I'm missing the point of renormalization in QFT I am a qft noob studying from Quantum Field Theory: An Integrated Approach by Fradkin, and in section 13 it discusses the one loop corrections to the effective potential $$U_1[\Phi] = \sum^\infty_{N=1}\frac{1}{N!}\Phi^N\Gamma^{N}_1(0,...,0)$$
And how the first $D/2$ terms are divergent where $D$ is the dimensionality.
The book then discusses that the solution to this is to define the renormalized mass $\mu^2 = \Gamma^{(2)}(0)$ and renormalized coupling constant $g = \Gamma^{(4)}(0)$, and then expressions relating the bare mass to the renormalized mass, and expressions relating the bare coupling constant to the renormalized constant are found, where the integrals now only run to a UV cutoff $\Lambda$. The effective potential is then written in terms of the renormalized mass and renormalized coupling constant, and the result is magically finite.
Somewhere in this process I am a bit lost. First, I am not really intuitively seeing the motivation for defining the renormalized mass and coupling constant as the two- and four-point vertex function at zero external momenta. What is the motivation behind this? Second I feel I am a bit lost about how the resulting effective potential after all of this becomes finite. I suppose I can see mathematically that the result is finite, but I do not at all understand it. At what point in our scheme does the finiteness really come out of? What is the point in all of this?
 A: Here's some complementary perspective to the excellent answer by AccidentalFourierTransform. This turned out very long, but this is a huge topic which can't be entirely summarized in one answer. An important point I want to make is that renormalization is conceptually independent of either the quantum nature of QFT or the field theory nature of QFT, and is rather a ubiquitous and foundational concept that really gets to the core of many-body physics (i.e. physics of many interacting degrees of freedom). QFT just so happens to be the "ultimate" many-body problem (infinitely many interacting quantum harmonic oscillators filling spacetime)
Renormalization and Coarse-Graining
The original clearest interpretation of renormalization was Kadanoff's block-spin construction for the Ising model, which can be referred to as "direct-space renormalization". The basic idea is that we have a microscopic model describing the physics at the smallest scales (e.g. a spin model on a lattice) and we perform consecutive coarse-grainings, spatially averaging the degrees of freedom over some region of size $\ell$, and rewriting the theory in terms of the original model Hamiltonian but now with "renormalized" parameters, and repeating this again with a slightly larger size region, etc., until we converge to a "fixed point". The idea here is that at each step we are "integrating out" microscopic degrees of freedom, and it is this process of tracing over degrees of freedom and "zooming out" to a macroscopic scale that is what renormalization is all about. A beautiful demonstration of this for the Ising model is shown in this video.
Note that in practice, it is much easier to perform this coarse-graining in "reciprocal/momentum space", where we integrated out high-momentum modes (corresponding in direct-space to short-wavelength modes). This is called "momentum-space RG".
From a physical experimental perspective, the basic idea of RG is that when you perform a measurement, you are necessarily resolution-limited to some smallest accessible lengthscale $\ell$ probed by your experiment (e.g. a collider can be thought of as being analogous to a gigantic microscope). You can't actually observe the physics at smaller lengthscales than $\ell$, it all appears "fuzzy" to your apparatus. Thus you are really probing the coarse-grained description. As you vary $\ell$, e.g. by ramping up your collider energy to make $\ell$ smaller, you are changing the coarse-grained lengthscale at which you probe the theory, thus the observables "flow" with $\ell$ (or with energy).
Renormalization and Probability Distributions
There is a perspective by which renormalization corresponds to a generalization of the central limit theorem. The idea here is that we have some physical system comprised of a large (potentially infinite, more on that at the end) number of degrees of freedom which dynamically fluctuates among a large number of possible configurations, with some probability distribution on this space of configurations. The coarse-graining process described above corresponds to averaging over more and more degrees of freedom, thought of as random variables. The central limit theorem essentially says that this process should converge to a Gaussian distribution, which in RG corresponds to the trivial fixed point (all couplings flow to zero). But it is possible that we flow to a non-trivial fixed point, which corresponds to flowing to a non-Gaussian probability distribution. This is explained in this paper.
Renormalization and Fluctuations
In a statistical mechanics problem, we often start with a microscopic model specified by an energy functional $E$ on the space of configurations, and all of the macroscopic statistical behavior is captured by the partition function, which computes the averages over microscopic degrees of freedom,
$$Z = \sum_{\text{configurations}} e^{-\beta E}$$
where $\beta$ is inverse temperature. In QFT the analog of the partition function is just the path integral, replacing $\beta \sim i\hbar^{-1}$ and the energy functional $E$ by the action functional. It is convenient to describe the same information contained in the partition function in terms of an "effective energy functional" ("effective action"), by writing
$$Z = e^{-\beta F}$$
where $F$ is often called the free energy. In thermodynamics $F$ takes into account both the internal energy $E$ and also the "fluctuations", encoded in the entropy $S$, and can be written $F = E - TS$ where $T$ is temperature, meaning that there is a balance between minimizing the energy and maximizing the entropy. In a microcanonical picture, $S$ is just the logarithm of the multiplicity of states at a given energy, and so is a measure of "how much room there is to fluctuate". While the system wants to stay in low-energy configurations, it also wants to fluctuate between as many configurations as possible, meaning it may explore higher-energy configurations if there are a lot of them available (i.e. the density of states is large).
In the above description we averaged over all of the microscopic configurations. A slightly more refined perspective would be something like the following. Let $\phi$ be an effective variable (an "effective field") that describes the macroscopic configurations of a system. As an analogy, imagine flipping hundreds of coins. The microscopic degrees of freedom are individual coins, and microscopic configurations are particular realizations of each coin being heads or tails. Then $\phi$ can describe the macroscopic "total number of heads" or "total number of tails". For each value of $\phi$ there are many microscopic configurations which yield the same macroscopic description. We write a generalized partition function and free energy as
$$Z[\phi] = \sum_{\text{configs. combatible with }\phi} e^{-\beta E} = e^{-\beta F[\phi]}$$
The free energy $F[\phi]$ now describes the combination of the energy of configurations compatible with $\phi$ and their multiplicity. This is the purely statistical origin of renormalization: it describes how one flows from a description of many microscopic degrees of freedom to a description of macroscopically observable ones.
$F[\phi]$ is directly analogous to the quantum effective action or effective potential in QFT. The system wishes to minimize the action while it explores a large number of higher-action configurations via quantum (rather than thermal) fluctuations (this is where instantons and anomalies begin to enter the story of QFT, which correspond to local, rather than global minima of the action). In QFT generally we start with an underlying microscopic field theory and we wish to describe it in terms of a macroscopic "fluctuation averaged" field theory which has "renormalized" couplings. (To some people this is what they mean by "renormalization", i.e. that the microscopic and macroscopic theories have the same form/Lagrangian, and it is the parameters of the microscopic theory which flow to those of the macroscopic theory.)
While in the condensed matter/statistical mechanics case we often have access to some microscopic model which may have fixed parameters that can be controlled, in QFT this is generally not the case. There are many underlying microscopic models ("UV completions") which can all flow to the same RG fixed point ("IR limits"), which intuitively makes sense, since we have in any case integrated out the microscopic details. This means that it is more useful to specify the physical macroscopic theory in terms of macroscopic quantities, i.e. the parameters appearing in the effective action, rather than in terms of the microscopic parameters (in continuum QFT the microscopic parameters may not even be sensible to specify at all, see later).
Anharmonicity and Interactions: Quasiparticles
While I just described renormalization due to fluctuations/statistical averaging, another important source of renormalization is interactions (note: even non-interacting QFT's can have nontrivial behavior). Often the starting point to an RG analysis is some mean-field theory. We assume some equilibrium configuration for our system and perform some sort of perturbative expansion around that. Examples of mean-fields would be

*

*A free field description plus weak interactions (such as in weakly-interaction QFT)

*A uniform ground state plus weak fluctuations (e.g. a low temperature expansion around the ground state of a ferromagnet)

*A Ginzburg-Landau expansion of the free energy in terms of an order parameter in the vicinity of a critical point.

Roughly speaking, one is performing an expansion around a minimum of some potential. Expanding the potential to quadratic order yields a free theory (i.e. a bunch of decoupled harmonic oscillators with plane waves being the basic excitations), while expanding beyond quadratic order introduces anhamonicities, i.e. interactions between the traveling wave solutions (generally the quartic term being the important one, describing two-particle interactions). In a free theory we interpret the plane waves (or perhaps Gaussian wavepackets) as being particles, at least in so far as they have well-defined rest masses. A basic question is whether or not the renormalized, macroscopic theory also has a particle description, i.e. can be approximated as a free theory. If that is the case then the resulting particles are (in condensed matter) referred to as quasiparticles. The origin of this term comes from Fermi liquid theory which describes a gas of interacting electrons (i.e. an ordinary metal). The quasiparticles in that case look like electrons, but have renormalized masses. In general quasiparticles may also be unstable with a finite lifetime.
The general picture of quasiparticles is that they are equivalent to the underlying particles of the theory (e.g. electrons) but they are "dressed" by interactions. In the QFT description, a charged electron point particle will polarize the virtual electron-positron vacuum fluctuations around it, which renormalizes the charge of the electron. In other words, the charge is surrounded by a cloud of charge/anti-charge dipole pairs, which polarize in the field of the core charge and thus attenuate the field when viewed from large distances. Thus the charge of the electron is renormalized by its interactions with other charged particles. This is exactly equivalent to the perspective that RG is "averaging over fluctuations", and is why the fine structure constant $\alpha$ flows.
At the field theory level, mass renormalization due to interactions corresponds to the renormalization of the quadratic (i.e. Gaussian) term in the action. At the level of probability distributions we are talking about approximating an anharmonic, non-Gaussian distribution by a Gaussian one.
Theories with a renormalized quasiparticle description are generally referred to as weakly interacting. Theories which do not admit a renormalized quasiparticle description are called strongly interacting. It may be possible, e.g. through a duality transform, to turn a strongly interaction theory into a weakly interacting one, though the dual weakly-interacting quasiparticles may have a very complex description in the original theory.
Divergences: UV and IR
Keep in mind the following equivalences of concepts

*

*high energy $\approx$ high momentum $\approx$ short-wavelength $\approx$ microscopic

*low energy $\approx$ low momentum $\approx$ long-wavelength $\approx$ macroscopic (coarse-grained)

Divergences in QFT are of two types: ultraviolet (UV, short-lengthscale) or infrared (IR, long-lengthscale). It's common to talk about renormalized theories as being being the "IR limit/theory", i.e. coarse-grained limit, and the microscopic theory as the "UV limit/theory" (or, given an IR theory, people talk about "UV completions", i.e. microscopic theories which flow to the given IR limit).
IR divergences generally come from "soft modes", such as massless particles. The basic issue is that whereas massive ("gapped") excitations are suppressed at low energy (long wavelength), requiring a minimum energy to be created, massless ("gapless") excitations can destabilize the theory because they can be excited at arbitrarily low energy (long wavelength) and thus have non-negligible contribution to the IR physics. Very roughly speaking, for a fixed energy, one could have a single massless particle at that energy, or two with half the energy, or three with a third the energy, etc (because massless particles have a linear energy-momentum relationship). Massless particles can sort of "infinitely proliferate" by splitting up into lower and lower energy (longer and longer wavelength) excitations. This is the origin of infrared divergences in QFT calculations. They generally show up as terms which are proportional to the "volume of space(time)". These divergences can be effectively cured by putting the entire system in a large box (or on a torus) of lengthscale $L$, thus limiting the longest wavelengths/frequencies to $L$ and the minimum energies to $1/L$.
UV divergences on the other hand are the ones which appear in the RG calculation. These divergences essentially come from the fact that in QFT we treat spacetime as continuous. As AccidentalFourierTransform's answer points out, the solution to this is to realize that the microscopic field configurations should be treated as distributions rather than functions. Treated as functions, they can become arbitrarily short wavelength (large momentum), meaning that the space of configurations averaged over includes extremely wild, non-differentiable ones (like a delta function, but everywhere), not just "smooth" configurations. Equivalently, the problem is that we have an infinite number of degrees of freedom per unit volume. A field theory is microscopically a bunch of coupled harmonic oscillators, but in the continuum there is one harmonic oscillator for every one of the uncountably many points in spacetime. These UV divergences are cured by introducing a minimum lengthscale (e.g. a lattice, which naturally gives us a finite number of degrees of freedom per unit volume and a minimum wavelength), or equivalently a maximum momentum/energy/frequency scale (an ultraviolet cutoff). If the cutoff can be removed by adding only a finite number of counterterms to absorb the divergences to the Lagrangian, the theory is renormalizable, or "UV complete". Otherwise the theory can still be considered an effective field theory valid below the cutoff energy.
Note that we don't know what the microscopic physics of the universe is, or whether or not spacetime is truly continuous. I think most people would say that it is not, and there is a natural UV cutoff provided by the Planck scale (e.g. arbitrarily short-wavelength + high-energy fluctuations will collapse into a Planck-scale black hole). I don't think it's controversial to say that we can regard the Standard Model to some extent as an effective field theory for (the IR limit of) some underlying theory of quantum gravity. Such a theory (e.g. string theory) would be a "UV completion" of the Standard Model.
A: Here's the thing: renormalization and divergences have nothing to do with each other. They are conceptually unrelated notions.
Renormalization
Simply put, renormalization is a consequence of non-linearities. Any theory (other than those that are linear) requires renormalization. Even in classical Newtonian mechanics.
Renormalization means: when you change an interaction, the parameters of your theory change. The simplest example is the classical (an)harmonic oscillator. Say you begin with $L=\frac12m\dot q^2-\frac12kq^2$. If you prepare this system in a laboratory, you will observe that $q$ oscillates with frequency $\omega^2=k/m$. Now, say you add the (nonlinear) interaction $\gamma q^4$ to your Lagrangian. The frequency that you will measure in a laboratory is no longer $\omega^2=k/m$, but rather $\omega^2\sim k/m+\gamma$.
This trivial phenomenon also occurs in quantum mechanics, in particular QFT. You typically have a set of measured parameters, such as $\omega$ above, and a set of coefficients in your Lagrangian, such as $k,m$. The latter are not directly observable. Solving your theory gives you a function $\omega=f(k,m,\dots)$ for some $f$. You can use the measured value of $\omega$, and the calculable function $f$, to fix the value of your Lagrangian parameters $m,k,\dots$. If you change your Lagrangian by adding a new term, the function $f$ will change, and therefore the value of your parameters $m,k,\dots$ will change. Of course, $\omega$ stays the same, as this is something you measure in a laboratory (and does not care about which Lagrangian you are using to model the dynamics).
For example, say you have a QFT that describes a particle. The system will have several parameters $c_1,c_2,\dots$, which multiply the different terms in your Lagrangian, say $L=(\partial\phi)^2-c_1\phi^2-c_2\phi^4+\cdots$. This Lagrangian predicts that there is a particle with some mass $m=f(c_1,c_2,\dots)$, where $f$ is a function that you obtain by solving the theory (say, using Feynman diagrams). The value of $m$ is fixed by experiments, you can measure it in a lab. From this measured value, and the known form of the function $f$, you can fix the value of your model-dependent parameters $c_i$. For example, if you begin with the model $L=(\partial\phi)^2-c_1\phi^2$, then the function $f$ takes the form $m=c_1$, and therefore the value of $c_1$ is identical to what you measure $m$ to be. If you take, instead, $L=(\partial\phi)^2-c_1\phi^2-c_2\phi^4$, then you now have $m=c_1+\frac{3}{16\pi}c_2+\mathcal O(c_2^2)$. Using this (and some other measured observable, such as a cross section), you can fix the value of $c_1,c_2$. Note that $c_1$ will not in general take the same value as before, namely $c_1\neq m$. This is what we mean by "interactions renormalize your couplings". We just mean that, by adding interactions to your model, the numerical value of your coupling constants will change.
This is all true even if your theory is finite. Theories without divergences still require renormalization, i.e., the value of the coupling constants are to be determined by comparing to observable predictions, and changing interactions changes the numerical value of your coupling constants. (The exception is, of course, linear theories and some other special cases such as those involving integrability).
Renormalizing a theory typically means: fix the value of your model-dependent parameters $\{c_i\}$ as a certain function of the measurable parameters, such as $m$ and other observable properties of your system. The value of $m$ is fixed by nature, and we have no control over it. The value of $\{c_i\}$ depends on which specific Lagrangian we are using to model the physical phenomenon, and it changes if we change the model.
Divergences
In QFT divergences are the result of mishandling distributions. In $d>1$, quantum fields are not functions of spacetime, but rather distributions. As such, you cannot manipulate them as if they were regular functions, e.g. you cannot in general multiply distributions. When you do, you get divergences.
This is manifested in the fact that the function $f$ from before typically has divergent terms inside. The cute thing is: for a large class of theories, if you write $c_i=\tilde c_i+\delta_i$, with $\delta_i$ a (specifically constructed) divergent contribution, and $\tilde c_i$ a finite part, then the relation $m=f(c_i)=\tilde f(\tilde c_i)$ is rendered finite, i.e., the function $\tilde f$ no longer has divergent terms inside.
But note that renormalization did not actually cure the divergences. The divergences were eliminated by writing $c_i=\tilde c_i+\delta_i$ and carefully compensating the divergences with another divergent part, $\delta_i$. This is not what renormalization is. Renormalization is the statement that, if you were to change the model, the numerical value of the constants $c_i$ (and, by extension, that of $\tilde c_i$), change accordingly.
For a general theory, you need to perform both steps: 1) cancel the divergences by splitting your model-dependent parameters as $c_i=\tilde c_i+\delta_i$, with finite $\tilde c_i$ and a suitable divergent $\delta_i$, and 2) renormalize your theory, i.e., fix the value of your model-dependent parameters $\tilde c_i$ as a function of observable parameters such as $m$.
n-point functions
What measurable parameters can we use beyond $m$? in general there will be multiple parameters $c_i$, so you need as many observables in order to fix the former as a function of the latter. For a completely general theory, it is hard to come up with a concrete list of observable parameters. For specific systems this is easy, e.g. for a thermodynamic material we could use the susceptibility at criticality, while for QCD we could use the pion decay constant. Both of these are measurable in a laboratory, and they can both be predicted as a function of the parameters in the Lagrangian.
But what if we are dealing with a more general QFT, one for which we do not have a clear picture of what it is describing in real life? Which observables can we use then, if we don't even know what the theory is modeling in the first place? In this situation, it is convenient to use correlation functions at specific values of the external momenta as "observables". So for generic theories, instead of using a decay constant as a measureable parameter, we use $\Gamma^{(n)}(0)$, as if this were something we could measure. Often enough, it actually is, but this really depends on which QFT you are working with.
