What is Renormalization group: Definition and 43 Discussions

In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle.
A change in scale is called a scale transformation. The renormalization group is intimately related to scale invariance and conformal invariance, symmetries in which a system appears the same at all scales (so-called self-similarity).As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller scale, with different parameters describing the components of the system. The components, or fundamental variables, may relate to atoms, elementary particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be variable couplings which measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the self-same components as one goes to shorter distances.
For example, in quantum electrodynamics (QED), an electron appears to be composed of electrons, positrons (anti-electrons) and photons, as one views it at higher resolution, at very short distances. The electron at such short distances has a slightly different electric charge than does the dressed electron seen at large distances, and this change, or running, in the value of the electric charge is determined by the renormalization group equation.

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  1. S

    A Relationship between bare and renormalized beta functions

    I'm looking at a proof of beta function universality in ##\phi^4## theory, and at one point they do the following step: after imposing that the renormalized coupling ##\lambda## is independent of the cutoff ##\Lambda##, we have $$0= \Lambda \frac {\text{d} \lambda}{\text{d} \Lambda} = \Lambda...
  2. qft-El

    A Solving renormalization group equation in QFT

    I'm learning about the RG equation and Callan-Symanzik equation. In ref.1 they claim to solve the RG equation via the method of characteristics for PDE. Here's a picture of the relevant part: First, the part I don't understand - the one underlined in red. What does "compatible" mean here...
  3. S

    A Renormalized vertex functions in terms of bare ones

    Let ##\Gamma[\varphi] = \Gamma_0[\sqrt{Z}\varphi ] = \Gamma_0[\varphi_0]## be the generating functional for proper vertex functions for a massless ##\phi##-##4## theory. The ##0## subscripts refer to bare quantities, while the quantities without are renormalized. Then $$\tilde{\Gamma}^{(n)}(p_i...
  4. ohwilleke

    I New QCD Parameter Determinations

    A pair of new papers (here and here) make precision determinations of the quark masses and the strong force coupling constant using the renormalization group summed perturbation theory (RGSPT). For comparison purposes, I have followed each value with the Particle Data Group (PDG) value, and then...
  5. G

    A Relationship between Wilson's RG and the Callan-Symanzik Equation

    I have taken a Quantum Field Theory course recently in which we first derived the Callan-Symanzik equation and then discussed Wilson's Renormalization. However, I don't think I have a clear understanding of the procedures and how they relate to each other. For the sake of this question, let's...
  6. DaniV

    The 1-loop anomalous dimension of massless quark field

    I tried as first step to find Z_q the renormalization parameter, to do so I did the same procedure to find the renormalization parameter of the gauge field of the gluon A^a_\mu when a is representation index a \in {1,2,...,N^2-1} such that A^{a{(R)}}_{\mu}=\frac{1}{\sqrt{Z_A}}A^{a}_{\mu}...
  7. H

    A Is Block Spin Still ±1 in Ising Model?

    When one applies block spin scheme on Ising model, what's the value for block spin? If we set the average value for spin in a block, it should not be ±1 anymore. But if so, is the transfer matrix is still two dimensional?
  8. Giulio Prisco

    A Did nature or physicists invent the renormalization group?

    Or in other words: The renormalization group is a systematic theoretical framework and a set of elegant (and often effective) mathematical techniques to build effective field theories, valid at large scales, by smoothing out irrelevant fluctuations at smaller scales. But does the...
  9. diegzumillo

    A Conformal Window: Understanding the Basics

    I'm reading about extensions of standard model and this pops up frequently but it's not very clear. I understand it's a region in parameters space so renormalization group naturally becomes relevant and that's about it for my understanding. I can't connect any of this to the beta function of the...
  10. S

    A Wilsonian RG and Effective Field Theories

    Years ago after reading Ch. 12 of Peskin and Schroeder (and the analogous discussion in Zee), I thought I fully understood the modern Wilsonian view of renormalization, and how it explains why non-renormalizable field theories still have meaning/predictive power at energies well below the...
  11. J

    A How are renormalizability and locality connected?

    In his paper Quantum Field Theory: renormalization and the renormalization group Zinn-Justin states: Low energy physics does not depend on all the details of the microscopic model because some RG has an IR fixed point or at least a low dimension fixed surface. Of course at this stage the next...
  12. J

    A Why do we need to renormalize in QFT, really?

    There are several reasons given in the literature, why UV infinities arise in QFT in the first place. My problem is putting them together, i.e. understand how they are related to each other. So... UV divergences arise and thus we need to renormalize, because: We have infinite number of...
  13. C

    I Scaling and the renormalization (sub)group

    I am aware of only two fields where the renormalization (sub)group ideas can be systematically and unambiguously applied: particle physics and equilibrium critical behaviour. 1.- Are there any others? 2.- What are these ideas used for in fluid mechanics? 3.- When cosmologists speak about...
  14. T

    A On the renormalization group flow

    I'm currently studying the Landau-Wilson model for critical phenomena (Statistical Mechanics, Kerson Huang) where the renormalization group is a central object. In the end, the calculations lead to a set of coupled differential equations that describe the (metaphorical) evolution of the...
  15. unknown1111

    A Computing the pole mass from a given MS mass?

    Given a Yukawa coupling as a function of scale and a vev, how can I compute the corresponding pole mas? Understandably most paper explain how from a measured pole mass one can compute the running mass, for example, Eq. 19 here. However I want to compute the pole mass from the running mass. In...
  16. unknown1111

    Top quark mass mt at energy scales μ<mt?

    Does it make sense to talk about the top mass at energies below mt, although in all processes the corresponding energy scale is above mt because of the rest mass energy of the top quark? Using an effective field theory approach, the top quark decouples at energies below the top quark mass and...
  17. Shahrokh

    Can the universe become stable with a new electroweak vacuum

    The top and Higgs mass determination arose the old discussion about electroweak vacuum metastablity. There is an interesting fact that with available data the universe places in the edge of stable and meta-stable zone tends to be inside the meta-stable region. This conclusion confirms up to...
  18. H

    Error in Srednicki renormalization?

    On page 164-165 of srednicki's printed version (chapter 27) on other renormalization schemes, he arrives at the equation $$m_{ph}^{2} = m^2 \left [1 \left ( +\frac{5}{12}\alpha(ln \frac{\mu^2}{m^2}) +c' \right ) + O(\alpha^2)\right]$$ But after taking a log and dividing by 2 he arrives at...
  19. A

    P28 of phase transitions and the renormalization group

    Hi, I'm confused about the discussion on p28 of Nigel Goldenfeld's "Lectures on phase transitions and the renormalization group" (this question can only be answered by people who have access to the book.) The goal is to compute the potential energy of a uniformly charged sphere where the...
  20. nikosbak

    Dimension of interaction in a QFT theory

    The problem statement. When an exercises say " the interaction in a QFT has dimensions Δ" , what does it mean?, it means the field or the Lagrangian has this mass dimension? In this exercise I'm trying to find the classical beta function (β-function) for the assciated couling.
  21. M

    The definition of mass of an electron (after the renorm group)

    Hi there, I have a question about the rest mass of an electron. As we all know, the charge of an electron is a function of the energy at which the system is probed. When defining the charge, we typically use as our reference scale the charge measured in Thompson scattering at the orders of...
  22. ChrisVer

    What group is renormalization group?

    What type of group is the Renormalization Group? All I've seen is people giving a (differential) equation for beta-function when they teach for the RG... Also I haven't been able to find an algebra characterizing the RG... Any clues?
  23. marcus

    Renormalization Group Approaches to Quantum Gravity (conference at PI)

    Perimeter conference http://pirsa.org/C14020 Here are links to the talks' videos and slides PDF Recent developments in asymptotic safety: tests and properties Tim Morris http://pirsa.org/14040085/ What you always wanted to know about CDT, but did not have time to...
  24. S

    Help Solving Renormalization Group Equations

    This isn't a homework problem, but something from a set of notes that I'd like to better understand. My confusion starts on page 23 here: http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-9-RenormalizationGroup.pdf. I'm having trouble reproducing his calculation for the...
  25. N

    Why can you only contract one field in the Wilson approach to renormalization?

    When I am reading about the Wilson approach to renormalization in Chapter12.1 of Peskin & Shroeder I am wondering why are you allowed only to contract the \hat{\phi} field (this is the field that carries the high-momentums degrees of freedom)as they show in equation 12.10, I thought that we...
  26. C

    Understanding the renormalization group

    From what I now understand of renormalization it is really a reparametrization of the theory in terms of measurable quantities instead of the 'inobservable bare quantities' that follow the Lagrangian; at least that is one interpretation of what is going on. The originally divergent physical...
  27. A

    What is the purpose of the renormalization group?

    Hello, I've been reading a book on QCD on I have a question: what is the purpose of the renormalization group? Is it to remove the large logs so that we can use pertubation theory (at least for large -q^2)? And what is the physical significance of the renormalization scale \mu^2?
  28. K

    Effective field theory and Wilson's renormalization group

    I have just read my first course on Quantum Field Theory (QFT) and have followed the book by Srednicki. I have peeked a bit in the books by Peskin & Schroeder and Ryder also but mostly Srednicki as this was the main course book. Now, I have to do a project in a topic not covered in the course...
  29. G

    Two separate renormalization group equations?

    Are there two separate renormalization group equations? One for how the physical coupling constants change with time, and one for how the bare parameters/coupling constants change with cutoff? Is there a relationship between the two? It just seems that textbooks use the term renormalization...
  30. W

    Renormalization Group for dummies

    Renormalization Group concept is rarely given in laymen book on QM and QFT and even Quantum Gravity book like Lisa Randall Warped Passages. They mostly described about infinity minus infinity and left it from there. So if you were to write about QFT for Dummies. How would you share it such...
  31. N

    Quantum field theory and the renormalization group

    The following statements are from the paper with the above title, recommended in another thread, are from here: http://fds.oup.com/www.oup.co.uk/pdf/0-19-922719-5.pdf An interpretion of these statements would be appreciated: 1. [first paragraph, page 3] What is 'conservation of...
  32. N

    Stat. Phys. : Renormalization Group and scaling hypothesis

    Hello everyone, I am currently studying the renorm. group in Stat. physics, more precisely how a rescaling (of space) leaves the partition function unchanged, at the price of having an infinite space of parameters due to the interaction proliferation at each rescaling. Let K be our...
  33. tom.stoer

    Renormalization group and universality

    I remember an argument which says that closed to critical points all systems are universal in the sense that their behavior is described by the critical exponents and that these critical exponents depend only on the dimension of the system and the dimension of the order parameter. I remember...
  34. N

    Why the renormalization group flow depends only the basic symmetry,but not Lagrangian

    Please teach me this: Why the renormalization group flow and the fix-point depends only on the basic symmetry but not on the Lagrangian form.In general speaking,the physics laws depend only the basic symmetries?By the way,the Klein-Gordon,linear sigma,nonlinear sigma Lagrangian flow to one...
  35. A

    Book on renormalization group in fluid dynamics?

    I found between my family's books (cousins mostly) 4 books for fluid mechanics, and since next semester i ll be taking it it d be cool if i could just chose between them. Oh btw its for mechanical engineering i currently have: Fluid Mechanics. Robert A. Granger...
  36. N

    What is the objective of renormalization group theory?

    Please teach me this: It seem to me that the objective of renormalization were the exclusion the infinities.But in renormalization group theory,they consider the dependence of physics parameters(e.g the interaction constant lamda,the mass parameter) on momentum p.Then I do not understand what...
  37. Q

    Searching the phenomenological renormalization group equation for 2d ising model

    hellow everybody i have a problem in styding the critical bihaviour of the tow dimensional ising model when i use periodiques boundary conditions i found that the fixed point for this case is the PRG equation that mean the following recursion relation...
  38. Z

    Book on Feynmann diagrammatica and renormalization group

    i would need some good books (with examples) in the following subjects - Feynman diagramas (how to calculate them) - Renormalization group my background: i have got a degree on physics, so i know what ODE , PDE , or even the Feynman integral and propagator, but i did not study the part...
  39. Z

    Does Renormalization group tell you if a theory is Renormalizable or not ?

    Does Renormalization group tell you if a theory is Renormalizable or not ?? the idea is this, using the Renormalization group equation for our theory (QED, Gravity, Gauge theories..) can tell this RG equation if our theory is renormalizable or not for big or small energies ??
  40. P

    Question about Renormalization Group Equation

    Hello, in our lecture, we computed \beta^{\overline{MS}} , \gamma_m^{\overline{MS}} for - \frac{\lambda}{4!} \phi^4 Theory. These are: \beta^{\overline{MS}} (\lambda_{\overline{MS}}) = b_1 \lambda_{\overline{MS}}^2 + O(\lambda_{\overline{MS}}^3) \gamma_m^{\overline{MS}}...
  41. M

    The idea behind renormalization group.

    What is the idea behind renormalization group ?? i believe you begin with an action S[\phi] =\int d^{4}x L(\phi , \partial _{\mu} \phi ) then you expand the fields into its Fourier components upto a propagator.. \phi (x) =C \int_{ \Lambda}d^{4}x e^{i \vec p \vec x} + c.c but...
  42. S

    Renormalization group and cut-off

    Hi.. in what sense do you intrdouce the cut-off inside the action \int_{|p| \le \Lambda} \mathcal L (\phi, \partial _{\mu} \phi ) then all the quantities mass m(\Lambda) charge q(\Lambda) and Green function (every order 'n') G(x,x',\Lambda) will depend on the value of cut-off...