An Overview of the Fundamental Interactions
Fundamental physics aims to distill natural phenomena into basic laws and theories that can quantitatively predict experimental observations At the microscopic level, matter and radiation are governed by three fundamental interactions: strong, electromagnetic, and weak forces While gravitational force is significant for larger bodies and cosmic phenomena, it is negligible in atomic and nuclear contexts Electromagnetic forces bind electrons to atomic nuclei, influencing the behavior of atoms and molecules Light, as an electromagnetic wave, is a specific manifestation of these forces Strong interactions are crucial for holding protons and neutrons together in atomic nuclei, overcoming the repulsive forces between protons due to their like charges Protons and neutrons themselves are made of quarks, which are held together by strong interactions mediated by gluons Weak interactions play a vital role in processes such as beta radioactivity, nuclear reactions that power stars, and the decay of certain particles, contributing to the dynamic nature of the universe.
The decay processes of the top quark and heavy charged leptons, including muons and tau particles, are governed by weak forces, which also account for all observed neutrino interactions.
Interactions, except possibly gravity, are explained through quantum mechanics and relativity, specifically within local relativistic quantum field theory Each particle, considered as pointlike, is linked to this theoretical framework.
G Altarelli, Collider Physics within the Standard Model,
The interactions of particles are fundamentally described by gauge invariance, which refers to the invariance under transformations that rotate internal degrees of freedom depending on spacetime points This principle is exemplified in the classical Maxwell equations of electrodynamics, where the concept of gauge invariance was first introduced Quantum electrodynamics (QED), developed between 1926 and 1950, serves as the prototype for quantum gauge field theories with a single gauged charge, representing the quantum adaptation of Maxwell's theory Theories exhibiting gauge symmetry in four-dimensional spacetime are renormalizable and fully defined by their symmetry group and the representations of the interacting fields The Standard Model (SM) of particle interactions encompasses strong, electromagnetic, and weak forces through a gauge theory involving 12 non-commuting charges However, only a subset of the SM symmetry is observable in the physical state spectrum, with part of the electroweak symmetry concealed by the Higgs mechanism, which facilitates spontaneous symmetry breaking.
The theory of general relativity offers a classical understanding of gravity, surpassing Newton's static approximation by incorporating dynamic phenomena such as gravitational waves A significant challenge in contemporary theoretical physics is the formulation of a quantum theory of gravitational interactions, as quantum effects in gravity become relevant only at energy concentrations not typically accessible in laboratory experiments Consequently, the search for an accurate theory relies on speculative approaches Attempts to describe quantum gravity through a well-defined local field theory akin to the Standard Model have not yielded satisfactory results Currently, the most promising description of quantum gravity involves non-pointlike objects known as "strings," which exist in a spacetime framework of 10 or 11 dimensions, with additional dimensions compactified at a curvature radius comparable to string dimensions This comprehensive string theory aims to unify all interactions, including gravity, positioning the Standard Model as a low-energy or large-distance approximation.
The Heisenberg uncertainty principle, a key concept in quantum mechanics, indicates that to study particles with spatial dimensions or interactions at a certain scale, a particle beam with momentum proportional to that scale is required Current particle accelerators, such as the Large Hadron Collider (LHC) at CERN, enable the examination of particle collisions with a total center of mass energy reaching up to 14 TeV.
These machines can, in principle, study physics down to distancesx&10 18 cm.
Experimental results from current accelerators confirm that at distances around 10^-33 cm, electrons, quarks, and all fundamental particles of the Standard Model exhibit no significant internal structure, appearing elementary and point-like Additionally, we anticipate that quantum effects in gravity will become significant at these small distances, correlating with high energy levels.
At energy levels around \( E \sim M_{\text{Planck}} \sim 2 \times 10^{19} \) GeV, the Planck mass, which is linked to Newton's gravitational constant by \( G_N \sim \frac{c}{M_{\text{Planck}}^2} \), suggests that particles, typically viewed as point-like, may exhibit an extended structure akin to strings This implies the need for a more comprehensive theoretical framework, where the local quantum field theory description of the Standard Model (SM) serves merely as a low-energy or large-distance approximation.
From the first few moments of the Universe, just after the Big Bang, the temperature of the cosmic background gradually went down, starting fromkT
The Boltzmann constant, k = 8.617 x 10^-5 eV/K, plays a crucial role in understanding the evolution of the Universe, beginning from M Planck c^2 to the current temperature of T ≈ 2.725 K This journey encompasses various stages of high energy physics, ranging from the speculative realm of string theory to the well-tested Standard Model (SM) phenomenology, which is directly accessible through experimental validation.
Big Bang This is the basis for the ever increasing connection between high energy physics and cosmology.
The Architecture of the Standard Model
The Standard Model (SM) is a gauge field theory based on the symmetry group
U.1/ The transformations of the group act on the basic fields.
This group has8C3C1 D 12generators with a nontrivial commutator algebra
(if all generators commute, the gauge theory is said to be “Abelian”, while the SM is a “non-Abelian” gauge theory).SU.2/N
The electroweak (EW) interactions are described by the gauge group U(1), where the electric charge Q is derived from the sum of the SU(2) generator T3 and the U(1) generator Y, expressed as Q = T3 + Y/2 Additionally, SU(3) represents the "color" group in quantum chromodynamics (QCD), which governs the strong interactions.
In gauge theory, each generator \( T \) is associated with a vector boson, known as a gauge boson, which shares the same quantum numbers as \( T \) When the gauge symmetry remains unbroken, these vector bosons possess zero mass Acting as mediators for their respective interactions, these spin-1 vector bosons play a crucial role in quantum electrodynamics (QED), where the photon is the vector boson linked to the generator \( Q \) In QED, the interaction between charged particles, such as electrons, occurs through the exchange of these gauge bosons.
This chapter revises and updates previous material, focusing on the interactions between photons and electrons, where one or more photons can be emitted by one electron and reabsorbed by another Additionally, in the Standard Model, there are eight gluons linked to the SU(3) color generators, alongside the SU(2) symmetry for weak interactions.
U.1/there are four gauge bosonsW C ,W ,Z 0 , and
Among the fundamental particles, only gluons and photons are massless, while the masses of the W and Z bosons are significantly large, with values of approximately 80.4 GeV for the W boson and 91.2 GeV for the Z boson This mass is comparable to that of intermediate-sized atoms, such as rubidium and molybdenum, due to the spontaneous symmetry breaking associated with the other three generators.
In electroweak theory, spontaneous symmetry breaking occurs when the vacuum state is not unique, leading to a continuum of degenerate states that still adhere to the symmetry This phenomenon, which took place in the early Universe for the Standard Model (SM), results in the violation of symmetry in the spectrum of states The Higgs mechanism facilitates this symmetry breaking in gauge theories, introducing scalar Higgs bosons that create a set of degenerate vacuum states The presence of one or more Higgs particles is essential, with the Higgs boson recently discovered at the LHC at a mass of 126 GeV, marking a significant milestone in validating the SM The Higgs boson serves as a mediator for new interactions, with coupling strengths varying according to particle masses, differentiating interactions between particles like electrons and top quarks.
In the Standard Model (SM), fermionic matter fields consist of quarks and leptons, both of which have a spin of 1/2 Quarks are categorized as colour triplets, meaning each flavour exists in three colours, and they possess electroweak charges, specifically +2/3 for up-type quarks and -1/3 for down-type quarks, making them subject to all SM interactions In contrast, leptons are colourless and do not participate in strong interactions; they carry electroweak charges, with charged leptons (such as electrons and muons) having a charge of -1, while neutrinos are neutral Both quarks and leptons are organized into three families or generations that share identical quantum numbers but differ in mass, though the reason for this repetition of fermion families remains unexplained.
The QCD sector of the Standard Model features a straightforward structure but is rich in dynamics, exemplified by its complex spectroscopy and numerous hadrons Key characteristics of QCD include asymptotic freedom and confinement, where the effective coupling at interaction vertices is influenced by the interaction itself Consequently, the intensity of the force measured varies with the square of the four-momentum transferred among participants In QCD, the significant coupling parameter in physical processes is denoted as αs = e²s / 4, where es represents the coupling constant for the fundamental interactions of quarks and gluons.
. Asymptotic freedom means that the effective coupling becomes a function of
Q 2 , and in fact˛s.Q 2 /decreases for increasing Q 2 and vanishes asymptotically.
In processes with large momentum transfer, known as hard or deep inelastic processes, the QCD interaction weakens significantly It can be demonstrated that all pure gauge theories based on a non-commuting symmetry group are asymptotically free in four spacetime dimensions The effective coupling decreases slowly at high momenta, following the relationship ˛s(Q²) = 1/(b log(Q²/2)), where b is a constant and is approximately a few hundred MeV In quantum mechanics, large momenta correspond to short wavelengths, leading to a potential between two color charges that resembles the Coulomb potential at short distances, characterized by an effective color charge that is small.
At large distances or small transferred momenta (Q < 0), the interaction strength in quantum chromodynamics (QCD) becomes significant All observed hadrons, such as baryons (composed of three quarks) and mesons (made of a quark and an antiquark), are tightly bound composite states that maintain overall colour neutrality through compensating colour charges A key feature of QCD is confinement, which prevents the separation of colour charges, including individual quarks and gluons, due to the linearly increasing interaction potential between them at long distances.
When attempting to separate a quark and an antiquark in a color-neutral meson, the interaction energy increases, leading to the creation of quark-antiquark pairs from the vacuum This results in new neutral mesons that coalesce, rather than observing free quarks For instance, in high-energy processes like C e !qqN, the final state quark and antiquark, moving rapidly apart, generate new pairs due to color confinement forces The outcome is typically two back-to-back jets of colorless hadrons, accompanied by slow pions that complicate the exact separation of the jets Occasionally, a third, distinct jet of hadrons is detected, indicating the radiation of an energetic gluon from the original quark-antiquark pair.
In the electroweak (EW) sector, the Standard Model (SM) builds upon the successful phenomenology of the four-fermion low-energy description of weak interactions It offers a coherent theoretical framework that unifies weak interactions with quantum electrodynamics The term "weak interactions" refers to their comparatively low strength, with the effective four-fermion interaction of charged currents characterized by its strength at low energy.
Fermi coupling constantG F For example, the effective interaction for muon decay is given by
In natural units, the Fermi coupling constant \( G_F \) has dimensions of \( (mass)^2 \), indicating that the strength of weak interactions at low energy is characterized by \( G_F E^2 \), where \( E \) represents the energy scale relevant to the process, such as \( E \sim m \) for muon decay.
In the context of weak interactions, the effective four-fermion couplings for neutral current interactions exhibit comparable intensity and energy behavior, particularly at low energies, which can reach up to a few tens of GeV However, the quadratic increase in energy cannot persist indefinitely, as this would violate unitarity At high energies, the effects of propagators become significant, leading to a resolution of the current–current interaction into current–W gauge boson vertices linked by a W propagator The strength of weak interactions at elevated energies is quantified by the W coupling, denoted as g_W, or more effectively by the parameter α_W, which is defined as g²_W/4, drawing an analogy to the fine-structure constant α of quantum electrodynamics (QED) In the framework of the standard electroweak theory, this relationship is expressed as α_W = p.
That is, at high energies the weak interactions are no longer so weak.
The range of weak interactions, denoted as r_W, is notably short, only becoming evident with the experimental discovery of the W and Z gauge bosons, which confirmed that r_W is non-vanishing It is now established that r_W is approximately 2.510^(-16) cm, correlating to a W boson mass of 80.4 GeV This significant mass for the W and Z bosons contrasts sharply with the massless photon, resulting in the finite range of the weak force compared to the infinite range of quantum electrodynamics (QED) Additionally, current experimental limits indicate that the photon mass is less than a certain threshold.
The photon is confirmed to be massless, with energy levels around 10 to 18 eV, while the weak bosons possess significant mass This stark contrast presents a challenge for developing a unified theory of electroweak interactions.
The Formalism of Gauge Theories
This section provides an overview of Yang–Mills gauge theory, detailing its definition and structure We will outline the fundamental principles for constructing this type of theory and subsequently apply these findings to the Standard Model (SM).
Consider a Lagrangian densityLŒ; @ which is invariant under aDdimen- sional continuous group of transformations:
In the context of internal symmetries, the quantities A represent numerical parameters, such as angles associated with a rotation group in an internal space For infinitesimal A, the approximate expression is applicable, where g denotes the coupling constant and T A serves as the generators of the transformation group in the representation of the fields It is important to note that the T A matrices are independent of spacetime coordinates, ensuring that the arguments of the fields remain consistent in the given transformation context.
If the unitary operator \( U \) is present, the generators \( T_A \) are Hermitian; however, this is not universally applicable, even though it holds true for the Standard Model (SM) Additionally, for a group of matrices with a unit determinant, the traces of the generators \( T_A \) equal zero, expressed as \( \text{tr}(T_A) = 0 \) Generally, the generators adhere to the commutation relations given by \( [T_A, T_B] = iC_{ABC} T_C \).
In gauge theory, the structure constants \( C_{ABC} \) are completely antisymmetric in their indices, meaning that the arrangement of indices does not affect their values, such as in \( T^A D T^A \) When all generators commute, it signifies a specific condition within the theory.
“Abelian” (in this case all the structure constantsC ABC vanish), while the SM is a
We normalize the generators \( t_A \) of the group so that, for the lowest dimensional non-trivial representation, the trace of the product of the generators \( t_A \) and \( t_B \) is defined as \( \text{tr}(t_A t_B) \).
A normalization convention is needed to fix the normalization of the couplinggand the structure constantsC ABC In the following, for each quantityf A , we define fDX
For example, we can rewrite (1.9) in the form
If we now make the parameters A depend on the spacetime coordinates, whence
A D A x /; thenLŒ; @ is in general no longer invariant under the gauge transformationsUŒ A x /, because of the derivative terms Indeed, we then have
@ 0 D@ U/¤U@ Gauge invariance is recovered if the ordinary derivative is replaced by the covariant derivative
D D@ Cig V ; (1.14) whereV A are a set ofDgauge vector fields (in one-to-one correspondence with the group generators), with the transformation law
For constant A ,Vreduces to a tensor of the adjoint (or regular) representation of the group:
V 0 DU V U 1 V CigŒ;V C ; (1.16) which implies that
V 0C DV C gC ABC A V B C ; (1.17) where repeated indices are summed over.
As a consequence of (1.14) and (1.15), D has the same transformation properties as:
The gauge fields \( V^A \) in the Lagrangian \( \mathcal{L} \) are currently treated as external fields that do not propagate, yet \( \mathcal{L} \) remains invariant under gauge transformations To develop a gauge invariant kinetic energy term for the gauge fields \( V^A \), we examine the expression \( \mathcal{D} ; \mathcal{D} D \).
@ V @ V CigŒV ;V ig F ; (1.20) which is equivalent to
From (1.8), (1.18), and (1.20), it follows that the transformation properties ofF A are those of a tensor of the adjoint representation:
The complete Yang–Mills Lagrangian, which is invariant under gauge transforma- tions, can be written in the form
The kinetic energy term is an operator with a dimension of 4, indicating that if L is renormalizable, then L YM is also renormalizable However, if we abandon renormalizability, it opens the door to the inclusion of additional gauge-invariant higher-dimensional terms It is important to note that gauge invariance prohibits the inclusion of mass terms for gauge bosons in the form of m²VV.
Application to QED and QCD
In Abelian theories such as Quantum Electrodynamics (QED), the gauge transformation is expressed as U(x) = exp[ieQ(x)], where Q represents the charge generator For theories with multiple commuting generators, the transformation involves a product of similar exponential factors Consequently, as outlined in equation (1.15), the gauge field, which is the photon, undergoes a specific transformation in response to these gauge transformations.
V 0 DV @ x/ ; (1.24) and the familiar gauge transformation is recovered, with addition of a 4-gradient of a scalar function The QED Lagrangian density is given by
HereD= D D , where are the Dirac matrices and the covariant derivative is given in terms of the photon fieldA and the charge operator Q by
Note that in QED one usually takese to be the particle, so thatQD 1and the covariant derivative isD D @ ieA when acting on the electron field In the
In the Abelian case, the F tensor exhibits linearity in the gauge field V, resulting in a free theory when matter fields are absent Conversely, in the non-Abelian case, the F_A tensor incorporates both linear and quadratic terms in V_A, leading to a non-trivial theory even without the presence of matter fields.
Quantum Chromodynamics (QCD) is recognized as a renormalizable gauge theory grounded in the SU(3) group, which incorporates color triplet quark matter fields This framework establishes the QCD Lagrangian density, defining the fundamental interactions among quarks and gluons within the theory.
Hereq j are the quark fields withn f different flavours and massm j , andD is the covariant derivative of the form
D D@ Cie s g ; (1.29) with gauge couplinge s Later, in analogy with QED, we will mostly use ˛sD e 2 s
In the context of quantum chromodynamics, the gluon fields \( A^A_\mu \) are represented by the indices \( A \) ranging from 1 to 8, while \( t^A \) denotes the SU(3) group generators in the triplet representation of quarks, represented as 3x3 matrices acting on the quark field \( q \) These generators satisfy the commutation relations given by \( [t^A, t^B] = i C^{ABC} t^C \), where \( C^{ABC} \) are the completely antisymmetric structure constants of the SU(3) group The normalization of \( C^{ABC} \) and the coupling constant \( e \) is defined in relation to the generators \( t^A \), specifically through the trace relation \( \text{Tr}[t^A t^B] = \frac{1}{2} \delta^{AB} \).
Chapter 2 provides an in-depth exploration of Quantum Chromodynamics (QCD) as the foundational theory of strong interactions It highlights key physical vertices, such as the gluon-quark-antiquark vertex, which parallels the photon-fermion-antifermion coupling found in Quantum Electrodynamics (QED).
In Quantum Chromodynamics (QCD), the presence of 3-gluon and 4-gluon vertices, which arise from interaction orders of e s and e² s respectively, distinguishes it from Abelian theories such as Quantum Electrodynamics (QED) Unlike QED, where the neutral photon interacts with charged particles, gluons in QCD carry color charge and are self-coupled This self-coupling is evident as the field strength tensor F in QED is linear in the gauge field, resulting in a purely kinetic term F² in the Lagrangian Conversely, in QCD, the tensor F A is quadratic in the gauge field, leading to cubic and quartic interaction vertices in F A² beyond the kinetic term Exploring a scalar version of QED further illustrates these differences.
In quantum electrodynamics (QED), charged bosons exhibit a kinetic term that is quadratic in the derivative, leading to a gauge–gauge–scalar–scalar vertex of order e² In quantum chromodynamics (QCD), the presence of a 3-gluon vertex arises from the color charge of gluons, while the 4-gluon vertex exists due to the bosonic nature of gluons.
Chirality
Chirality is a fundamental concept essential for understanding the Electroweak (EW) Theory, as it pertains to the behavior of fermion fields These fields can be characterized by their right-handed (RH) and left-handed (LH) components, each exhibiting distinct chirality properties.
L ; RDŒ.1 5 /=2 ; N L ; R D N Œ.1˙ 5 /=2 ; (1.34) where 5 and the other Dirac matrices are defined as in the book by Bjorken and Drell [102] In particular, 5 2 D1, 5 D5 Note that (1.34) implies
The matricesP ˙ D 1˙ 5 /*re projectors They satisfy the relationsP ˙ P ˙ D
P ˙,P ˙ P D0,P CCP D1 They project onto fermions of definite chirality For massless particles, chirality coincides with helicity For massive particles, a chirality
C1state only coincides with aC1helicity state up to terms suppressed by powers ofm=E.
The 16 linearly independent Dirac matrices () can be divided into 5 -even (E) and 5 -odd (O) according to whether they commute or anticommute with 5 For the5-even, we have
N E D N L E RC N R E L E1;i5; / ; (1.35) whilst for the 5 -odd,
In gauge Lagrangians, fermion kinetic terms and gauge boson interactions maintain chirality, whereas fermion mass terms invert it For instance, in Quantum Electrodynamics (QED), an electron's chirality remains unchanged when it emits a photon In the ultrarelativistic limit, where the electron mass is negligible, chirality and helicity align, indicating that photon emission does not alter the electron's helicity In massless gauge theories, left-handed (LH) and right-handed (RH) fermion components are independent and can be transformed separately When LH and RH components transform under different representations of the gauge group, it is termed a chiral gauge theory; conversely, if they share the same gauge transformations, it is classified as a vector gauge theory Both QED and Quantum Chromodynamics (QCD) are vector gauge theories, as they treat LH and RH fermions with identical electric charge and color.
The standard Electroweak (EW) theory is characterized as a chiral theory, meaning that left-handed (L) and right-handed (R) fermions respond differently to the gauge group This property allows for the potential violation of parity and charge conjugation symmetries, which is significant in understanding fermion mass terms.
In the EW gauge-symmetric limit, L R+ h.c terms are prohibited Specifically, within the framework of the Minimal Standard Model (MSM), which incorporates all observed particles along with a single Higgs doublet, all left-handed particles (L) are SU(2) doublets, while all right-handed particles (R) are singlets.
Quantization of a Gauge Theory
The classical theory is comprehensively described by the Lagrangian density L YMin (1.23) Transitioning to a quantum framework necessitates defining quantization, regularization, and renormalization procedures Establishing Feynman rules poses challenges, particularly due to issues common in all gauge theories, including the Abelian case of Quantum Electrodynamics (QED) This complexity is evident when examining the free equations of motion for V A, derived from equations (1.21) and (1.23).
The propagator of the gauge field is typically defined by the inverse of the operator @ 2 g @ @, but it lacks an inverse since it acts as a projector on transverse gauge vector states This issue can be addressed by selecting a specific gauge By opting for a covariant gauge condition, such as @ V A D 0, a corresponding gauge fixing term can be introduced.
A j@ V A j 2 (1.38) has to be added to the Lagrangian (1ts as a Lagrangian multiplier) The free equations of motion are then modified as follows:
This operator now has an inverse whose Fourier transform is given by
The propagator in this class of gauges is represented by the equation D AB q/D i q 2 Ci g C.1/ q q q 2 Ci ı AB The parameter in this equation can assume any value, ultimately vanishing from the final expression of any gauge-invariant physical quantity Two widely used specific cases of this propagator are the Feynman gauge (D 1) and the Landau gauge (D 0).
In non-Abelian theories, the quantization process requires more than just a gauge fixing term; it necessitates the introduction of D fictitious ghost fields, known as Faddeev–Popov ghosts, which are included as internal lines in closed loops This is essential because gauge fields related by a gauge transformation represent the same physical reality, leading to fewer physical degrees of freedom than the number of gauge field components Ghosts emerge in the functional integral as a transformation Jacobian, addressing the redundancy of variables associated with fields on the same gauge orbit By manipulating the path integral, the ghost contributions can be incorporated into an additional term in the Lagrangian density The ghost Lagrangian is derived from the impact of an infinitesimal gauge transformation on the gauge fixing condition, ultimately providing a comprehensive framework for the complete Feynman rules in non-Abelian gauge theories.
A ; (1.41) where the gauge condition@ V C D 0has been taken into account in the last step. The ghost Lagrangian is then given by
In the context of ghost fields, denoted as A (1.42), each indexA is treated as a scalar field However, it is important to include a factor of one for each closed loop, similar to the treatment of fermion fields.
Beginning with non-covariant gauges, it is possible to develop ghost-free gauges A significant example of this is the "axial" gauges represented by \( n V A D 0 \), where \( n \) is a fixed reference 4-vector Specifically, when \( n \) is spacelike, it corresponds to a proper axial gauge; for \( n \) being light-like, it is termed a light-like gauge, and when \( n \) is timelike, it refers to a Coulomb or temporal gauge The gauge fixing term takes a specific form in this context.
With a procedure that can be found in QED textbooks [102], the corresponding propagator in Fourier space is found to be
In this scenario, there are no ghost interactions present because V 0A, derived from a gauge transformation of n V A, lacks gauge fields after considering the gauge condition n V A D 0 Consequently, the ghosts are decoupled and can be disregarded.
The introduction of a suitable regularization method that maintains gauge invariance is crucial for defining and calculating loop diagrams, as well as for the theory's renormalization program Currently, dimensional regularization is employed, which involves formulating the theory in n dimensions This approach allows all loop integrals to have an analytic expression valid for non-integer values of n When considering results in 4 dimensions, the loops are ultraviolet finite for n > 0, with divergences manifesting as poles.
Spontaneous Symmetry Breaking in Gauge Theories
The gauge symmetry of the Standard Model (SM) is challenging to identify due to its subtle presence in nature Among the gauge bosons, only the photon has been directly observed as massless, while gluons are believed to be massless as well, although they cannot be directly detected due to confinement effects.
W and Z bosons possess significant mass, which presents challenges in unifying weak and electromagnetic interactions While electromagnetic forces have an infinite range, weak forces operate over a very short distance due to the mass of W and Z bosons The resolution to this issue is found in the concept of spontaneous symmetry breaking, a principle adapted from condensed matter physics.
Fig 1.1 The potential V D 2 M 2 =2 C M 2 / 2 =4 for positive (a) or negative 2 (b) (for simplicity, M is a 2-dimensional vector) The small sphere indicates a possible choice for the direction of M
Consider a ferromagnet at zero magnetic field in the Landau–Ginzburg approxi- mation The free energy in terms of the temperatureTand the magnetizationMcan be written as
This expansion is applicable at low magnetization, where higher-order terms in M² are disregarded, reflecting the renormalizability criterion Additionally, stability requires that T/ > 0, and the function F remains invariant under rotations, indicating that all directions of M in space are treated equally The minimum condition for F is established accordingly.
In the analysis presented, two scenarios are identified When the condition is met that 2 & 0, the sole solution is MD0, indicating the absence of magnetization and the preservation of rotational symmetry This situation leads to a unique lowest energy state, or vacuum in quantum theory, that remains invariant under rotations Conversely, if the condition changes to 2 < 0, an alternative solution emerges, characterized by jM 0 j 2 D 2 =
In this case there is a continuous orbit of lowest energy states, all with the same value ofjMj, but different orientations A particular direction chosen by the vector
M 0 leads to a breaking of the rotation symmetry.
When iron is heated to high temperatures in the presence of an external magnetic field (B), it experiences a breaking of rotational symmetry, resulting in a nonzero magnetization (M) aligned with the field As the temperature (T) is lowered while maintaining a constant magnetic field, the behavior of the material changes; specifically, a parameter, denoted as 2, transitions from positive to negative values The critical temperature, known as the Curie temperature (T crit), is identified as the point where 2(T) equals zero, indicating a change in the material's magnetic properties.
For pure iron,T crit is below the melting temperature So atT DT crit iron is a solid.
In the absence of a magnetic field, the mobility of magnetic domains in a solid is restricted, resulting in a persistent magnetization (M0) The free energy maintains a rotationally invariant form, similar to previous equations, yet the system can achieve a minimum energy state with a non-zero magnetization aligned with the magnetic field (B) This alignment breaks the symmetry by selecting a specific vacuum state from an infinite array of possibilities.
We now prove the Goldstone theorem [228] It states that when spontaneous symmetry breaking takes place, there is always a zero-mass mode in the spectrum.
In a classical context this can be proven as follows Consider a Lagrangian
The potentialV./can be kept generic at this stage, but in the following we will be mostly interested in a renormalizable potential of the form
The potential described is represented by a column vector with real components, allowing for the decomposition of complex fields into real ones It is symmetric under an N x N orthogonal matrix rotation, denoted as O, which corresponds to an SO(N) transformation By excluding odd powers, we imply an additional discrete symmetry Importantly, for positive values of the parameter, the mass term in the potential has a negative sign, indicating the condition for a non-unique lowest energy state Additionally, the potential is symmetric under infinitesimal transformations.
In the context of symmetry groups and field representations, the equation \(0 = D C \cdot A_{ij} \cdot (1.50)\) illustrates the role of infinitesimal parameters \(A\) and the matrices \(t_{ij}^A\) These matrices are essential for representing the symmetry group associated with the fields \(i\), where a summation over \(A\) is implied The minimum condition on the potential \(V\) is crucial for determining the equilibrium position within this framework.
(or the vacuum state in quantum field theory language) is
By taking a second derivative at the minimum i D i 0 of both sides of the previous equation, we obtain that, for eachA,
The second term vanishes owing to the minimum condition (1.51) We then find
The second derivativesM 2 ki D @ 2 V=@ k @ i / i D i 0 /define the squared mass matrix Thus the above equation in matrix notation can be written as
In the absence of spontaneous symmetry breaking, the ground state is unique and invariant under all symmetry transformations, resulting in all t A 0 equating to zero However, when certain values of t A lead to non-vanishing vectors, indicating the presence of a generator that can shift the ground state to another state with the same energy, the vacuum becomes non-unique Each non-zero t A corresponds to an eigenstate of the squared mass matrix with a zero eigenvalue, signifying a massless mode linked to each broken generator The quantum numbers of these massless modes differ from the vacuum's, typically considered zero, by the values of the t A charges, meaning that the massless modes share the same quantum numbers as the broken generators that do not annihilate the vacuum.
The Goldstone theorem, initially proven in the classical context, can be adapted to the quantum case by recognizing that the classical potential aligns with the tree level approximation of the quantum potential In this framework, higher-order loop diagrams introduce necessary quantum corrections The functional integral formulation of quantum field theory serves as an ideal method to define and compute the quantum potential through a loop expansion, effectively detailing the vacuum properties of the quantum theory In weakly coupled theories, such as the electroweak theory with a moderately light Higgs particle, the tree level expression for the potential remains a reliable approximation of the true quantum potential.
In a quantum system characterized by a finite number of degrees of freedom, such as those described by the Schrödinger equation, there are no degenerate vacua; the vacuum state is always unique This is exemplified in the one-dimensional Schrödinger problem, where the potential leads to a singular vacuum state.
In the context of a 4x4 system, two degenerate minima exist at specific points denoted as jCi and ji The potential is not diagonal in this basis, as indicated by the non-zero off-diagonal matrix elements, which highlight the tunneling effect between the two vacua This tunneling is characterized by an amplitude that is exponentially dependent on the distance between the vacua and the height of the barrier Following the diagonalization process, the resulting eigenvectors can be expressed as a linear combination of the two minima.
2, with different energies (the difference being proportional to ı) Suppose now that we have a sum of n equal terms in the potential, i.e.,
The transition amplitude in VDP i V.x i / is proportional to ı n, which approaches zero as n approaches infinity, indicating that the likelihood of all degrees of freedom simultaneously overcoming the barrier diminishes This scenario features a finite number of minimum points.
The case of a continuum of minima is obtained, still in the Schrửdinger context, if we take
The Schrödinger potential V(x) is analogous to the Higgs potential, featuring a unique ground state characterized by a total orbital angular momentum of zero, represented by an s-wave state This wave function is solely dependent on the radial distance |r| and remains constant across all angles, embodying a uniform superposition of directions However, when substituting a single vector r with a vector field M(x)—where each point in space has a distinct vector—the probability of transitioning from one minimum state to another diminishes to zero as the volume approaches infinity Consequently, the vectors across all spatial points exhibit an increasingly negligible capacity for simultaneous rotation In the infinite volume limit, all vacua along each direction maintain equal energy, paving the way for spontaneous symmetry breaking.
A massless Goldstone boson is associated with long-range forces, which would be detectable unless the particles are confined, as seen with gluons in Quantum Chromodynamics (QCD) In constructing the Electroweak (EW) theory, physical massless scalar particles cannot be accepted However, during spontaneous symmetry breaking in a gauge theory, massless Goldstone modes arise but are considered unphysical and vanish from the spectrum Each Goldstone mode transforms into the third helicity state of a gauge boson, which acquires mass through the Higgs mechanism.
[189, 236, 243, 261] (it should be called the Englert–Brout–Higgs mechanism, because of the simultaneous paper by Englert and Brout) Consider, for example, the simplest Higgs model described by the Lagrangian [243,261]
Quantization of Spontaneously Broken Gauge Theories
In Section 1.6, we addressed the challenges in quantizing gauge theories and establishing the correct Feynman rules, including gauge fixing terms and ghost fields This section provides a concise overview of the results for spontaneously broken gauge theories, focusing on the R gauge formalism This formalism highlights the relationship between transverse and longitudinal gauge boson degrees of freedom, demonstrating how their combination results in the cancellation of the gauge parameter from physical quantities We will also explore an Abelian example in detail, which can be easily extended to the non-Abelian case.
The Abelian model discussed in (1.59) with Q set to 1 involves a nonlinear field transformation that effectively removes the would-be Goldstone boson from the Lagrangian This transformation resembles a gauge transformation linked to the U(1) symmetry of the Lagrangian As a result, the theory is reformulated in new variables, yielding a model that exclusively features physical fields, including a massive gauge boson A with mass MDp.
The "unitary" gauge is characterized by the presence of only physical fields; however, when examining the propagator of a massive gauge boson, it reveals problematic ultraviolet behavior due to a term in the numerator This indicates that the unitary gauge may not be the most effective choice for analyzing the ultraviolet behavior of the theory An alternative approach involves retaining the would-be Goldstone boson in the Lagrangian, which, despite introducing extra degrees of freedom, ensures that all propagators exhibit good ultraviolet behavior, making them "renormalizable." To achieve this, the nonlinear transformation can be substituted with its linear equivalent, as perturbation theory focuses on small oscillations around the minimum.
In this analysis, we applied a shift of the amount v and separated the real and imaginary components of the resulting field, ensuring the vacuum expectation value remains zero By retaining A in its original form and substituting the linearized expression, we derive the essential quadratic terms that are crucial for propagators.
The mixing term between A and @ prevents the direct writing of diagonal mass matrices However, this mixing term can be removed by suitably modifying the covariant gauge fixing term as outlined in (1.38) for the unbroken theory.
By addingL GF to the quadratic terms in (1.67), the mixing term cancels (apart from a total derivative that can be omitted) and we have
We see that thefield appears with a massp
Mand its propagator is iD D i k 2 M 2 Ci : (1.70)
The propagators of the Higgs fieldhand of gauge fieldA are iD h D i k 2 2 2 Ci ; (1.71) iD k/D i k 2 M 2 Ci g 1/ k k k 2 M 2
All propagators exhibit favorable behavior in the large k limit, categorized as "R gauges." Specifically, in the case of D1, there is a generalization of the Feynman gauge that incorporates a Goldstone boson with mass M, resulting in the gauge propagator defined as iD(k) = ig / (k² - M² + i).
In the unitary gauge description, the would-be Goldstone propagator disappears, leading to a gauge propagator that aligns with the unitary gauge as shown in equation (1.65) It is essential that all dependencies in individual Feynman diagrams, including the unphysical singularities of the and A propagators at k² = DM², cancel out when summing all contributions to any physical quantity.
In R gauges, the presence of a Faddeev–Popov ghost adds complexity, a contrast to unbroken Abelian gauge theories where such ghosts are absent Specifically, when applying an infinitesimal gauge transformation characterized by a parameter, the transformations can be expressed accordingly.
The gauge fixing condition@ A M D0undergoes the variation
In an Abelian gauge theory, the ghost does not couple to the gauge boson; instead, it has a coupling to the Higgs field \( h \) The ghost Lagrangian reflects this unique interaction.
The ghost mass is seen to bem ghDp
Mand its propagator is iD gh D i k 2 M 2 Ci : (1.77)
The Feynman rules for basic vertices involving gauge bosons, Higgs bosons, would-be Goldstone bosons, and ghosts can be derived from the total Lagrangian with gauge fixing and ghost additions While the generalization to non-Abelian cases is straightforward in principle, it involves formal complications related to projectors over the spaces of would-be Goldstone bosons and Higgs particles Each massive gauge boson is accompanied by a corresponding would-be Goldstone boson and a ghost, with the latter having mass.
M a The Feynman diagrams,both for the Abelian and the non-Abelian case, are listed explicitly, for example, in the textbook by Cheng and Li [250].
The renormalizability of non-Abelian gauge theories, including scenarios with spontaneous symmetry breaking, was established in the seminal research by t'Hooft and Veltman This topic is explored in depth in reference [278].
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QCD: The Theory of Strong Interactions
Introduction
This chapter provides a succinct introduction to quantum chromodynamics (QCD), the fundamental theory governing strong interactions Emphasizing conceptual understanding over technical details, it offers a broad overview of strong interactions, with references to dedicated literature for further exploration.
This chapter will first provide a brief overview of non-perturbative Quantum Chromodynamics (QCD) methods, including both analytic techniques and lattice simulations The primary focus will then shift to the principles and applications of perturbative QCD, which will be explored in detail.
The QCD theory of strong interactions, as outlined in Chapter 1, is an unbroken gauge theory founded on the color group SU(3) It features eight massless gauge bosons known as gluons (g^A), while the matter fields consist of color triplets of quarks (q^a_i) in various flavors (i) Within the Standard Model, quarks and gluons are recognized as the only fundamental fields.
The Quantum Chromodynamics (QCD) Lagrangian, as presented in Section 1.4, must be extended for quantization to include gauge fixing and ghost terms The Feynman rules for QCD outline the essential interactions, featuring physical vertices such as the gluon-quark-antiquark coupling, similar to the photon-fermion interactions in Quantum Electrodynamics (QED) Additionally, QCD introduces unique 3-gluon and 4-gluon vertices, which are of orders \(e_s\) and \(e_s^2\) respectively, and do not have counterparts in Abelian theories like QED.
WhySU.N C D3/colour? The choice ofSU.3/as colour gauge group is unique in view of a number of constraints:
The group must accommodate complex representations to effectively differentiate between quarks and antiquarks Notably, there exist meson states comprised of quark-antiquark pairs, yet there are no similar bound states This limitation is particularly relevant among simple groups.
G Altarelli, Collider Physics within the Standard Model,
Lecture Notes in Physics 937, DOI 10.1007/978-3-319-51920-3_2
The Feynman rules for Quantum Chromodynamics (QCD) are illustrated in Fig 2.1, where solid lines represent quarks, curly lines depict gluons, and dotted lines indicate ghosts The gauge parameter is denoted by a specific symbol The 3-gluon vertex is expressed assuming all gluon lines are outgoing, and the framework considers the groups SU(N) with N=3 and SO(4N) with N=2, recognizing that SO(6) shares the same algebra as SU(4).
The group must recognize a fully antisymmetric color singlet baryon composed of three quarks, denoted as qqq Research in hadron spectroscopy reveals that the low-lying baryons form an octet and a decuplet based on flavor.
The SU(3) symmetry describes the rotation of the three light quarks: up (u), down (d), and strange (s) Particles composed of these quarks are color singlets, meaning they are neutral in color charge To comply with Fermi statistics, the wave function of three quarks (qqq) must be entirely antisymmetric in color.
In quantum chromodynamics (QCD), a particle with spin component +3/2, such as a baryon, has a wave function that is fully symmetric in terms of space, spin, and flavor To comply with Fermi statistics, this necessitates complete antisymmetry in color This requirement is effectively met in QCD by the expression abc q a q b q c, where a, b, and c represent SU(3) color indices.
• The choice ofSU.N C D 3/colour is confirmed by many processes that directly measureN C Some examples are listed here.
The data on R D e C e ! hadrons shows a notable comparison with QCD predictions, particularly highlighted by the data points above 10 GeV, which indicate the b b N threshold, and at 40 GeV, where the increase attributed to the Z 0 resonance becomes significant.
The total rate for hadronic production ine C e annihilation is linear inN C More precisely, if we considerR D R e C e D e C e ! hadrons/=point.e C e !
In the range above the bbN threshold and below the Z boson mass, neglecting minor computable radiative corrections, we observe a sum of individual contributions proportional to Q², where Q represents the electric charge in units of the proton charge, arising from qqN final states with quark flavors u, c, d, s, and b.
The data neatly indicateN C D 3, as can be seen from Fig.2.2 [306] The slight excess of the data with respect to the value 11/3 is due to QCD radiative corrections (see Sect.2.7).
In the Born approximation, the branching ratio B.W !e /N can be analyzed, focusing on the fermion–antifermion (ffN) final states, which include e, µ, τ, d, and s Notably, the top quark is excluded from these states, as its mass prevents the occurrence of bNt Each channel contributes equally to the overall results, with the exception of the quark channels.
ForN C D3, we obtainBD11% and the experimental number isBD10:7%. Another analogous example is the branching ratioB.£ ! e N e £ / From the final state channels withf De , ,d, we find
In the case of ForN C D 3, we achieve a theoretical value of B D 20%, while the experimental measurement stands at B D 18% The discrepancy in accuracy can be attributed to the significant radiative and phase-space corrections, primarily due to the mass of £ being considerably smaller than that of m W.
An important process that is quadratic inN C is the rate 0 !2”/ This rate can be reliably calculated from a theorem in field theory which has to do with the chiral anomaly:
2 eV; (2.4) where the prediction is obtained forf D 130:7˙0:37/MeV The experimental result is D.7:7˙0:5/eV, in remarkable agreement withN C D3.
There are many more experimental confirmations thatN C D3 For example, the rate for Drell–Yan processes (see Sect.2.9) is inversely proportional toN C
Non-perturbative QCD
Confinement
In this article, we discuss a potential represented by a column vector with real components, where complex fields can be decomposed into real fields The potential is symmetric under an N x N orthogonal matrix rotation, specifically an SO(N) transformation We have excluded odd powers, indicating an additional discrete symmetry Notably, for positive values, the mass term in the potential has a "wrong" sign, suggesting the presence of a non-unique lowest energy state Additionally, we assume that the potential maintains symmetry under infinitesimal transformations.
In the context of symmetry groups and field representations, the equation \(0 = D C \cdot A_t^{ij} (1.50)\) illustrates the relationship between infinitesimal parameters \(A\) and the corresponding matrices \(t^{ij}_A\) This formulation is essential for determining the minimum condition on the potential \(V\), which identifies the equilibrium position of the system.
(or the vacuum state in quantum field theory language) is
By taking a second derivative at the minimum i D i 0 of both sides of the previous equation, we obtain that, for eachA,
The second term vanishes owing to the minimum condition (1.51) We then find
The second derivativesM 2 ki D @ 2 V=@ k @ i / i D i 0 /define the squared mass matrix Thus the above equation in matrix notation can be written as
In scenarios without spontaneous symmetry breaking, the ground state is singular and invariant under symmetry transformations, leading to the condition that all generators leave the state unchanged However, when certain values of the generators yield non-zero vectors, it indicates the existence of multiple states with the same energy, resulting in a non-unique vacuum Each non-zero generator corresponds to an eigenstate of the squared mass matrix with a zero eigenvalue, signifying the presence of massless modes linked to the broken symmetries These massless modes possess quantum numbers that differ from the vacuum's, which is typically assigned zero values, reflecting the charges of the broken generators that do not annihilate the vacuum.
The Goldstone theorem has been previously demonstrated in the classical context, but in quantum field theory, the classical potential relates to the tree level approximation of the quantum potential Higher-order diagrams with loops introduce necessary quantum corrections The functional integral formulation serves as the most suitable framework for defining and calculating the quantum potential through loop expansion, which outlines the vacuum properties of the quantum theory In weakly coupled theories, such as those with a small coupling constant, the tree level expression for the potential remains a close approximation, making the classical scenario valid This situation is particularly evident in the electroweak theory, especially when considering a moderately light Higgs particle.
In a quantum system characterized by a finite number of degrees of freedom, such as those described by the Schrödinger equation, there are no degenerate vacua, meaning the vacuum state is always unique For instance, in the one-dimensional Schrödinger problem involving a specific potential, this uniqueness of the vacuum is clearly illustrated.
In a system with two degenerate minima represented by states |C⟩ and |j⟩, the potential is not diagonal in this basis, leading to non-zero off-diagonal matrix elements due to tunneling effects between the two vacua This tunneling amplitude is influenced by the distance between the vacua and the height of the barrier, expressed through an exponential decay factor After diagonalizing the potential, the eigenvectors can be expressed as a superposition of the degenerate states.
2, with different energies (the difference being proportional to ı) Suppose now that we have a sum of n equal terms in the potential, i.e.,
The transition amplitude in the VDP and V.x models is proportional to the variable n, and it approaches zero as n becomes infinite, indicating that the probability of all degrees of freedom simultaneously overcoming the barrier diminishes This scenario illustrates the presence of a discrete number of minimum points.
The case of a continuum of minima is obtained, still in the Schrửdinger context, if we take
The Schrödinger potential V(x) is analogous to the Higgs potential, with a unique ground state characterized by total orbital angular momentum zero, represented by an s-wave state dependent solely on the radial distance |r| This state reflects a uniform superposition of all directions However, when substituting a single vector r with a vector field M(x) that varies at each spatial point, the probability of transitioning from one minimum state to another diminishes to zero in an infinite volume scenario Essentially, the vectors across space exhibit a negligible chance of collectively rotating simultaneously In this infinite volume limit, all vacua along different directions possess equivalent energy, paving the way for spontaneous symmetry breaking.
A massless Goldstone boson is associated with long-range forces, which would be detectable unless the particles are confined, as seen with gluons in Quantum Chromodynamics (QCD) In constructing Electroweak (EW) theory, physical massless scalar particles cannot be accepted However, during spontaneous symmetry breaking in a gauge theory, massless Goldstone modes emerge but are unphysical and vanish from the spectrum Each Goldstone mode effectively transforms into the third helicity state of a gauge boson that acquires mass, illustrating the Higgs mechanism.
[189, 236, 243, 261] (it should be called the Englert–Brout–Higgs mechanism, because of the simultaneous paper by Englert and Brout) Consider, for example, the simplest Higgs model described by the Lagrangian [243,261]
The presence of a "wrong" sign in front of the mass term for the scalar field is crucial for enabling spontaneous symmetry breaking Additionally, the Lagrangian described maintains invariance under U(1) gauge symmetry.
In our analysis of the U(1) charge Q, we adopt Q D, similar to Quantum Electrodynamics (QED), where the particle is the electron (e) We define the ground state, denoted as 0 D v6D 0, with v being a real number that minimizes the potential and triggers spontaneous symmetry breaking; specifically, v is represented by v 2 D 2 = By utilizing gauge invariance, we implement a change of variables to further our exploration.
Then the position of the minimum at 0 D v corresponds to h D 0, and the
In the context of the Lagrangian, the field \( \phi(x) \) represents the would-be Goldstone boson, particularly when the term \( A = 0 \) is considered In this scenario, the kinetic term \( \partial_\mu \phi \) remains, but the mass term is absent Conversely, in the gauge case, after variable transformation in the Lagrangian, the field \( \phi(x) \) is entirely eliminated, resulting in the absence of even the kinetic term, while the mass term \( \frac{1}{2} m^2 A^2 \) persists.
A is now present: the gauge boson mass isM D p
2ev The fieldhdescribes the massive Higgs particle Leaving a constant term aside, the last term in (1.62) is now
The equation 2@ h@ hh 2 2 C ; (1.63) indicates that the dots represent cubic and quartic terms in h Notably, the h mass term possesses the correct sign, resulting from the interplay of the quadratic terms in h, which emerge from both the quadratic and quartic terms after a shift Consequently, the mass of h is expressed as m 2 h D2 2.
The Higgs mechanism is prominently observed in various physical contexts, initially identified in condensed matter physics by Anderson In the Landau–Ginzburg framework, the free energy of a superconductor can be expressed in a specific form.
The magnetic field interacts with the Cooper pair, characterized by its charge and mass An incorrect sign in the equation results in a minimum value of 6D 0, mirroring the non-relativistic version of the Higgs model This mechanism indicates that massless phonons, which have a dispersion relation of ! D kv with a constant velocity v, do not propagate.
Moreover, the mass term forAis manifested by the exponential decrease ofBinside the superconductor (Meissner effect) However, in condensed matter examples, the
Higgs field is not elementary, but rather a condensate of elementary fields (like for the Cooper pairs).
1.8 Quantization of Spontaneously Broken Gauge Theories:
Chiral Symmetry in QCD and the Strong CP Problem
In the QCD Lagrangian (1.28), the quark mass terms are of the general form
L RCh:c:] (recall the definition of L;R in Sect.1.5and the related discussion).
The presence of specific terms is crucial for demonstrating a chirality flip in the QCD Lagrangian Without these terms, particularly when formD equals zero, the Lagrangian remains invariant under independent unitary transformations applied separately to left-handed (L) and right-handed (R) components This highlights the significance of these terms in understanding the mass dynamics within the framework of quantum chromodynamics.
N f lightest quarks are neglected, the QCD Lagrangian is invariant under a global
In the context of N f D 2, the symmetry SU(2)/V reflects the observed approximate isospin symmetry, while U(1)/V pertains to the baryon number associated with up and down quarks The absence of approximate parity doubling in light quark bound states indicates that the U(2)/A symmetry is spontaneously broken, as evidenced by the lack of opposite parity counterparts to protons and neutrons.
MeV separation in mass from the ordinary nucleons) The breaking of chiral symmetry is induced by the VEV of a quark condensate For N f D 2 this is
A recent lattice calculation has determined the value of the condensate to be 234 ± 18 MeV in the MS scheme with N_f = 2 + 1, using the physical mass value at a scale of 2 GeV This scalar operator is an isospin singlet, which maintains U(2)V symmetry while breaking U(2)A symmetry, and it transforms as (1/2, 1/2) under U(2)L.
NU.2/R, but is a singlet under the diagonal groupU.2/V.
Pseudoscalar mesons, particularly the three pions, are prime examples of would-be Goldstone bosons linked to the breaking of the axial group They possess the quantum numbers of the broken generators, making them approximately massless Goldstone bosons, and they become exactly massless in the limit where the u and d quark masses vanish, reflecting the symmetry breaking of three generators of U(2)/L.
NU.1/A The couplings of Goldstone bosons are very special: in particular only derivative couplings are allowed The pions as pseudo-
Goldstone bosons have couplings that satisfy strong constraints An effective chiral
The Lagrangian formalism effectively captures the low-energy theorems associated with Goldstone particles, particularly in the case of pions It provides a successful framework for describing Quantum Chromodynamics (QCD) at energy scales below 1 GeV.
The breaking mechanism for the remaining U(1)A is attributed to a subtle process, as a state in the -0 space cannot correspond to the Goldstone particle due to excessively large masses, and the 0 mass does not approach zero in the chiral limit Instead, the conservation of the singlet axial current, j₅ = DPqN i = q i, is disrupted by the Adler–Bell–Jackiw anomaly.
F A t A and the normalization is Tr.t A t B / D 1=2ı AB , with
In equation (1.31), the singlet axial current is represented by the factor \( N_f \), which accounts for the involvement of \( N_f \) flavors, specifically \( N_f = 2 \).
An important point is that the pseudoscalar quantityI.x/is a four-divergence More precisely, one can check that
As a consequence the modified currentQj 5 and its associated chargeQQ 5 still appear to be conserved, viz.,
The modified chiral current and charge, which includes an additional gluonic component, is influenced by the QCD vacuum's topological structure, specifically instantons However, this charge is not conserved, highlighting the complexities of quantum chromodynamics For further insights, refer to the introduction in [308].
The configuration where all gauge fields are zero A A D 0 can be called
In gauge theories, all configurations connected to a specific vacuum by a gauge transformation must represent the same physical vacuum In Abelian theories, gauge fields expressed as the gradient of a scalar are equivalent to a reference gauge field However, in non-Abelian gauge theories, certain large gauge transformations can be topologically nontrivial, associated with non-zero integer values of a topological charge known as the winding number For instance, considering SU(2) as a simplified example, one can analyze a time-independent gauge transformation, which highlights the complexity of gauge configurations in non-Abelian contexts, where the relationship between different gauge fields is more intricate than in Abelian theories.
A a a *nd recalling the general expression of a gauge transformation in (1.15), the gauge transform of the potential by˝ 1 is
For the vector potentialA 1 / , which is a pure gauge and hence part of the “vacuum”, the winding numbern, defined in general by nD ie 3 s
A i x/A j x/A k x/ ijk ; (2.14) is equal to 1, i.e.,n D 1 Similarly, forA m / obtained from˝ m = [˝1 m , one has n Dm Given (2.9), we might expect the integrated four-divergence to vanish, but instead one finds ˛s
(2.15) for a configuration of gauge fields that vanish fast enough on the space sphere at infinity, and the winding numbers aren at timetD 1(“instantons”).
In Quantum Chromodynamics (QCD), gauge fields are categorized into distinct sectors characterized by different quantum numbers Each sector has its own vacuum state, denoted as |jni, where the relationship |jni D |jn+1i indicates a non-gauge invariant structure However, the true vacuum must maintain gauge invariance, which is achieved through a superposition of all states |jni.
In fact, ˝ 1 ji DX e in jnC1i De i ji: (2.17)
If we compute the expectation value of any operatorOin thevacuum, we find hjOji DX m ; n e i mn / hmjOjni: (2.18)
The path integral describing the O vacuum matrix element at D 0 must be modified to reproduce the extra phase, taking (2.15) into account: hjOji D
Z dAd d ON exp iS QCDCi˛s
: (2.19) This is equivalent to adding a term to the QCD Lagrangian:
The term is parity (P) odd and charge conjugation (C) even, so it introduces
CP violation and time reversal (T) violation are expected to yield a value of Q around O(1) However, this would significantly affect the neutron electric dipole moment, with the relationship d_n (ecm) = 3 x 10^-16 Q Given the stringent experimental limits on d_n, approximately 3 x 10^-26, it follows that Q must be exceedingly small.
The "strong CP problem" involves the challenge of explaining the unexpectedly small value of CP violation in quantum chromodynamics A key aspect for potential solutions is that a chiral transformation results in a consistent, fixed change.
By recalling (2.11), we have e i• Q Q 5 ji D j2N fıi: (2.21)
To prove this relation we first observe thatQQ 5 is not gauge invariant under˝1, because it involvesk 0 : ˝ 1 QQ 5 ˝ 1 1 DQ 5 ˝ 1 2N f˛s
In a chiral invariant theory, the possibility of eliminating certain terms hinges on the condition that at least one quark mass is zero, with the up quark (m_u) being a prime candidate However, this scenario has been ruled out in previous studies For quarks with non-zero masses, a transformation is required to render the mass matrix Hermitian and diagonal, which necessitates a chiral transformation that impacts the system This is characterized by the relationship U(N) = U(1) × U(N-1).
SU.N/and that for Hermitianmthe argument of the determinant vanishes, i.e., arg detmD 0, the transformation from a genericm 0 to a real and diagonalmgives arg detmD0Darg det U L Carg detm 0 Carg detU R
From this equation one derives the phaseıRıL of the chiral transformation and then, by (2.21), the important result for the effective value: effDCarg detm 0 : (2.25)
The small empirical value of the effective potential poses a significant naturalness issue for the Standard Model (SM) of particle physics A compelling solution to this problem is the mechanism proposed by Peccei and Quinn, which offers an intriguing approach to addressing these naturalness concerns.
The Standard Model (SM) incorporates an enlarged theory that remains invariant under an additional chiral symmetry, U(1)/PQ, which acts on its fields This symmetry is spontaneously broken by the vacuum expectation value of a scalar field, denoted as vPQ The resulting Goldstone boson, known as the axion, is not massless due to the chiral anomaly The anomaly leads to the cancellation of the parameter by the vacuum expectation value of the axion field, influenced by the characteristics of the associated potential Axions may play a role in dark matter composition if their mass resides within a specific narrow range.