Effective Theories
The central concept of the RG (Renormalization Group) is the relationship between phenomena across various distance and energy scales To effectively describe phenomena at a length scale of μ − 1, we can utilize variables defined at this scale For instance, hadrons and mesons, which are composed of quarks, serve as the best description for processes occurring at low energies However, as we examine higher energy processes, or shorter distances, the significance of quark degrees of freedom becomes essential At low energies, quarks play a crucial role in determining coupling constants and masses within the effective theory, where hadrons and mesons are the primary focus.
The concept of an effective theory is crucial to our discussion, as it posits that the physical world is best described on distance scales greater than μ − 1 by focusing on degrees of freedom defined at this scale This approach eliminates unnecessary degrees of freedom, resulting in an optimal description However, this effective theory tends to break down for length scales smaller than μ − 1, necessitating the development of a new effective theory for these smaller scales.
1 In the following μ is a momentum scale.
T J Hollowood, Renormalization Group and Fixed Points, SpringerBriefs 1 in Physics, DOI: 10.1007/978-3-642-36312-2_1, © The Author(s) 2013
The Renormalization Group concept introduces new degrees of freedom, where the parameters of the effective theory, such as coupling constants and masses, are determined by the properties of a more fundamental theory This involves establishing matching conditions at a specific momentum scale between the two theoretical layers A key assumption is the separation of scales, which necessitates identifying the relevant physical variables and understanding their interactions at the desired scale These interactions, represented by various couplings, can theoretically be derived from the underlying microscopic theory When the number of effective variables and couplings is limited, experimental data can be used to refine these details, resulting in an effective theory capable of making accurate predictions.
In the functional integral formalism, a quantum field theory (QFT) is characterized by the action S[φ;g i ], which serves as a functional of the fields and depends on an infinite number of parameters known as coupling constants These coupling constants encompass all mass terms and interaction strengths, effectively representing coordinates in theory space Consequently, a QFT is defined through a functional integral that encapsulates these elements.
All the difficulties in defining a QFT are lurking in the definition of the measure on the fields
Defining a classical field is complex due to its infinite degrees of freedom, making integration over such an infinity challenging In perturbation theory, this difficulty manifests as ultraviolet (UV) divergences in loop integrals, which arise when internal momenta become large These divergences are closely related to the infinite fluctuations of the field across all scales.
RG Flow
At least initially, in order to properly define the measure
To effectively manage the ultraviolet (UV) degrees of freedom in the theory, it is essential to implement a cut-off procedure This approach serves to regulate the divergences that arise in loop diagrams within the framework of perturbation theory.
High-energy divergences in quantum field theories arise from the fields' fluctuations at extremely small distances, necessitating the suppression of these modes for proper regulation This process introduces a new momentum scale, known as the cutoff (μ), into the theory Wilson's innovative approach to renormalization group (RG) theory cleverly transforms this seemingly disadvantageous aspect into a beneficial tool for understanding the behavior of quantum fields.
There are several methods to introduce a cut-off or regulator in Quantum Field Theory (QFT), such as defining the theory on a spatial lattice after Wick rotation to Euclidean space, where μ − 1 represents the physical lattice spacing Alternatively, high momentum modes can be suppressed in the Euclidean theory by modifying the action or measure, with μ acting as the momentum or energy scale Upon returning to Minkowski space, μ becomes a space-like momentum scale Another regularization method used in perturbation theory is dimensional regularization, which involves analytically continuing integrals over Euclidean loop momenta by altering the space-time dimension After isolating and removing divergences, the space-time dimension in loop integrals can be fixed to its physical value While this approach may not be intuitive, it is straightforward to implement and is widely accepted in particle physics.
In the context of effective theories, a physical quantity F(g i ;) μ is influenced by a length scale, necessitating that the scale be greater than μ − 1 This quantity is also dependent on the cut-off scale μ The renormalization group (RG) theory asserts that one can adjust the cut-off in such a way that the physics remains unchanged for length scales greater than μ − 1 Consequently, this implies that the couplings in the action must vary with μ, which is encapsulated in the RG equation.
The functions g i (μ) define the RG flow of the theory in the space of couplings.
In theoretical physics, the physical mass is represented as m phys = m phys (g i (μ)), indicating that it is independent of any length scale It is crucial to differentiate between the physical mass defined by the pole position in the field's propagator and the mass term in the Lagrangian, as the latter requires renormalization like other coupling constants.
The RG flow is typically understood as progressing from the ultraviolet (UV) to the infrared (IR), indicating a decrease in the parameter μ However, it is also useful to consider the flow in the opposite direction, where μ increases towards the UV For the RG equation to be valid, the coupling space must encompass all possible couplings, which is inherently infinite The complexity of the RG arises from the need to "integrate out" the theory's degrees of freedom as the cutoff is lowered.
2 If F is a correlation function then there will be additional wave-function renormalizations of the operator insertions required: see (1.12) below.
The concept of the Renormalization Group (RG) relates energy scales μ and μ', presenting a challenging yet advantageous situation in Quantum Field Theory (QFT) due to the unique focusing properties of RG flows The first RG equation highlights the significance of the RG momentum scale μ As discussed regarding effective theories, the ideal cutoff choice for μ is at the scale μ = -1, aligning with the momentum scale relevant to the phenomena being studied.
The RG equation can be utilized in a second manner by applying the RG transformation to reduce the cutoff from μ to μ/s, where s is greater than 1 Subsequently, we can rescale the entire theory's length by s − 1 to restore the cutoff back to μ By defining the couplings g i as dimensionless through appropriate scaling with powers of μ, we derive an alternative RG equation.
The second renormalization group (RG) equation, expressed as F(g i (μ);s) μ = s − d F F(g i (μ/s);) μ, is crucial for understanding the infrared (IR) and ultraviolet (UV) behaviors of a theory with a fixed cutoff This equation allows for the analysis of physical quantities at different scales, depending on whether the scale factor \( s \) is greater or less than one Specifically, when \( s > 1 \), the left-hand side represents a physical quantity at scale \( s \), while the right-hand side reflects the same quantity with couplings evolving towards the IR as \( μ → μ/s \) This perspective is particularly relevant in particle physics, where it is essential to compare the behavior of theories across varying momentum scales The RG transformation is derived from how the action changes with the cutoff, with the Wilsonian Effective Action \( S[φ;μ,g i ] \) encapsulating the dependence on fields, couplings, and the cutoff Thus, the RG framework provides a systematic approach to understanding how physical observables evolve with energy scale changes.
In quantum field theory (QFT), the wave-function renormalization, denoted as Z(μ), plays a crucial role in the behavior of fields In scenarios with multiple fields, Z(μ) becomes a matrix that can interrelate all the fields involved For the simplest QFT model, which consists of a single scalar field, the action is expressed as the sum of a kinetic term and a linear combination of operators O_i(x).
1.2 RG Flow 5 are powers of the fields and their derivatives,e.g.φ n ,φ n ∂ μ φ∂ μ φ, etc: 3
In the context of renormalization group (RG) analysis, the mass dimension of the operator \( O_i(x) \) is denoted as \( d_i \), and we have opted for dimensionless couplings, necessitating the inclusion of an appropriate power of the cutoff for dimensional consistency Unlike the first approach, the second perspective on RG maintains the same cutoff, rendering the re-scaling irrelevant Additionally, the wave-function renormalization factor \( Z(\mu) \) can be interpreted as the coupling to the kinetic term.
, (1.6) although Zalso appears in the other terms in the action as a multiplicative factor.
In practice one thinks about infinitesimal RG transformations which are encoded in thebeta functionsof the couplings defined as β i (g j )=μdg i (μ) dμ (1.7)
The beta functions are solely dependent on the couplings and their relationship to the cutoff is implicit The running couplings can be determined by integrating the beta-function equations Consequently, the solution to the flow equations connects the couplings at two different scales, expressed as g_i(μ) = g_i g_j(μ), μ/μ.
The fact that couplings always appear as combinationsμ d − d i g i in the action (1.5) means that the beta functions have the form μdg i dμ =(d i −d)g i +β i quant (g j ) (1.9)
The initial term originates from the classical scaling associated with the powers of μ in the action, while the second component is derived from the complex quantum aspect of the renormalization group (RG) transformation This quantum aspect entails a significant integration of degrees of freedom within the functional integral, which will be elaborated upon in Chapter 2.
Ifwere re-introduced exp[i S] →exp[i S/], then the quantum piece would indeed vanish in the limit→0 We also define theanomalous dimensionof a fieldφin
The term "operator" or "composite operator" originates from a canonical quantization method in quantum field theory (QFT), where a Hilbert space is constructed, allowing fields to be represented as operator-valued entities This terminology persists in the functional integral approach, even though the quantities involved are not technically operators.
6 1 The Concept of the Renormalization Group terms of the quantityZ(μ)as γ φ (g i )= −μ
The reason for the terminology is explained below (1.13).
In particle physics, the S-matrix elements serve as the fundamental physical observables, dictating the probabilities of various processes These elements can be derived from the more fundamental Green's functions, which act as correlation functions of fields Green's functions are evaluated through insertions into the functional integral Γ(n)(x₁, , xₙ; μ, gᵢ(μ)).
UV and IR Limits and Fixed Points
1.3 UV and IR Limits and Fixed Points
The RG scale μ is crucial for understanding the momentum scale of physical interest, particularly through its IR (infrared) and UV (ultraviolet) limits as μ approaches 0 and ∞, respectively These limits indicate how the theory behaves at very long and short distances As we flow towards the IR, all physical masses relative to the cutoff, m/μ, increase In theories without massless particles, known as having a mass gap, decreasing μ beyond the mass of the lightest particle results in no physical degrees of freedom to propagate at low momentum scales Consequently, in the IR limit as μ approaches 0, the theory becomes empty with no propagating states Alternatively, the RG flow may begin on the critical surface.
In the realm of theoretical physics, there exists an infinite dimensional subspace where the mass gap is absent In these specific theories, the infrared (IR) limit becomes significant, leading to a scenario where only the massless degrees of freedom persist.
As the energy scale approaches zero (μ→0), massless degrees of freedom persist, and in many traditional theories, the coupling constants evolve towards a fixed point in the renormalization group flow, where the beta functions become zero: 4 μdg i/dμ = g ∗ j.
Notice that the wave-function renormalization and anomalous dimensions do not need to vanish at a fixed point.
Fixed point theories are unique due to their massless degrees of freedom and lack of couplings with non-vanishing mass dimensions, making them scale invariant This scale invariance is inherently elevated to conformal transformations, leading to the designation of these theories as conformal field theories (CFTs) The conformal group, particularly its connected part, includes Poincaré transformations, scale transformations (dilatations), and special conformal transformations, represented mathematically as x μ → sx μ and x μ → x μ + x²b μ, respectively.
4 There are some exotic situations where the couplings flow to a limit cycle.
The concept of the Renormalization Group for a vector \( b_\mu \) involves infinitesimal transformations related to Lorentz, dilatations, and special conformal transformations, represented as \( \delta x^\mu = \epsilon^{\mu\nu} x_\nu \), \( \delta x^\mu = s x^\mu \), and \( \delta x^\mu = x^2 b^\mu - 2 x^\mu (x \cdot b) \), where \( \epsilon^{\mu\nu} = -\epsilon^{\nu\mu} \) In any local quantum field theory (QFT), an energy-momentum tensor \( T^{\mu\nu} \) is present, which plays a crucial role in the Ward identity expressed as \( n \, p = 1 \, \phi_1(x_1) \, \frac{\delta \phi_p(x_p)}{\delta \phi_n(x_n)} \).
The invariance of quantum field theory (QFT) under Lorentz transformations necessitates a symmetric energy-momentum tensor, expressed as T μν = T νμ Additionally, the requirement for invariance under dilatations leads to the condition that the tensor is traceless, T μ μ = 0 Together, these two conditions are sufficient to ensure invariance under infinitesimal special conformal transformations.
(1.19) Returning to the the RG flows, in the neighbourhood of a fixed pointg i =g i ∗ + δg i , we can always linearize the beta function: μdg i dμ g ∗ j +δ g j
In a suitable basis for{δg i }, which we denote by{σ i }, the linear term is diagonal: μdσ i dμ =(Δ i −d)σ i +O(σ 2 ) (1.21) and so to linear order the RG flow is simply σ i (μ) μ μ Δ i − d σ i (μ ) (1.22)
The scaling dimension, denoted as Δ i, refers to the operator associated with σ i in quantum field theory (QFT) In interacting QFTs, this dimension typically differs from the mass dimension The anomalous dimension of the operator is defined as γ i = Δ i − d i, paralleling the definition of the anomalous dimension for the field itself.
1.3 UV and IR Limits and Fixed Points 9
Couplings in the neighbourhood of a fixed point flow as in (1.22) and are accordingly classified in the following way:
(i) If a coupling hasΔ i dthe coupling flows back into the fixed point and is known as an irrelevantcoupling.
In the case of Δ i = d, the coupling is considered marginal, necessitating higher-order analysis to understand its behavior If higher-order terms cause the coupling to diverge from or converge towards the fixed point, it is classified as marginally relevant or irrelevant, respectively Alternatively, if the coupling does not evolve at all orders, it is deemed a truly marginal coupling, indicating that the original fixed point is part of a continuum of fixed points.
In a conformal field theory (CFT), the Green's functions exhibit covariance under scale transformations, imposing significant constraints on their behavior For instance, the two-point Green's function, represented as Γ(2)(x) = φ(x)φ(0), adheres to the broader renormalization group (RG) equation outlined in equation (1.12).
At a fixed point, the equations g i (μ) = g i (μ) = g i ∗ and Z(μ) = (μ /μ) 2 γ φ ∗ Z(μ) hold true, where γ φ ∗ = γ φ (g ∗ i ) Through dimensional analysis, we derive Γ ( 2 ) (x;μ,g i ∗ ) = μ 2d φ G(xμ) By substituting into the RG equation, we can solve for the unknown function G, resulting in Γ ( 2 ) (x;μ,g i ∗ ) = c μ 2 γ φ ∗ x 2d φ + 2 γ φ ∗ ∝ 1 x 2 Δ φ, where c is a constant This outcome illustrates the typical power-law behavior found in correlation functions within a conformal field theory (CFT).
When tracing an RG flow backward toward the UV, the physical masses of particles relative to the cutoff, denoted as m_phys/μ, decrease, indicating that a theory with a mass gap approaches the critical surface In general scenarios, whether a mass gap exists or not, the trajectory tends to diverge.
5 This is true in both versions of the RG equations (1.2), where m phys is fixed and μ decreases, and(1.3), where μ is fixed and m phys increases.
The concept of the Renormalization Group (RG) involves analyzing the behavior of physical systems as the energy scale changes In the context of finite μ, it is possible to fine-tune the trajectory of the system so that it approaches a fixed point on the critical surface as μ approaches infinity This process allows for the examination of RG flows around a fixed point characterized by two irrelevant directions and one relevant direction, highlighting the intricate relationships between different scales in the system.
The flows off the critical surface converge towards the "renormalized trajectory," which is characterized by the flow emerging from the fixed point This convergence, combined with the limited number of significant directions, highlights the concept of Universality, a fundamental aspect of RG flows Additionally, Universality is observed in flows initiating from the critical surface, as these flows also progress towards the fixed point.
CFTs possess a limited number of relevant couplings, leading to an infinite-dimensional domain-of-attraction for fixed points in theory-space Consequently, the renormalization group (RG) flows of theories situated off the critical surface converge onto finite-dimensional subspaces defined by these relevant couplings This indicates that the infrared (IR) behavior of a theory is influenced primarily by a small set of relevant couplings, rather than the infinite array of couplings {g i } As a result, the IR characteristics of a theory are categorized into a few "universality classes," which correspond to the domains of attraction of one or more fixed points.
The Continuum Limit
The Renormalization Group (RG) plays a crucial role in addressing the challenge of taking a continuum limit of a Quantum Field Theory (QFT), which involves removing the cutoff by increasing its original finite value μ to infinity while preserving the physics at any momentum less than the original μ This process is non-trivial, and the existence of such a limit is uncertain Notably, taking a continuum limit requires the inverse RG flow, where the coupling constant g_i(μ) evolves as μ increases, posing a significant theoretical hurdle.
The RG Equation (1.2) illustrates the process of achieving a well-defined UV limit of g_i(μ) as μ approaches infinity, where g_i(∞) serves as a fixed point of the RG This leads to the emergence of a "renormalized trajectory," which characterizes the theory across all length scales Although finding a renormalized trajectory may seem daunting due to the need for fine-tuning an infinite set of couplings, universality simplifies this search It is not necessary to be precisely on the renormalized trajectory to define a continuum theory; instead, a one-parameter set of theories with a cutoff μ and couplings g_i = ˜g_i(μ) suffices, provided that g˜_i(∞) is situated on the critical surface within the fixed point's domain of attraction related to the CFT.
IR à increasin g ˜ gi(à ′ ) ˜ gi(∞) fixed point renormalized trajectory critical surface
The limit as μ approaches infinity is defined to ensure that the infrared (IR) physics at the original cut-off scale μ remains constant Specifically, the number of parameters needed to establish a continuum limit—determining the IR physics—is equal to the number of relevant couplings in the conformal field theory (CFT) However, there are various methods to fix these relevant couplings at the scale μ and to define the behavior of the irrelevant couplings g˜ i (μ) as μ approaches infinity This flexibility allows for multiple approaches to achieving a continuum limit, known as renormalization group (RG) schemes, all of which yield the same continuum theory In the realm of particle physics, this freedom can be effectively utilized.
The Renormalization Group concept simplifies the action in quantum field theory (QFT) when dealing with a limited number of terms, corresponding to the relevant couplings of the ultraviolet conformal field theory (UV CFT) This approach also enables the description of the same QFT using a lattice cutoff, resulting in an action that appears significantly different from the original continuum action.
In particle physics, while some may argue that a continuum limit is unnecessary as effective theories remain valid with a cutoff above the relevant momentum scale, exploring the limits of these theories can provide valuable insights Understanding how far we can extend an effective theory into the ultraviolet (UV) region before inconsistencies arise offers a glimpse into the underlying fundamental physics at shorter distance scales.
The renormalization group (RG) concept, as introduced by Wilson, is thoroughly detailed in his reviews (Wilson and Kogut, 1974; Wilson, 1983), with the latter serving as a comprehensive summary of his Nobel Prize Lecture in 1982 This work provides an insightful overview of the history of renormalization in statistical physics and quantum field theory, along with an intuitive explanation of the RG For those interested in statistical mechanics, notable introductions include the works of Pfeuty and Toulouse (1977), Cardy (1996), and Fisher (1998) From a quantum field theory perspective, Zinn-Justin (2002) and Amit (1984) offer valuable insights, while Peskin and Schroeder's textbook also covers RG extensively.
(1995) Another interesting book about the RG and critical phenomena which links
RG ideas to more general non-equilibrium problems, is by Goldenfeld (1992).
Amit, D.J.: Field Theory, the Renormalization Group, and Critical Phenomena World Scientific, Singapore (1984)
Cardy, J.L.: Scaling and Renormalization in Statistical Physics Cambridge University Press, New York (1996)
Fisher, M.E.: Renormalization group theory: its basis and formulation in statistical physics Rev. Mod Phys 70, 653 (1998)
Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group, Frontiers in Physics, vol 85 Addison-Wesley, New York (1992)
Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory Westview Press, New York (1995)
Pfeuty, P., Toulouse, G.: Introduction to the Renormalization Group and Critical Phenomena Wiley, Chichester (1977)
Wilson, K.G., Kogut, J.B.: The renormalization group and the epsilon expansion Phys Rept 12,
Wilson, K.G.: The renormalization group and critical phenomena Rev Mod Phys 55, 583 (1983)
Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena Clarendon Press, Oxford (2002)
In this chapter, we apply the Renormalization Group (RG) concept to the fundamental quantum field theory (QFT) involving a single real scalar field This example demonstrates how the seemingly straightforward RG idea can present significant challenges when put into practice.
In particle physics we often write down simple actions like 1
To align with the principles of renormalization group (RG) theory, we initially permit all terms that adhere to space-time symmetries, particularly for a scalar field This includes all powers of the field and its derivatives, arranged in a Lorentz invariant manner For clarity, we focus on operators that remain even under the transformation φ → −φ, demonstrating how symmetry can limit possible couplings while ensuring that the symmetry is maintained throughout the RG flow A straightforward scaling analysis reveals that a composite operator O, which incorporates p derivatives and 2n powers of the field, can be represented as ∂^p φ^(2n), yielding a classical mass dimension of 2 d_O = n(d−2) + p.
At the classical level, the quantity of relevant and marginal couplings, defined by d O ≤ d, is notably limited The following table categorizes various operators into relevant, marginal, and irrelevant classes based on their space-time dimensions.
1 In our notation, the scalar product in Minkowski space is a μ b μ = η μν a μ b ν = − a 0 b 0 + a ã b.
2 Note that the mass dimension of the field itself is fixed by the kinetic term to be d − 2 2
T J Hollowood, Renormalization Group and Fixed Points, SpringerBriefs 13 in Physics, DOI: 10.1007/978-3-642-36312-2_2, © The Author(s) 2013
O d > 4 d = 4 d = 3 d = 2 φ 2 rel rel rel rel φ 4 irrel marg rel rel φ 6 irrel irrel marg rel φ 2n irrel irrel irrel rel
(∂ μ φ) 2 marg marg marg marg φ 2n (∂ μ φ) 2 irrel irrel irrel marg
The classical scaling suggests that, at least in dimensions d>2, we only need to keep track of the kinetic term along with a completely general potential energy term, that is
In the above, we have used the powers of the cut offμin order to have dimensionless couplings g 2n
Finding the RG Flow
To address the core issue of locating the RG flow, we need to utilize the RG Equation (1.4) in conjunction with the Wilsonian Effective Action.
In the context of a scalar field theory with a cutoff μ, the observables at momentum scales below this cutoff remain constant as μ changes To relate theories with varying cutoffs, it is essential to establish a specific cutoff procedure A fundamental and straightforward method to regularize the theory involves applying a sharp momentum cutoff to the Fourier modes following the Wick rotation to Euclidean space In this framework, the Lagrangian takes on a specific form, facilitating further analysis of the theory.
2(∂ μ φ) 2 +V(φ) (2.5) with S E d d xL E and the functional integral becomes
[dφ]exp(−S E ) 3 The momentum cut off involves Fourier transforming the field
3 We take it as established fact that one can transform between the Minkowski and Euclidean versions of the theory without difficulty In our conventions, the Wick rotation involves η μν a μ b ν =
− a 0 b 0 + a ã b → a μ b μ = a 0 b 0 + a ã b In Euclidean space the functional integral
The expression [d φ] e − S E [φ] serves as a probability measure on the field configuration space when appropriately normalized This connection highlights the close relationship between Euclidean Quantum Field Theory (QFT) and statistical physics systems.
To find the RG flow, we start with the expression φ(x) = (1/(2π)^d) ∫ d^d p φ(˜p)e^(ip·x) and apply a sharp cutoff, limiting the momentum vector to |p| ≤ μ This approach ensures that the resulting theory is UV finite, as loop momenta do not reach infinity We perform the RG transformation by decomposing the field φ, defined with the cutoff μ, into two components: ϕ, which contains modes with |p| ≤ μ, and ˆφ, which encompasses modes in the range μ ≤ |p|.
To extract the beta functions, we analyze the infinitesimal transformation defined by μ = μ + δμ By examining the changes in the action after integrating out the Fourier modes φ, we can derive the renormalization group (RG) flow This approach allows us to concretely express the results in terms of the exponential function.
The Wilsonian Effective Action is defined at the scales μ and μ + δμ on both sides, with the field on the right-hand side scaled to ensure that Z(μ + δμ) equals 1, maintaining generality.
Expanding the action on the right-hand side in powers ofφˆ 4 :
In this discussion, we observe that the cross term \( d d x \partial_\mu \phi \partial_\mu \hat{\phi} \) vanishes, which is clearly evident when expressed in momentum space modes The key aspect of the analysis is that we will omit the subscript E for "Euclidean," as the context will clarify whether we are operating within Minkowski or Euclidean space.
In our analysis of the action, we have omitted the term φ ˆ V (ϕ) because it does not influence the effective potential we will calculate It is important to recognize that we can treat ϕ as a constant and adjust it to correspond with the minimum of V (ϕ).
16 2 Scalar Field Theories to integrate over the modesφ One way to do this is to work in terms of Feynmanˆ diagrams:
The terms that one gets in the effective action, after integrating out φ, canˆ be interpreted in terms of Feynman diagrams with only φˆ on internal lines contributing a propagator
In the context of quantum field theory, the interaction vertices are derived by expanding the potential V(ϕ + ˆφ) in terms of the fields ϕ and ˆφ, leading to terms such as g6 The analysis focuses on configurations with two external lines of ϕ and ˆφ, without the presence of propagators, indicating a specific structure of the interactions within the theory.
Each loop involves an integral over the momentum ofφˆ which lies in a shell between radiiμandμ in momentum space: μ≤| p |≤μ d d p
If we are only interested in an infinitesimal RG transformationμ =μ+δμthen the integrals over loop momenta (2.11) become much simpler: μ≤| p |≤μ+δμ d d p
(2π) d d d − 1 Ωˆ f(μΩ),ˆ (2.12) where Ωˆ is a unit d vector to be integrated over a unit d −1-dimensional sphere
In the calculation involving S d − 1, the crucial aspect is that each loop integral introduces a factor of δμ, which means only one-loop diagrams are necessary to linear order in δμ The momentum dependence in the loop is exclusively through the invariant p² = μ², leading to a constant integral over the solid angle Ωˆ, which represents the volume of a d−1-dimensional sphere, Vol(S d − 1) Despite this significant simplification, we still face the complex combinatorial challenge of summing an infinite series of one-loop diagrams, as illustrated in the example below.
Fortunately, there is a simple way to sum all such one-loop diagrams in one go The trick is to keep only the terms quadratic inφˆin (2.10),
(2.13) and then perform the resulting Gaussian integral overφˆ e −δ S
In order to extract the RG transformation, we identify the change in the Wilsonian effective action as
In this analysis, we observe that there is no wave-function renormalization involved Since our primary focus is on the effective potential, we can simplify our approach by treating ϕ as a constant By denoting the Fourier transform of φ(x)ˆ as φ(˜ p), we can proceed with our calculations.
(2π) d d d − 1 Ωˆ φ(μ˜ Ω)ˆ φ(μ˜ Ω) ,ˆ (2.16) using the fact that p ν p ν = μ 2 for the modesφ˜ Hence, performing the Gaussian integral over the modesφ˜yields the result e −δ S =C π μ 2 +V (ϕ)
In the context of quantizing field theories, the number of modes in the momentum shell can be determined by introducing an infra-red cut-off through a large box of size L with periodic boundary conditions This approach quantizes momentum components as \( p^\mu = \frac{2\pi n^\mu}{L} \), where \( n^\mu \) are integers, in natural units Consequently, each volume element in Euclidean phase space contains one mode per volume of \( (2\pi)^d \) Assuming a sufficiently large L, the total volume of space-time is represented as \( V = L^d \), leading to a calculation of the number of modes within the momentum space shell.
In this theory, at one-loop order, the sole Feynman diagram featuring two external legs lacks any external momentum flowing through the loop, which prevents the emergence of a term proportional to it.
In this case, we can write the contribution in (2.17), up to an unimportant overall constant, as exp(−δS)=exp
In the context of the Wilsonian effective action, we define a parameter \( a \) as \( a = \frac{Vol(S^{d-1})}{2(2\pi)^d} = 2^{-d} \frac{\pi^{-d/2}}{\Gamma(d/2)} \) This formulation replaces the volume factor \( V \) with an integral over space-time, allowing us to eliminate the temporary assumption that \( \phi \) remains constant Consequently, we can accurately identify the changes in the Wilsonian effective action.
Expanding the right-hand side in powers ofϕ, allows us to extract the beta functions of the couplings directly: μdg 2n dμ =(n(d−2)−d)g 2n −aμ n ( d − 2 ) d 2n dϕ 2n log(μ 2 +V (ϕ)) ϕ= 0 (2.21)
If we expand in powers of the coupling constants, the contributions on the right- hand side can be identified with individual one-loop diagrams like the one shown previously.
From (2.21), the first few beta functions in the hierarchy are μdg 2 dμ = −2g 2− ag 4
Quantum contributions are influenced by inverse powers of the factor 1 + g², which represents m²/μ² + 1, where m denotes the mass of the field These factors originate from the propagators of the modes in the loop Consequently, when m is much smaller than μ, the quantum terms become suppressed, aligning with the expected behavior of decoupling.
Mapping the Space of Flows
The beta functions play a crucial role in mapping the renormalization group (RG) flow within theory space, beginning with the identification of RG fixed points linked to conformal field theories (CFTs) Among these, the "Gaussian fixed point" stands out as the trivial fixed point where all couplings are zero (g²n = 0) By linearizing around this Gaussian fixed point, the beta functions can be expressed as μdg²n/dμ = (n(d−2)−d)g²n − ag²n + 2.
At the Gaussian fixed point, the scaling dimensions are given by the classical dimensions Δ2n = d2n = n(d - 2), indicating that the anomalous dimensions are zero The couplings that diagonalize the scaling dimensions matrix, σ2n, do not exactly equal g2n due to additional terms Specifically, σ2 = g2 is always relevant, while σ4 = g4 + ag2/(2 - d) is relevant for d < 4, irrelevant for d > 4, and marginally irrelevant for d = 4 In the case of d = 4, a non-linear approach is necessary Since g6 is irrelevant in d = 4, it can be disregarded, and with a = 1/16π², we find that μdg4/dμ = 3.
At the Gaussian fixed point, g4 becomes marginally irrelevant as μ decreases, indicating its diminishing significance We can express the integration constant using a mass-dimensioned parameter, Λ, represented as g4(μ) = 16π².
In the context of quantum theory, dimensional transmutation occurs when the freedom to specify a dimensionless coupling g₄ in the action transforms into a mass dimension quantity, Λ, under the condition μ < Λ This Λ represents the momentum scale at which the running coupling g₄(μ) diverges, indicating that perturbation theory will fail as this critical scale is approached, as elaborated in Chapter 3.
Finding non-trivial fixed points is challenging, but progress can be made by perturbatively analyzing the couplings This involves considering the renormalization group (RG) flow equations in arbitrary non-integer dimensions, treating ε = 4−d as a small parameter It is anticipated that the results for small ε will qualitatively apply to physical cases where ε equals 1, 2, and so on Through this approach, a new non-trivial fixed point, referred to as the Wilson-Fisher fixed point, is identified.
2.2 Mapping the Space of Flows 21 g ∗ 2 = −1
The Wilson-Fisher fixed point is deemed physically acceptable only when ε > 0 or d < 4; otherwise, the couplings g 2n ∗ become negative, resulting in an unbounded potential and instability in the theory Near the fixed point in the (g 2, g 4) subspace, the linear approximation in terms of ε reveals that μ d dμ δg 2 δg 4 equals ε/3−2 − b(1+ε/6).
So the scaling dimensions of the associated operators and the associated couplings at the Wilson-Fisher fixed point are Δ 2=2−2ε
Therefore at this fixed point only the mass couplingσ 2is relevant.
The flows in the(g 2 ,g 4 )subspace of scalar QFT for smallε >0 are shown below: g4 g 2
The Gaussian and Wilson-Fisher fixed points are illustrated, highlighting that all other couplings are irrelevant, which allows us to focus solely on the flows within the (g2, g4) subspace It is crucial to note, as indicated in equation (2.28), that the irrelevant couplings do not disappear within this subspace Additionally, the critical surface intersects this subspace along the line connecting the two fixed points.
The Wilson-Fisher fixed point has been established for small ε and is known to exist in both dimensions d = 2 and d = 3 This fixed point belongs to the universality class of the Ising Model, highlighting its significance in the realm of statistical physics.
7 The Ising Model is a statistical model defined on a square lattice with spins σ i ∈{+ 1 , − 1 } at each site and with an energy (which we identify with the Euclidean action)
22 2 Scalar Field Theories fails to show is that in d =2 there are actually an infinite sequence of additional fixed points 8
The continuum limits of scalar field theories can be understood through the qualitative analysis of RG flows In four dimensions (d=4), there exists only the Gaussian fixed point, characterized by a single relevant direction, specifically the mass coupling g² This results in a unique renormalized trajectory where g²(μ) = (μ/μ₀)² g²(μ₀), with all other couplings approaching zero This trajectory effectively describes the behavior of a free massive scalar field.
If we sit precisely at the fixed point we have a free massless scalar field In particular, according to our crude analysis there is no interacting continuum theory in d =4.
To ensure meaningful interactions, it is essential to maintain a finite cut-off, a conclusion supported by advanced analyses This concept, referred to as “triviality,” holds significant implications for the Higgs sector within the standard model, as discussed in Chapter 3 In three dimensions, assuming the presence of the Wilson-Fisher fixed point, there exist two fixed points and a two-dimensional space of renormalized trajectories defined by the couplings g2 and g4 Specifically, when we characterize our continuum theories using g2 and g4, they are confined to designated regions in the parameter space.
In particular, the line of theories A is free and massive (and must have g 2 >0); the theories in regions B and D are interacting and massive and in the UV becomes non-
The sum is calculated over all nearest-neighbour pairs (i, j), with T representing the temperature At low temperatures, the energy promotes the alignment of spins, while high temperatures lead to significant thermal fluctuations that disrupt long-range order, highlighting the competition between energy and entropy A second-order phase transition occurs at the critical temperature T = T_c, where long-distance power-law correlations emerge This critical point aligns with the Wilson-Fisher fixed point in the same universality class.
8 In d = 2 there are powerful exact methods for analyzing CFTs because in d = 2 the conformal group is infinite dimensional as it consists of any holomorphic transformation t ± x → f ± ( t ± x )
The continuum theories are characterized by two key parameters: a mass scale and an interaction strength In scenario D, where g2 is less than zero and the field possesses a vacuum expectation value (VEV), the line of theories emerges from the Gaussian fixed point, indicating a dynamic interaction between the elements in the space of flows.
The theories represented by points C and E illustrate significant interactions that culminate in the Ising model conformal field theory (CFT) in the ultraviolet (UV) regime In case E, where g2 is less than zero and a vacuum expectation value (VEV) is present, the theories along line F depict a massless interacting theory that transitions from a free theory in the UV to the Ising model CFT in the infrared (IR) Point G corresponds to a free massless theory, known as the Gaussian fixed point, while point WF represents the Ising model CFT, referred to as the Wilson-Fisher fixed point Notably, any theories located to the right of line C–E lack continuum limits.
In three-dimensional RG flows, the concept of RG crossover is exemplified through two theories linked to RG trajectories, labeled B and D Theory B represents a scenario with interacting heavy mass approximations, while Theory D corresponds to a conformal field theory (CFT) characterized by weakly interacting light mass.
The trajectories originating from the Gaussian fixed point exhibit asymptotic freedom in the far ultraviolet (UV) region, a characteristic commonly associated with gauge theories At intermediate energy scales, these trajectories approach the Wilson-Fisher fixed point, where the spectrum features a light interacting scalar particle, denoted as m with mμ In this regime, the theory exhibits approximate conformal invariance, with its physics largely governed by the Wilson-Fisher conformal field theory (CFT) However, as the energy scale μ decreases to and below m, the renormalization group (RG) trajectory diverges from the Wilson-Fisher fixed point, leading to an interacting and massive theory.
The implementation of the Wilsonian RG in scalar field theories with a sharp momen- tum cut off is a well-studied problem The approximation of the full effective action
The Background Field Method
The renormalized couplings are influenced by the dimensional regularization mass scale μ, similar to the sharp momentum cut-off scheme By employing the background field method, we can capture the essence of Wilson's renormalization group (RG) This approach involves expanding the field φ in the renormalized action into a slowly varying background field ϕ and a rapidly fluctuating component, represented as φ = ϕ + φ̂.
This is analogous to the decomposition (2.7) in the sharp momentum cut-off scheme.
We then treatφˆas the field to integrate over in the functional integral whilst treating ϕas a fixed background field.
Given that ϕ is slowly varying, we can expand it in terms of its derivatives Following the approach outlined in Chapter 2, we can treat ϕ as a constant to calculate the effective potential Consequently, the effective potential in d = 4 at one loop is derived.
Subtracting the divergence with a counter-term in the MS scheme gives
4!λφ 4 , notice that the counter-term is proportional to
The effective potential can be expressed with terms proportional to ϕ² and ϕ⁴, resembling the original potential To achieve this, it is necessary to incorporate counter-terms for m² and λ Through further analysis, it is determined that the appropriate counter-terms correspond to equations (3.11) and (3.14) The renormalization group (RG) flow of the dimensionless couplings g₂ = m²/μ² and g₄ = λ can be derived by ensuring that the effective potential adheres to an RG equation.
V eff (Z(μ) 1 / 2 ϕ;μ,g i (μ))=V eff (Z(μ ) 1 / 2 ϕ;μ ,g i (μ )), (3.21) where hereμis not the Wilsonian cut off, but is a variable that plays an analogous rôle, the mass scale introduced in dimensional regularization.
At one-loop order, there is no wave-function renormalization, leading to the equations μdg²/dμ = −2g² + ag²g⁴ and μdg⁴/dμ = 3ag²⁴, where a = 1/16π² Comparing these results with the beta functions from the momentum cut-off scheme reveals that the latter provides exact one-loop results, while the former includes contributions from all loop orders Additionally, the momentum cut-off scheme shows clear decoupling at mμ(g²₁), unlike the dimensional regularization with the MS subtraction scheme, which requires manual implementation of decoupling These differences highlight that the renormalization group (RG) flows are sensitive to the chosen scheme, yet the fundamental "topological" characteristics, such as the existence of fixed points and the relevance of couplings, remain scheme-independent Notably, repeating the analysis in dimensions d < 4 confirms the presence of the Wilson-Fisher fixed point.
Triviality
Our examination of scalar field theories in four-dimensional space-time reveals a crucial finding: there exists only one fixed point, the non-interacting Gaussian fixed point Consequently, the only viable continuum theory is that of a free particle, either massive or massless This absence of an interacting continuum scalar theory in four dimensions is referred to as triviality.
The conclusion presented appears contradictory to perturbation theory, which suggests the possibility of defining an interacting theory with a finite renormalized coupling However, issues arise within perturbation theory due to the necessity of maintaining a finite renormalized coupling while allowing the cutoff μ to approach infinity Analyzing the flow of the coupling reveals a singularity at μ = μe 16 π² / 3g₄(μ), where the bare coupling g₄(μ) diverges, indicating a Landau pole This phenomenon, commonly associated with Quantum Electrodynamics (QED), confirms expectations regarding the behavior of the coupling in such theories.
In the context of scalar quantum field theory (QFT) in four dimensions, the triviality of the coupling constant \( g_4(\mu) \) indicates that it becomes irrelevant at the Gaussian fixed point As we trace the renormalization group (RG) trajectory back toward the ultraviolet (UV) regime, the flow diverges and becomes infinite at a finite value of \( \mu \) This suggests that, despite appearing renormalizable in a perturbative sense, scalar QFT lacks a non-trivial fixed point and is therefore not truly renormalizable in four dimensions An effective interacting theory can be defined, but it will have an explicit cutoff, making it valid only for momenta \( p < \mu \) This finding is particularly relevant to the Higgs sector of the standard model, which ultimately predicts its own breakdown at high energies when momenta approach the cutoff scale \( \mu \) determined by the low momentum coupling \( g_4(\mu) \).
We now sketch how to use perturbation theory (when valid) and the background field method to calculate the RG flows of the relevant couplings.
(1) Write down a LagrangianL r (φ)with all the “relevant” couplingsg r i
(2) Splitφ=ϕ+ ˆφand calculate loop diagrams with externalϕand internalφˆ (more precisely only include diagrams which areone particle irreducible
To effectively manage divergences in 1PI, it is essential to introduce counter-terms that align with the form of L_r(ϕ) If the counter-terms deviate from this form, a suitable generalization of L_r(ϕ) must be implemented.
(3) The analogue of the Wilsonian effective action in the special subspace of theories parametrized by the relevant coupling is then
Note that this action will include all the irrelevant couplings as well; how- ever, these will depend on the relevant couplings.
(4) Extract the RG flows of the relevant couplings inL r (ϕ)by imposing the
RG Improvement
In this problem we consider the issue of RG improvement in the context of the effective potential of aφ 4 theory ind =4 The one-loop effective potential in the
MS scheme is given in (3.19) TakingV = 1
The key inquiry revolves around the nature of the couplings \( m \) and \( \lambda \) The fundamental principle dictates that the effective potential \( V_{\text{eff}} \) must adhere to the renormalization group (RG) equation (3.21) Consequently, the couplings \( m \) and \( \lambda \) should vary with the scale \( \mu \) to ensure that \( V_{\text{eff}} \) remains invariant with respect to \( \mu \).
The running couplingsm(μ) 2 = μ 2 g 2 (μ)andλ(μ)= g 4 (μ)are the solutions of the beta function equations (3.22) and to one-loop order there is no wave-function renormalization.
The effective potential, V_eff, is expected to be independent of the renormalization scale, μ; however, the selection of μ becomes significant due to the limitations of perturbation theory While a complete summation of the perturbative series would render μ irrelevant, working to a specific order necessitates truncating the series in powers of the running coupling, resulting in a truncated effective potential that is dependent on μ.
In order to simplify the remaining discussion, let us specialize to the massless case and point out an interesting—but in retrospect, incorrect—conclusion Setting m=0, we have
This apparently has a minimum away from the origin at a value ofϕfor which ϕ 2 = 2μ 2 λ exp
The one-loop correction results in an effective potential that shifts its minimum away from the origin, generating a vacuum expectation value (VEV) for the scalar field, which could have significant implications for spontaneous symmetry breaking However, caution is warranted, as the scalar field's value is non-perturbative in the coupling, leading to tree-level and one-loop contributions to the effective potential being comparable in magnitude This indicates a violation of the perturbative expansion, raising doubts about the reliability of the conclusions drawn.
As ϕ approaches zero, the log term in the effective potential V_eff (3.26) becomes large, which disrupts the perturbative expansion around the tree-level minimum at ϕ=0 To address the behavior of V_eff (ϕ) for small values of ϕ, we can express it as V_eff (ϕ; μ, λ(μ)) and select an appropriate renormalization group (RG) scale μ Given that ϕ is the only additional scale in the problem, a natural choice for μ is to set it equal to ϕ.
In perturbation theory, the effective potential remains unchanged at all orders; however, when truncated to one loop, its functional behavior significantly diverges from equation (3.26) Notably, as the limit approaches ϕ → 0, the running coupling at one-loop order satisfies the equation μdλ/dμ = 3.
As the field variable ϕ approaches zero, the perturbation theory improves significantly, leading to a scenario where the "RG improved" effective potential achieves a minimum at ϕ = 0, indicating the absence of spontaneous symmetry breaking This highlights the importance of applying perturbation theory in a coupling that varies with the energy scale relevant to the problem, ensuring its validity in regions where the coupling remains small.
Coleman and Weinberg (1973) demonstrated that a consistent model of spontaneous symmetry breaking driven by quantum loop effects can be developed in theories with multiple coupling constants Specifically, when a scalar field interacts with the electromagnetic field, there are two couplings: the coupling constant λ and the electric charge e of the scalar field This necessitates that the scalar field be complex Consequently, the effective potential is influenced by contributions from a photon loop, resulting in an expression akin to (3.26), but with λ in the second term substituted by e² and an additional factor of 3 accounting for the extra degrees of freedom associated with the photon.
The contribution from the photon loop dominates the original scalar loop and the latter is ignored Now the minimum of (3.31) lies at ϕ 2 = 2μ 2 e 2 exp
At the one-loop level, wave-function renormalization can be disregarded Furthermore, it is important to recognize that selecting μ = aϕ, where a is a constant equal to 1, is equivalent to using a = 1 through a redefinition of the mass and coupling.
The presence of a minimum requires balancing tree-level and one-loop contributions across two distinct perturbative expansions in λ and e² As a result, unlike earlier scenarios, we can now reliably conclude that a vacuum expectation value (VEV) develops.
The renormalization group (RG) emerged in perturbation theory to address ultra-violet divergences Key texts, such as Collins (1984) and Peskin and Schroeder (1995), provide insights into this aspect of RG, with the latter also detailing the process of RG improvement.
Coleman, S.R., Weinberg, E.J.: Radiative corrections as the origin of spontaneous symmetry break- ing Phys Rev D 7, 1888 (1973)
Collins, J.C: Renormalization An Introduction to Renormalization, the Renormalization Group, and the Operator Product Expansion, p 380 University Press, Cambridge (1984)
Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory Addison-Wesley, Reading(1995)
Gauge Theories and Running Couplings
In this chapter, we turn our attention to the RG properties of gauge theories includingQED along with the strong and weak interactions.