enumeration:

1st. Matter fermionic fields.

2nd. Bosonic force fields.

and the additional listing of the interactions among those two kind of fields

1st. Boson-Boson interactions.

2nd. Fermion-Boson interactions.

Interactions in the SM can be understood from Feynmann diagrams. The SM releveant Feynman diagrams are

Note that I have not included the Higgs boson interaction vertices and its Feynman diagrams here. The following graph summarizes every SM interaction and how particles interact (I include the Higgs particle as well in this case):

Thus, in this sense, there is two fundamental classes of “fields” in the SM, as the previous particle properties show:

Bosons are force carriers fields and they have entire spin, i.e., $s=0,1,2,...,\infty$ . However, generally, some principles restrict the possible values of spin in “physical theories” like the SM. We will see that the SM fields have spin 1 ( photons, gluons and massive photons like the W, Z bosons) and spin 0 ( the Higgs field/particle).

Fermions are matter fields obeying Fermi’s exclusion principle. They have spin $s=1/2,3/2,\ldots$ Fermions feeling only the electromagnetic and weak (electroweak, EW) interaction are called leptons. Fermions feeling the strong color force are called hadrons and they are made of constituent quarks. Quarks are confined into hadrons. They appear in couples of quark-antiquark groups named mesons and clustered groups of three quarks called baryons. There are some hypothetical objects in strong interaction theories named exotics. They are multiquark states like tetraquarks, pentaquarks,…and purely gluonic matter-energy sometimes referred as glueballs. There is no conclusive proof of exotics in the experimental aside, although we do know that theoretically are predicted by the Quantum Chromodynamics (QCD) and the Lattice Field Theories of strong interactions. In the SM there are only 1/2 fermions, called generally (Dirac) spinors. Fermions are also classified according to the kind of force/interaction they are able to feel or “smell”.

The main properties of these particles and interactions in the SM can be also summarized in a nice table:

The construction of the SM is a long procedure. It can be thought like some clever type of recipe ( imagine you are cooking some delicious dish, you need “cool ingredients”, aren’t yout?). The SM recipe is “complex” but “simple”. There are “many ingredients” but the prescription rules or “cooking procedures” are clear. There is two initial main steps:

1) Field enumeration. Thinking about the previous cooking analogy, fields are the SM “ingredients”.

$\mbox{FIELDS}\sim \mbox{INGREDIENTS}$

2) Gauging fields: the Local Gauge Theory procedure. In our cooking analogue model, gauging is equivalent to “the making” of the tasty “meal”, a.k.a., the SM!

$\mbox{Gauging}\sim \mbox{Cooking/Making the tasty dish}$

Firstly, the field enumeration begins with matter fields. Matter fields are spinorial fields. There are different kinds of spinors. However, without loss of generality at this moment, we can consider that every spinor field is of “Dirac Type/Class”. Dirac spinors are spin 1/2 fields. As we consider a 4D spacetime, Dirac spinors have 4 components. Dirac spinors also have fixed “helicity” ( eigenvalues of the helicity operator, the projection of spin along the direction of motion) and certain chirality ( eigenvalues of the chirality operator). The SM is a chiral theory. Since helicity is generally fixed, the SM is a sometimes say to be a chiral theory.

Remark: There are some relationships between chirality and helicity but we are not going to study it today. Essentially, there are Right-handed and Left-handed leptons/quarks excepting neutrinos. Observed neutrinos are known to be left-handed. There is no right-handed light neutrino. Then, we can think that chirality in the SM is related to the weak interactions and the neutrino fields, one of the most mysterious ingredients of the SM. Indeed, neutrinos could be Majorana spinors instead of Dirac spinors. Majorana spinors are essentially spinor fields being their own antiparticle, and thus, it is an option only available to electrically neutral particles like neutrinos. No other particle in the SM can be a Majorana spinor. We do know indeed that leptons and quarks are Dirac spinors.

Field enumeration

Being a relativistic theory, Lorentz invariance and causality allow the existence of pairs of particles and antiparticles, with the same mass but opposite electrical charge. Particles and antiparticles are described by the same spinor field. In the case of fields with definite helicity, the same spinor describes a left-handed (LH or L) fermion and its corresponding right-handed (RH or R) antifermion. Mathematically,

$\Psi_L (x)\approx \Psi_R (x)$

and where the L (R) symbols refer to the left and right chiral projectors

$\Psi_L\equiv \dfrac{1-\gamma_5}{2}$

$\Psi_R\equiv \dfrac{1+\gamma_5}{2}$

Therefore, as the R and L fields are related, we can treat only with the L part (as we do indeed with neutrinos).

Remark: In 1954, C.N.Yang and R.L. Mills introduced the hypothesis of isospin symmetry as a symmetry group of a local symmetry by the first time. Their main purpose was to apply the mathematical formalism they had invented for non-abelian gauge groups to the unified weak and electromagnetic interactions ( being the weak isospin and the hypercharge the gauge charges/symmetries). The GSW model was born in 1967 and the strong interaction theory based on local gauge group arised in the period 1968-1972. The whole YM gauge theory framework applies to any group with a finite number of generators. Then, theories for which the Lagrangian field has a complex group as a local symmetry are generally known in the literature as YM (gauge) theories, or gauge theories for short.

The matter fields in the SM can be classified into two larger groups. Each group is composed of 3 “generations” (G) or “families” (F). The counting of particles is as follows:

1) There are 45 fermionic elementary/fundamental particles, as far as we do know today divided between lepton and quark field species.

2) Leptonic fields. The electron, the positron and the electron neutrino, plus the muon, the antimuon and the muon neutrino, plus the tau particle, the antitau and the tau neutrino.

$N_{\mbox{leptons}}=3\times 3=9$

Note that we don’t include the antineutrino in the counting as a different particle!

3) Quark fields. There are 3 families as well. The up, down quark (u,d) familly. The charm, strange (c,s) quark family and finally the bottom, top (b, t) quark family. Moreover, there are 3 colors: red (r), green (g), and blue (b). Thus,

$N'_{\mbox{quarks-antiquarks}}=3\times 6\times 2=36$

and where the extra factor of 2 counts the quark field antiparticles.

Then, the free matter lagrangian can be easily determined. It consists of the kinetic part of the lagrangian including only fermionic fields ( leptons and quarks) without mass terms or interactions:

$\boxed{\displaystyle{\mathcal{L}_{m}^{\mbox{free}}=\sum_{j=1}^{45}\overline{\Psi}_{Lj}i\gamma^\mu \partial_\mu \Psi_{Lj}}}$

This part of the lagrangian does not include mass terms and has itself a $U_L(45)$ symmetry. Note that, we could have written more generally the free lagrangian:

$\displaystyle{\mathcal{L}_{m}^{\mbox{free}}=\sum_{j=1}^{45}\overline{\Psi}_{j}i\gamma^\mu \partial_\mu \Psi_{j}}$

and it would be invariant under the group $U(45)_L\times U(45)_R$ . That would be the maximal symmetry group we could create from these fermionic fields. The SM lagrangian is invariant only under the Left part since we can make the SM be a chiral theory in 4 dimensions. The final symmetry group has to be a subgroup of $U_L(45)$ or equivalently a $U_L(45)\times U_R(45)$ subgroup from the maximal symmetry viewpoint if we neglect mass terms. We will see how the mass terms dramatically change and reduce this large symmetry.

Gauging procedure

The second step into the first approaching to the SM consists of the gauging principle (or gauge principle). We promote the SM total group $G_{SM}$ , a subgroup of $U_L(45)$ , to a local symmetry group. It means that we “gauge” the theory. What does it mean? It is easy. The transformation laws of the matter fields or equivalently the irreducible group representations to which the matter fields belong to, must be the simplest possible transformations. Then, they are either in the trivial of the fundamental representation of the subgroups. The trivial representation of dimension 1 corresponds to the “singlet”. The fundamental representation corresponds to the representation whi no trivial of lowest dimension. The fundamental representation has dimension 2 ( they are “doublets”) in the case of $SU(2)$ and dimension 3 ( there are 2 of this, the 3 and $\overline{3}$ ) in the case of $SU(3)$ . The consequence is that fermion fields minimally couple to the gauge fields through a gadget called “covariant derivative”:

$\displaystyle{D_\mu=\partial_\mu 1+ig\sum_{j=1}^{N_G} A^j_\mu L^j}$

instead of coupling with an ordinary derivative. The covariant derivative $D_\mu$ is a matrix-valued differential operator, $A^j$ are “gauge” fields belonging to certain vectorial representation of the group and $L^j$ with $j=1,\ldots , N_G$ is a basis of generators of the Lie Algebra created by the Lie group. Generally, there are $N_G$ generators for a $N_G$ parameter (dimensional) Lie group. The sums is extended to the dimension of the algebra (group).

We can review these results in the following lines. The representations of matter fields for a single generation are:

A) Group $SU(3)_C$ . Multiplets:

$\begin{pmatrix}q_r \\ q_g\\ q_b\end{pmatrix}_L$ $\begin{pmatrix}q_r\\ q_g\\ q_b\end{pmatrix}_R$

Representation: color and anticolor triplets.

B) Group $SU(2)_L$ .

Lepton doublets: $\begin{pmatrix}\nu_l \\ l^{-}\end{pmatrix}_L$ Quark doublets: $\begin{pmatrix}u \\ \tilde{d}\end{pmatrix}_L$

Additionaly, there are lepton and quark singlets as well. The trick is mimicked for the 3 families.

Representation: weak isospin doublets. Left-handed (chiral) sector in the SM.

C) Group $U(1)_Y$ .

It acts on every particle with suitable cautions on neutral particles ( it is related to the quantum number assignments). It also means this force is truly “universal” and it acts on short and long scales.

Representation: Weak hypercharge singlets.

1) $q_c$ is any of the 6 quark flavours ( u, d, c, s, b, t) with colors $c=1,2,3$ or $c=r,g,b$ .

2) $l^-$ is any lepton with negative charge ( electron, muon, tau).

3) The tilde on $\tilde{d}$ or similar particles means that these quarks are linear combinations of the initial quarks, i.e., “tilded” quarks are rotated by some unitary matrix, commonly called and referred as the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The neutrinos can be also rotated with an analogue matrix, and the corresponding unitary matrix is called the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix.

Quantum fields are then related to some quantum numbers. The quantum numbers of the fields unders the gauge groups $SU(3)_C$ and $SU(2)_L$ transformations are determined by assignments of representations. The quantum numbers for the $U(1)_Y$ subgroup with hypercharge Y of each field is fixed useing the next equation:

$\boxed{Y=2Q-2T_3}$

$\boxed{Q=T_3+\dfrac{Y}{2}}$

and where we have defined

1st. $Q=\mbox{Electromagnetic charge}$ of the particle

2nd. $T_3$ is the 3rd generator of the Lie algebra $su(2)$ , i.e., the 3rd component of weak isospin.

3rd. Y is called the hypercharge generator.

It can be shown that the number Q is the generator of a new group, the $U(1)_{em}$ group. Clearly, $U(1)_{em}$ is a subgroup of $SU(2)_L\times U(1)_Y$ . The SM weak quantum number assignments can be represented in a vector space with 3 labels, $(Q, T_3, Y)$ , and they are summarized in the two groups:

1) Doublets. Notation for particles in the weak charge quantum number space

$(\mbox{Particle})=(Q,T_3,Y)$

Therefore, we have

$(\tilde{\nu}_l)_L=(0,1/2, -1)$ $(l^-)_L=(-1,1/2,-1)$

$(u)_L=(2/3,1/3,2/3)$ $(\tilde{d})_L=(-1/3,-1/2,1/3)$

These numbers in weak charge space are mimicked for the 3 families of leptons and quarks. The 3 families only differ by mass and not by any other quantum number.

2) Singlets. Notation for particles in the weak charge quantum number space

$(\mbox{Particle})=(Q,T_3,Y)$

Therefore, we have

$(\nu_l)_R=(0,0,0)$ $(l^-)_R=(-1,0,-2)$

$(u)_R=(2/3,0,4/3)$ $(d)_R=(-1/3,0,-2/3)$

Again, the numbers in weak charge space are mimicked for the 3 families of leptons and quarks for singlets too. The 3 families only differ by mass and not by any other quantum number. Why 3 “light” families? Nobody knows yet for sure…

Then, these two groups and the quantum number assignment show that the right-handed neutrinos, if they were proven to exist, have all their quantum numbers equal to zero, or equivalently, right-handed neutrinos are not “charged” under the SM gauge group! As a result, they don’t take part in any kind of interactions according to the SM since they would not provide any extra terms through the “minimal coupling” recipe using the covariant derivate as their couplings would be null. However, these hypothetical neutrinos have physical relevance due to the neutrino mass and that they could arise somehow in spite of the above facts through some kind of mechanism known as “neutrino oscillations”.

Gauge bosons and their interactions

In local gauge theories, the fermions interact between them only exchanging gauge bosons. SM gauge bosons are spin 1 fields. These bosonic particles are described by quantum fields in the representation group of the local symmetry group. The number of gauge bosons is the number of the gauge group generators. The gauge group generators match the number of generators of the corresponding Lie algebra. For instance, U(N) and SU(N) have, respectively, $N^2$ and $N^2-1$ generators. The gauge groups of the SM have:

$U(1)_Y$ : $1$ generator. It correspond to the photon field $A_\mu$ . Sometimes is called $B_\mu$ as well. The coupling constant is denoted by $g'$ and it is related to the electric charge and the electromagnetic fine structure constant.

$SU(2)_L$ : $2^2-1=3$ generators. They correspond to the massive photon-like fields of the $W^{\pm}, Z$ particles. They are usually represented as $A_\mu^j$ , with $j=1,2,3$ . Sometimes, they are also written as $W_{\mu j}$ and the coupling constant is denoted by $g$ or $g_w$ .

$SU(3)_C$ : $3^2-1=8$ generators. They correspond to the massless gluon fields. They are fields $A_\mu^j$ with $j=1,\ldots,8$ and coupling constant $g_s$ , the strong coupling constant. Gluon fields are also written as $G_{\mu j}$ .

Thus, the total gauge group of the SM ( $G_{SM}=SU(3)_C\times SU(2)_L\times U(1)_Y$ ) have $N_G=1+3+8=12$ generators.

Gauge boson masses and self-interactions

In any known local gauge theory of YM type, the interacting bosons are necessarily particle with zero mass, i.e., local gauge YM theories imply the existence of massless gauge bosons. Equivalently, there are not quadratic terms/mass terms for them in the lagrangian. If we include quadratic terms in the fields ( a.k.a. mass terms) the YM theory looses the local gauge symmetry! The gluon fields $G_{\mu j}$ and the bosonic field associated to the photon satisfy this massless requirement. That is, the gluon and the photon are consistently observed to be massless particles. However, we have a problem with the remaining gauge bosons $W^\pm$ and $Z^0$ . They appear to be massive particles in Nature. This is the reason we invented a new tool in the framework of the Stantard Model: the Higgs mechanism and the phenomenon known as Spontanous Symmetry Breaking (SSB). We will learn more about these in forthcoming posts.

By the other hand, we know that the SM gauge group is a non-abelian group. It iplies that gauge bosons have non-zero quantum numbers, i.e., they are charged under the gauge groups or equivalently, they are charged with respect to the fundamental interactions ( electromagnetic, weak and strong) and self-interactions of the non-abelian fields are indeed possible. Only the boson associated to the electromagnetic sector, the usual photon $\gamma$ , does not have any other charge and then the photon field does not own interactions with itself (selfinteractions). However, the gluons or the $W^\pm, Z$ bosons are charged under non-abelian groups with certain quantum numbers. Therefore, the W/Z bosons or the gluons experience self-interactions theirselves.

SM: the full EW+QCD lagrangian

The SM lagrangian can be separated in two independent pieces BEFORE SSB: the QCD sector and the electroweak (EW) sector.

Mathematically speaking, this decomposition is really simple:

$\boxed{\mathcal{L}_{\mathcal{SM}}=\mathcal{L}_{\mathcal{QCD}}+\mathcal{L}_{\mathcal{EW}}=\mathcal{L}_{SU(3)_c}+\mathcal{L}_{SU(2)_w\times U(1)_Y}}$

See you in the next SM blog post!

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Comments

LOG#059. Standard Model (I). — 1 Comment

Shawn Woolard on 2012/12/15 at 10:01 AM said:

What an informative post! This is truly incredible! Thanks for sharing your brilliant thoughts!

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The Spectrum Of Riemannium

Mobilis in mobili: Physmatics Chronicles!

LOG#059. Standard Model (I).

Field enumeration

Gauging procedure

Gauge bosons and their interactions

Gauge boson masses and self-interactions

SM: the full EW+QCD lagrangian

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LOG#059. Standard Model (I). — 1 Comment

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Field enumeration

Gauging procedure

Gauge bosons and their interactions

Gauge boson masses and self-interactions

SM: the full EW+QCD lagrangian

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