Editing Renormalization (section)

== Renormalized and bare quantities ==
The solution was to realize that the quantities initially appearing in the theory's formulae (such as the formula for the [[Lagrangian (field theory)|Lagrangian]]), representing such things as the electron's [[electric charge]] and [[mass]], as well as the normalizations of the quantum fields themselves, did ''not'' actually correspond to the physical constants measured in the laboratory.  As written, they were ''bare'' quantities that did not take into account the contribution of virtual-particle loop effects to ''the physical constants themselves''.  Among other things, these effects would include the quantum counterpart of the electromagnetic back-reaction that so vexed classical theorists of electromagnetism.  In general, these effects would be just as divergent as the amplitudes under consideration in the first place; so finite measured quantities would, in general, imply divergent bare quantities.

To make contact with reality, then, the formulae would have to be rewritten in terms of measurable, ''renormalized'' quantities.  The charge of the electron, say, would be defined in terms of a quantity measured at a specific [[kinematics|kinematic]] ''renormalization point'' or ''subtraction point'' (which will generally have a characteristic energy, called the ''renormalization scale'' or simply the [[energy scale]]).  The parts of the Lagrangian left over, involving the remaining portions of the bare quantities, could then be reinterpreted as [[counterterm]]s, involved in divergent diagrams exactly ''canceling out'' the troublesome divergences for other diagrams.

=== Renormalization in QED ===
[[Image:Counterterm.png|thumb|upright=1.1|Figure 3. The vertex corresponding to the {{math|''Z''<sub>1</sub>}} counterterm cancels the divergence in Figure 2.]]

For example, in the [[quantum electrodynamics|Lagrangian of QED]]
<math display="block">\mathcal{L}=\bar\psi_B\left[i\gamma_\mu \left (\partial^\mu + ie_BA_B^\mu \right )-m_B\right]\psi_B -\frac{1}{4}F_{B\mu\nu}F_B^{\mu\nu}</math>
the fields and coupling constant are really ''bare'' quantities, hence the subscript {{mvar|B}} above. Conventionally the bare quantities are written so that the corresponding Lagrangian terms are multiples of the renormalized ones:
<math display="block">\left(\bar\psi m \psi\right)_B = Z_0 \bar\psi m \psi</math>
<math display="block">\left(\bar\psi\left(\partial^\mu + ieA^\mu \right )\psi\right)_B = Z_1 \bar\psi \left (\partial^\mu + ieA^\mu \right)\psi</math>
<math display="block">\left(F_{\mu\nu}F^{\mu\nu}\right)_B = Z_3\, F_{\mu\nu}F^{\mu\nu}.</math>

[[Gauge invariance]], via a [[Ward–Takahashi identity]], turns out to imply that we can renormalize the two terms of the [[covariant derivative]] piece
<math display="block">\bar \psi (\partial + ieA) \psi</math>
together (Pokorski 1987, p.&nbsp;115), which is what happened to {{math|''Z''<sub>2</sub>}}; it is the same as {{math|''Z''<sub>1</sub>}}.

A term in this Lagrangian, for example, the electron–photon interaction pictured in Figure 1, can then be written
<math display="block">\mathcal{L}_I = -e \bar\psi \gamma_\mu A^\mu \psi - (Z_1 - 1) e \bar\psi \gamma_\mu A^\mu \psi</math>

The physical constant {{mvar|e}}, the electron's charge, can then be defined in terms of some specific experiment:  we set the renormalization scale equal to the energy characteristic of this experiment, and the first term gives the interaction we see in the laboratory (up to small, finite corrections from loop diagrams, providing such exotica as the high-order corrections to the [[magnetic moment]]).  The rest is the counterterm.  If the theory is ''renormalizable'' (see below for more on this), as it is in QED, the ''divergent'' parts of loop diagrams can all be decomposed into pieces with three or fewer legs, with an algebraic form that can be canceled out by the second term (or by the similar counterterms that come from {{math|''Z''<sub>0</sub>}} and {{math|''Z''<sub>3</sub>}}).

The diagram with the {{math|''Z''<sub>1</sub>}} counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2.

Historically, the splitting of the "bare terms" into the original terms and counterterms came before the [[renormalization group]] insight due to [[Kenneth G. Wilson|Kenneth Wilson]].<ref name=Wilson1975>{{cite journal | last=Wilson | first=Kenneth G. |author-link=Kenneth G. Wilson| title=The renormalization group: Critical phenomena and the Kondo problem | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=47 | issue=4 | date=1975-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.47.773 | pages=773–840| bibcode=1975RvMP...47..773W }}</ref> According to such [[renormalization group]] insights, detailed in the next section, this splitting is unnatural and actually unphysical, as all scales of the problem enter in continuous systematic ways.

=== Running couplings ===
To minimize the contribution of loop diagrams to a given calculation (and therefore make it easier to extract results), one chooses a renormalization point close to the energies and momenta exchanged in the interaction. However, the renormalization point is not itself a physical quantity: the physical predictions of the theory, calculated to all orders, should in principle be ''independent'' of the choice of renormalization point, as long as it is within the domain of application of the theory. Changes in renormalization scale will simply affect how much of a result comes from Feynman diagrams without loops, and how much comes from the remaining finite parts of loop diagrams. One can exploit this fact to calculate the effective variation of [[Coupling constant|physical constants]] with changes in scale. This variation is encoded by [[Beta function (physics)|beta-function]]s, and the general theory of this kind of scale-dependence is known as the [[renormalization group]].

Colloquially, particle physicists often speak of certain physical "constants" as varying with the energy of interaction, though in fact, it is the renormalization scale that is the independent quantity.  This [[Coupling constant#Running coupling|''running'']] does, however, provide a convenient means of describing changes in the behavior of a field theory under changes in the energies involved in an interaction.  For example, since the coupling in [[quantum chromodynamics]] becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes large – a phenomenon  known as [[asymptotic freedom]].  Choosing an increasing energy scale and using the renormalization group makes this clear from simple Feynman diagrams; were this not done, the prediction would be the same, but would arise from complicated high-order cancellations.

For example,
<math display="block">I=\int_0^a \frac{1}{z}\,dz-\int_0^b \frac{1}{z}\,dz=\ln a-\ln b-\ln 0 +\ln 0</math>
is ill-defined.

To eliminate the divergence, simply change lower limit of integral into {{mvar|ε<sub>a</sub>}} and {{mvar|ε<sub>b</sub>}}:
<math display="block">I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b}</math>

Making sure {{math|{{sfrac|''ε<sub>b</sub>''|''ε<sub>a</sub>''}} → 1}}, then {{math|''I'' {{=}} ln {{sfrac|''a''|''b''}}.}}