List of Questions in Modules
Questions for EFT Intro
1) Introduction to EFT (Week 1)
Pre Text: We will begin with an introduction to the paradigm of Effective Field Theory.
Video 1.
Post Box Question: Write down a list of the effective field theories you know or have heard of. Determine which of those in your list do NOT have a power counting expansion that can be determined by the mass dimensions of operators. Put a * beside them.
Answer: Some examples of EFTs that do not have a power counting expansion in mass dimensions are: NonRelativistic QED, NonRelativistic QCD, Soft-Collinear Effective Theory, EFT of a Fermi Surface, and an EFT for Cold Atoms or Nonrelativistic Nucleons at Unitarity. Some common incorrect responses are: Heavy Quark Effective Theory, Chiral Perturbation Theory
2) Introduction Example with Hydrogen (Week 1)
Pre Text: In this component I discuss the idea of having an effective field theory with power counting that is not in strict ratios of mass scales. I will also introduce the concepts of neglecting massive particles, and of couplings as fundamental parameters that are modified by short distance physics. The plethora of expansions used when treating Hydrogen with nonrelativistic Quantum Mechanics are used to illustrate these points.
Video 2.
Post Question: none
3) Bottom-Up and Top-Down EFT (Week 1)
Pre Text: In this component we will discuss two distinct methods of employing effective field theory, either from the "top down" or from the "bottom up". In the top-down case we construct the EFT from another theory, colloquially known as the "full theory". In the bottom-up case we formulate the effective theory on its own, based on symmetries, the relevant degrees of freedom, and establishing a power counting.
Video 3.
Post Box Question (distinct vertical): Classify each of the following theories as top-down versus bottom-up. To ensure definite answers here we will define a theory as top-down only if its construction can be carried out perturbatively. Show them a list: { Heavy Quark Effective Theory(T), The Standard Model (B), Chiral Perturbation Theory(B), Nonrelativistic QED(T), Soft-Collinear Effective Theory(T), Hard Thermal Loop Effective Theory(T), An EFT for Heavy Dark Matter(B), Nonrelativistic limit of General Relativity(T*), Effective Field Theory for Inflation(B), An EFT for Cold Atoms(B) } Include bullet selectors where they can choose one of 3 choices for each example: top-down, bottom-up, or I don't know. The answers are provided above T=top-down, B=bottom up and should be provided after they've attempted the question.
2nd Post Box Question: This question is more difficult. What is an example of an effective theory which expands in the inverse mass of a heavy particle, but is a bottom-up theory rather than a top-down theory?
Answer: Possible answers include: Heavy Baryon Chiral Perturbation theory for the proton and neutron interacting with pions. Or the description of the static proton in an EFT for the Hydrogen Atom.
Post Discussion Question: none
Chap.#1
4)Standard Model as an EFT (Week 1)
Pre Text: In this component I will discuss the standard model of particle and nuclear physics, and why the Lagrangian for the strong, weak, and electromagnetic interactions, as well as couplings with the Higgs boson should be thought of as an Effective Field Theory.
Video 4.
Post Box Question (same vertical): How does an EFT with an infinite number of operators retain predictive power? Please respond with a couple of sentences.
Answer: To make predictions at a given level of precision one only needs to work to a certain order in the power counting which can be predetermined. The EFT retains predictive power because it only has a finite number of terms when truncated at this order.
Post Discussion Question (distinct vertical): The concept that an effective field theory only needs to be renormalized order by order in an expansion parameter means that we can consider quantum gravity as an EFT. A Lagrangian theory that has graviton loops can be renormalized order by order in an expansion in energy over $M_{\rm plank}$. In this way we can derive predictions from quantum gravity using quantum field theory, without having a full theory of quantum gravity like string theory. You are encouraged to use the Discussion Board to talk further about the implications of this statement. How should we view renormalizability as a motivation for a quantum theory like the Standard Model, or for string theory in this context? Why would we want a "traditionally renormalizable" quantum field theory?
5) Power Counting in Operator Dimensions (Week 1)
Pre Text: In this lecture component I will prove a theorem relating power counting expansions in ratios of the masses of particles to the power counting associated with the dimension of quantum field theory operators.
Video 5.
Post Box Question (same vertical):
The proof that we just discussed, showing that power counting is equivalent to determining the dimension of operators, seems very general. Can you identify where we made an assumption that may or may not be valid in other situations? What is the range of validity of the theorem we proved?
Answer: The Answer can be found in the next section, here.
6) Short Video with Answer to the above Question (Week 1)
Pre Text: In this short component I answer the question: What were the assumptions we made in proving that power counting is equivalent to determining operator dimensions?
Video 6
Post Video Additional Answer(same vertical): Thus the proof we gave assumes that all spacetime components of our field scale in the same way. This assumption means that we assumed that our Effective Theory fields are "soft", with all spacetime components of order the smaller mass scale. The modes that are not in our EFT were "hard", with spacetime components scaling as the larger mass scale. EFTs that satisfy this requirement are sometimes called "Classic Effective Theories". Obviously the homogeneity of the spacetime components is related to the Lorentz symmetry of the EFT. If we instead have a non-relativistic EFT, then we will need to assume that the components for time and space scale differently. This leads to velocity power counting for nonrelativistic effective theories. We can also violate the assumptions in other ways. For example, our EFT could have more than one type of low energy mode, and each one could have different scalings for their spacetime coordinates. We will see examples of this later in the course.
7) Dimension 5 & 6 Operators in the SM (Week 1)
Pre Text: In this lecture component we will discuss higher dimensional operators that exist once we consider the Standard Model as an Effective Field Theory.
Video 7.
Post Box Question (distinct vertical): Why should the higher dimension operators added to lowest order "Standard Model" Lagrangian be invariant under the gauge symmetry SU(3) x SU(2) x U(1) of the Standard Model?
Box Answer: We are assuming that the Standard Model gauge symmetry is an exact symmetry of nature, without additional explicit breaking by small corrections. In the case of QCD and QED this has been tested to an extreme level of precision. In the case of Electroweak Symmetry which is spontaneously broken by the vacuum expectation value from the Higgs mechanism it has been tested to better than 1%. We also make an assumption when we write down only Lorentz Invariant operators, that Lorentz Symmetry is exact and not broken by small higher order terms. If we want to explore the breaking of either of these two types of symmetries then developing higher dimension operators that break the symmetries is the way to do it.
8)Field Redefinitions (Week 2)
Pre Text: An important question when developing any theoretical formalism is what are the relevant mathematical objects that we are manipulating. In an Effective Field Theory it is the fields which are the key variables. But which fields? In fact there is a redundancy in what field variables we use, and different choices can lead to exactly the same theory. This redundancy is common to all Effective Theories, and goes by the name of making "Field Redefinitions" to take us between equivalent formulations of an EFT. This will be the topic of our two lecture components today.
Video 8.
Post Box Question (distinct vertical): Did the assumption that our field redefinition preserves the single particle states enter into our proof so far (where we analyzed the impact of changing the Lagrangian)?
Answer: It did enter indirectly in our analysis of the change of variable in the Lagrangian. The Lagrangian with the new field has the same kinetic term as the original Lagrangian because the field redefinition is linear in the new field as that field goes to zero, which preserves the single particle states.
9)Field Redefinitions continued
Pre Text: We are half way through proving a theorem about our ability to find equivalent physical formulations of an EFT by making certain types of Field Redefinitions. We continue here.
Video 9
Post Video Comment: Thus we have learned that field redefinitions can be used to simplify the choice of field variables used to describe our theory. The standard thing to do is simplify the EFT by removing higher order Lagrangian terms that are proportional to the lowest order equations of motion, and hence we have phrased our theorem using this language.
Post Box Question (distinct vertical): Lets ask the question again. Where in the second part of our proof did it enter that single particle states were preserved by the field redefinition?
Answer: Both in the decoupling of the ghosts, where it was important in order that the ghosts got a mass of order Lambda_new, and in the use of LSZ formulation where we showed that the two theories have the same S-matrix elements. If the new and old fields are linearly related as the fields go to zero, then they will have the same single particle states. The constant of proportionality of this linear relation need not be unity, and in our use of the LSZ formulation we saw explicitly how the constant of proportionality drops out of S-matrix elements.
9.5) Dimension 6 Operators in SM
Pre Text: Having derived the theorem about field redefinitions, in this component we will see it in action through an application to determining the basis of dimension 6 (baryon number conserving) operators that are present when the standard model is treated as an EFT. This component is presented with slides.
Video 9.5
Post Video Comment:
Post Box Question: none
10) Loops, Renormalization, and Matching (Week 3, Section II starts here)
Pre Text:
We now start section II of our course material, where we will discuss loop graphs, and demonstrate how to carry out matching calculations.
Over the course of the next few video components I will discuss the implications of using different regulators and choosing different renormalization schemes for the formulation of objects in an Effective Field Theory. I will discuss Wilsonian EFT (with a hard cutoff) and Continuum EFT (with regulators like dimensional regularization, which only softly spoil power counting). Wilsonian and Continuum EFT are two aspects of the same thing (good ol' EFT). This has the implication that we can often be a bit cavalier and mix language that would otherwise be most naturally associated to one or the other formulation, like "removing high energy fluctuations" and "matching". This will be greatly beneficial since Wilsonian EFT is much easier to think about, and Continuum EFT is much easier to use for explicit calculations and to avoid breaking symmetries.
Video 10.
Post Box Question (distinct vertical): Are Wilsonian Renormalization Group Equations (RGE) always equivalent to the Continuum Renormalization Group Equations of Gell-Mann and Low?
Answer: No in general for power law RGE behavior which is captured by the Wilsonian approach, but can be missed if we formulate our Continuum EFT using MSbar and dimensional regularization (Dim.Reg.). But the answer is yes for the dominant logarithmic behavior. If we know what theory we are considering then it is often sufficient to treat only the logarithmic corrections, but power law corrections can be relevant for flows where a theory moves from one power counting regime into another. With a soft regulator like Dim.Reg. we can also formulate a renormalization group for power law terms, by noting that power divergences still show up as poles in other dimensions. An example we will talk about later on is the power divergence subtraction scheme. We will use this scheme to show an explicit example of a flow between regimes with two different power countings.
2nd Post Box Question: If we are sure about the power counting needed to define the EFT, are the Wilsonian and Continuum EFTs equivalent?
Answer: Actually, although they are closer, they are still not fully equivalent. In particular there is another type of power law behavior in quantum field theory that can show up, which leads to poor convergence of the perturbative series in the coupling constant. This poor convergence is particularly acute in the $\overline{\rm MS}$ scheme for higher order corrections in QCD, and is known as infrared renormalons. Later in the course I will discuss a method to properly treat these power divergences within Operator Product expansions that use Dim.Reg. For that discussion we will exploit our freedom to use a scheme other than $\overline{\rm MS}$, leading to RGE equations with power law behavior that perturbs the Continuum $\overline{\rm MS}$ RGE towards the Wilsonian cutoff RGE, while maintaining manifest gauge invariance.
Post Video Discussion(same vertical as above question): In the discussion forum you can talk further about ways in which Wilsonian and Continuum EFTs are equivalent, and ways in which they are different.
11)Dimensional Regularization in Effective Theories (Week 3) (end of L3 at end)
Pre Text: In this component I discuss the axioms which lead to Dimensional Regularization, and some of the subtleties encountered in using Dimensional Regularization. I will often refer to Dimensional Regularization as simply "Dim Reg".
Video 11.
Post Box Question: none here
Link to Collin's book Ch.4. Link to handout page on loop integrations.
12) Matching: Massive Particle Thresholds (L4 start) (decoupling in MSbar)
Pre Text: In this lecture component we continue our discussion of the good and bad features of Dim. Reg. We also discuss the decoupling theorem in Quantum Field Theory, and the way in which matching calculations are used to properly treat mass thresholds in MSbar.
Video 12.
Post Box Question (distinct vertical): How do we deal with multiple mass scales in an EFT? Consider the wide range of masses for standard model quarks and leptons, which are all much lighter than the W, Z, and H (Higgs) Bosons, and the top quark. Lets consider the common mass scale for the heavy particles to be $\sim M$. If we formulate the effective field theory that removes the light fermions, do we really have a complicated EFT with multiple power counting parameters: $m_\tau / M$, $m_b/M$, $m_c/M$, $m_e/M$, ... ?
Answer: In this situation we can treat the mass scales one at a time. The heaviest of the light fermions is the b-quark, so to describe physics at this scale we can count $p \sim m_b$, and hence do an expansion for all fermions in $m_b / M$. The fermions with masses $\ll m_b$ can even be treated as massless in this theory. You will get more practice with this in the problem set problems.
13) Integrating out Massive Standard Model Particles (b to cud, basis, same IR needed)
Pre Text: In this lecture component we will consider the EFT that arises from integrating out heavy standard model particles, namely the W, Z, Higgs, and top-quark. We will also consider the advantage we obtain by performing matching calculations for operators to describe the short distance physics with Wilson coefficients.
Video 13.
Post Box Question (distinct vertical): Here is a somewhat challenging question. Do you think there are constraints on the possible IR regulators one can use for matching?
Answer: Most often any reasonable IR regulator will do. The answer for matching calculations will then be independent of the choice made for this IR regulator as long as the same choice is made in the full and effective theories. In general the answer is yes, there are constraints on the IR regulator. A simple one is that we must be able to formulate it as a valid IR regulator in the effective theories. The treatment of the IR regulator could spoil power counting in the EFT, for example if we regulated with a mass scale that was hierarchically larger than the small mass scale associated to power counting in our EFT. Another example is that the choice of our IR regulator could introduces spurious scales in the EFT that are not present in the physical system that the EFT is describing, which again has an impact on the power counting. For the simplest EFTs, these extra complications usually do not arise.
14) Counterterms for Operators versus Counterterms for Coefficients
Pre Text: In this lecture component we setup the general structure of operator or coefficient renormalization in an EFT, and derive a relation between the renormalization constants and renormalization group equations for coefficients and operators.
Video 14.
Post Box Question: Why is the product of bare coefficient and bare operator equal to the product of the renormalized coefficient and renormalized operator?
Answer: We learned that there are two equivalent ways to carry out the renormalization, either by renormalizing operators or with counterterms in the coefficients. They are related, and we found that the change of variables that takes us from bare operators to renormalized operators is the inverse of the change of variables that takes us from bare coefficients to renormalized coefficients. Thus if we start with the bare coefficient times a bare operator, and change one of them to their renormalized counterpart, the transform matrix can then be used to convert the other one too:
$C^{\rm bare} O^{\rm bare} = C^{\rm ren} Z_c Z_{\rm wfn} O^{\rm bare} = C^{\rm ren} Z_O^{-1} Z_{\rm wfn} O^{\rm bare} = C^{\rm ren} O^{\rm ren}$
Chap.#3
Chap.#4
21)Chiral Lagrangians (L6: linear sigma model, field redefinitions and nonlinear representations)
Pre Text: Here we begin our discussion of Chiral Perturbation Theory, which is a very important example of a bottom-up effective field theory. As our main example for this material we will consider the classic Chiral Lagrangian for the light pions, kaons, and eta in QCD. I am going to assume that you have already learned about spontaneous symmetry breaking in a quantum field theory course. Some of the concepts that we will cover over the course of the next few lectures are linear and nonlinear representations for the fields describing the broken symmetry, chiral power counting, and the nature of renormalization in a momentum suppressed loop expansion. We begin our discussion with the first concept in the context of the linear and non-linear sigma models.
Video
Post Box Question:
22)CCWZ parameterization (L6)
Pre Text: Chiral Lagrangians provide a wonderful description of the dynamics of Goldstone bosons, but given that these bosons need not be fundamental scalar fields in some Lagrangian, it is reasonable to ask how do we pick the fields for these bosons? In this component we discuss the classic Callan-Coleman-Wess-Zumino procedure for introducing fields for the Goldstone bosons corresponding to broken generators for the coset space. Different choices for the Goldstone fields in our effective theory correspond to different coordinates for the coset.
Video
Post Box Discussion: For further discussion of the CCWZ parameterization see the review by Aneesh Manohar or the original CCWZ paper.
23)ChPT for QCD (L6, symmetry breaking, spurion analysis, start on loops)
ChPT = Chiral Perturbation Theory
Pre Text: We now consider chiral perturbation theory for pions, corresponding to the spontaneously broken chiral symmetry $SU(2)_L \times SU(2)_R \to SU(2)_V$ in QCD. The concept of a spurion analysis is discussed in order to include the explicit breaking from the light up and down quark masses, and to analyze the left handed current. We then discuss the Chiral Lagrangian Feynman rules and power counting for loop corrections.
Video
Post Box Question: Summarize in one or two sentences what the purpose of a spurion analysis is.
Post Box Answer: A spurion analysis provides a simple means of constructing operators in an effective Lagrangian that transform in a particular way under a symmetry group, by constructing symmetry invariants with the spurion field, rather than directly determining the objects that transform.
24) Loops in ChPT (f vs. Lam_chi) (L7 starts)
Pre Text: In this component we continue our discussion of loops in chiral perturbation theory (ChPT). We show how the use of dimensional regularization modifies the dimensions of ChPT parameters and introduces dependence on the scale $\mu$. Ultraviolet divergences are cancelled by counterterms for higher order terms in the chiral Lagrangian, and the $\mu$ dependence of loops cancels with the $\mu$ dependence in the renormalized coefficients of this higher order Lagrangian.
Video
Post Box Question: none
25) Naive Dimensional Analysis (L7)
Pre Text: Here we discuss how the cancellation of $\mu$ dependence between loops and coefficients of higher order operators in chiral perturbation theory leads to a natural expectation for the size of these coefficients. This allows us to account for factors of $(4\pi)$ when determining the natural size of the coefficients. Counting factors of $(4\pi)$ in this manner is known as Naive Dimensional Analysis.
Video
Post Box Question: In chiral perturbation theory, does it make sense to vary $\mu$ up and down by a factor of two like we did in our discussion of top-down EFTs that involved the gauge theory coupling $\alpha_s(\mu)$? Explain your answer.
Post Box Answer: In the gauge theory case we had two expansions, the power counting expansion and a coupling expansion. We varied $\mu$ up and down in order to have a means of getting an estimate about the size of terms that are higher order in the coupling expansion. In chiral perturbation theory without gauge bosons we only have the power counting expansion, and the $\mu$ dependence cancels out explicitly at each order in the momentum expansion. Since varying $\mu$ does not give information about higher order terms in the power expansion it is fine to pick a $\mu$ and stick with it. Varying $\mu$ simply changes the meaning of the parameters in our Chiral Lagrangian rather than giving information about terms we've left out. One context in which it would make sense to vary $\mu$ in ChPT is if we had calculated the chiral loop corrections and wanted to obtain an estimate for the numerical impact of the terms from the higher order Lagrangian.
26) ChPT UV and IR structure, SU(2) Phenomenology (L7)
Pre Text: Here we discuss what chiral perturbation theory would look like with a hard cutoff, and its behavior in the infrared
``$p\to 0$'' limit. We also discuss the classic example of making parameter free predictions for the $\pi\pi$ scattering phase shifts.
Video
Post Box Question: none
27) ChPT power counting theorem (L7)
Pre Text: In this component we derive an all orders power counting formula for chiral perturbation theory. This result allows us to determine the order of any diagram with contributions from operators of any structure, and hence allows us to enumerate which terms are relevant to the order we are working.
Video
Post Box Question (distinct vertical): To test your understanding of the power counting formula, lets apply it to an example. Consider the ChPT graph constructed by taking three terms from the $p^4$ Lagrangian, each of which is a 4-point pion vertex, and contracting them to yield a one-loop graph with 6 external pions. To what power of $p$ does this graph contribute? Carry out the calculation two ways, by using our power counting formula, and also by simply counting powers of $p$ for the vertices, propagators, and loop integration in this diagram, and check that you obtain the same result.
Post Box Answer: To use the power counting formula we note that $N_L=1$ and the only non-zero vertex factor is $N_4=3$. This gives $D= 2 + 2 N_L + \sum_n N_n (n-2) = 2 + 2 (1) + 3 (4-2) = 10$, so the graph contributes at $p^{10}$. If we instead count the factors of $p$ directly in the expression for the loop integral then from the vertices we have $(p^4)^3 = p^{12}$, from the loop integration we have another $d^4k \sim p^4$, while from the three pion propagators we get a $(p^{-2})^3= p^{-6}$. All together this is $p^{12+4-6} = p^{10}$ just as before.
28) ChPT for SU(3) and Decay Constants (L7)
Pre Text: In this component we extend our discussion of ChPT to the SU(3) case, including the Kaons and Eta as pseudo-Goldstone bosons. We also briefly discuss the decay constant example that you will treat in detail as a homework problem.
Video
Post Box Question (distinct vertical): Since the kaon and eta mesons are heavier than the pion we know that SU(3) ChPT has a larger expansion parameter than SU(2) ChPT. Considering just the meson masses what estimate would you give for the expansion parameter in each of these theories? What are some of the advantages/disadvantages to using SU(2) versus SU(3) ChPT?
Post Box Answer: For SU(2) we might estimate $m_\pi^2/\Lambda_\chi^2 = m_\pi^2 / (4\pi f)^2 \sim 1/(4\pi)^2 = 0.006$ or the more realistic $m_\pi^2 / (1\, {\rm GeV})^2 = 0.02$. (Note that the first estimate depends on our convention $f=0.13\,{\rm GeV}$, and would be closer to the second estimate if we used the other common convention where $f=0.093\,{\rm GeV}$.) For SU(3), using the larger $m_\eta = 0.55\, {\rm GeV}$, the same estimates would give $m_\eta^2/(4\pi f)^2 = 0.1$ (or $0.2$ with the smaller value for $f$) or the more realistic $m_\eta^2/(1\, {\rm GeV})^2 = 0.3$. It could also be appropriate to replace the denominator of these estimates by the mass of a heavier meson that is not in the low energy theory, such as the $\rho$ where $m_\rho = 0.77\, {\rm GeV}$.
An advantage of using SU(3) ChPT is that we are treating more particles as Goldstone bosons, and hence can make direct predictions for more processes, including for example those that only involve Kaons. The cost of using SU(3) is that we have to deal with a larger expansion parameter, so the convergence will be slower. This slow convergence even enters into pion observables, since for example there will be contributions from Kaon loops at some order. When we use SU(2) ChPT the expansion converges much more quickly, but we only make predictions based on the chiral dynamics for the pions. In this case the Kaon is treated as a heavy particle which we integrated out. This is seen explicitly in the relationship we wrote down between the SU(2) and SU(3) $L_i$ couplings, where the Kaon mass appears in a logarithm.
Insert Pset #2, part #2 (fPi and fK problem)
Chap.#6
15)Renormalization Group Equations
Pre Text: In this lecture component we discuss the renormalization group equations derived by composite operator renormalization in the EFT. We show that the anomalous dimensions of coefficients and operators are related in the EFT. This relation embodies the equivalence of either i) taking the operators and coefficients at a low scale and running the coefficients up to a high scale, or ii) taking the operators and coefficients at a high scale and running the operators down to a low scale.
Video 15.
Post Box Question: none
16) General Solution & Higher Orders
Pre Text: In this component we discuss the general structure of solutions to the Renormalization Group equations, and the form of the logarithmic series that are summed up by solving these equations.
Video 16.
Post Box Question: Often rather than taking $\mu_W = m_W$ people vary the scale $\mu_W$ about $m_W$ to estimate uncertainties. Why is this a reasonable thing to do?
Answer: The $\mu_W$ dependence cancels between the matching coefficient evaluated at $\mu_W$, and the evolution kernel $U(\mu,\mu_W)$ which runs from $\mu_W$ down to $\mu$. This cancellation occurs to the order one is working, but not beyond, so there will be left over $\mu_W$ dependence that is formally higher order in the resummed perturbative expansion. If you work to LL order then the left over $\mu_W$ dependence will be at NLL order, etc. Thus $\mu_W variation gives an estimate for the size of higher order terms in the perturbation theory. A standard choice people use is to vary between $\mu_W=m_W/2$ and $\mu_W = 2 m_W$ to get a theory uncertainty band. The size of this uncertainty band will decrease as one goes to higher orders: LL $\to$ NLL $\to$ NNLL.
17) Physical Application with Large Logs
Pre Text: In this component we use our log summation result for the weak quark level transition $b \to c \bar u d$ to carry out a physical application of in the transition between hadronic bound states $B\to D\pi$. The existence of large logs is discussed in physical matrix elements for this top-down application of effective field theory with the Weak Hamiltonian.
Video 17.
Post Box Question: none (note that this is a short video)
18)Comparison of EFT with Full Theory
Pre Text: In the next two components we will compare one-loop results in the full theory and effective theory in order to understand the physics of a matching computation in detail. Topics we will discuss here include why the UV divergences differ in the two theories, the agreement of IR divergences, and the physics behind scales in various logarithms.
Video 18.
Post Box Question (distinct vertical): Imagine that you have carried out a one-loop computation in a full theory (like the standard model), and by examining the scales in your result you are able to determine the logs that will show up in a one-loop matching calculation. Would this provide you with enough information to derive the RGE in the EFT for the 4-fermion operators we discussed? How about in a general EFT computation?
Answer: In general there is not enough information in a full theory calculation to determine anomalous dimensions. In particular, if there is operator mixing then your counterterm for a coefficient $C$ may be proportional to another coefficient $C^\prime$, and in the full theory calculation you will not see $C^\prime$, since it will be set to its tree level value. To properly determine the RGE, you must find an equation for C^\prime(mu) too, since it will appear on the RHS of the differential equation: $\mu d/d\mu C(\mu) = \ldots $. In the particular case of the four-quark operators we considered, the situation is simple enough that one could construct the form of the anomalous dimension matrix. But in more general situations the operator associated with $C^\prime(\mu)$ might be quite different from the operator associated with $C(\mu)$, and hence may not be evident directly from the full theory calculation.
Insert Pset #2, part #1 (operator renormalization for Electroweak Hamiltonian)
19) Massive particles: One Loop Matching & Next-to-Leading Log Summation (logs, scheme dependence)
Pre Text: Continuing with our discussion of matching, we now consider the non-logarithmic "constants" in the full and effective theory. The results for these constants are scheme dependent, they depend on whether we use the $\overline{\rm MS}$ scheme, or some other scheme to define our renormalized coefficients and operators. Nevertheless the results for physical observables are scheme independent, so there are cancellations of the scheme dependence that occur between the one-loop matching coefficients, the next-to-leading order anomalous dimensions, and the low energy matrix elements. In a simplified setup we will discuss exactly how these cancellations occur in combination of various scheme dependent quantities.
Video
Post Box Question (distinct vertical): In the electroweak Hamiltonian the mixing of the neutral $B$ mesons, so-called $B^0$--$\bar B^0$ mixing, is generated by 4-quark operators $(\bar d b)(\bar d b)$. These operators are generated by integrating out $W$ bosons from box diagrams in the standard model. Imagine that you have determined the Wilson coefficients of these operators in the $\overline {\rm MS}$ scheme at next-to-leading order, by a suitable matching calculation. Your friend works on lattice QCD, where ultraviolet divergences are regulated with the lattice spacing cutoff $a$, and has carried out a nonperturbative calculation of the matrix element $\langle B^0 | (\bar d b)(\bar d b) | B^0 \rangle$, with results for a few different $a$ values, and extrapolated to obtain a continuum $a\to 0$ result. Together you should be able to accurately predict the rate of $B^0$--$\bar B^0$ oscillations in the standard model. What problem(s) could arise if you and your friend simply multiplied your results?
Answer: In general the results for the Wilson coefficient and the matrix element of the operator will be in two different schemes. For the analysis to make sense either the Wilson coefficients must be converted to the lattice scheme, or the matrix elements must be converted to the $\overline{\rm MS}$ scheme. Only results in the same scheme can be multiplied to achieve a meaningful result, where the scheme dependence properly cancels between the coefficient and the operator. This is true whether or not large logs are summed with renormalization group methods. If large logs are summed, there are two scheme dependence cancellations, one at the high scale and one at the low scale. If large logs are not summed then there is a single cancellation at the scale one is working. A simple way to see that multiplying the original results given in two different schemes does not make much sense, is that the coefficient will have $\ln(\mu)$ dependence at ${\cal O}(\alpha_s)$, whereas the matrix element has none. Converting the matrix element to the $\overline{\rm MS}$ scheme will cause it to have $\mu$ dependence which then properly cancels with the $\mu$ dependence in the coefficient.
20)Phenomenology with the Electroweak Hamiltonian: $b \to s \gamma$ (L6 starts)
Pre Text: In this component we will study an important flavor changing process that involves a transition between quarks of equal electromagnetic charge, and hence which occurs only through loops in the Standard Model, namely $b\to s\gamma$. The agreement of measurements of this process with Standard Model calculations provides important constraints on new physics models that attempt to go beyond the Standard Model. Furthermore, perturbative corrections from matching and running for the Electroweak Hamiltonian play an important role in this analysis, so it provides a nice example of the formalism we have been discussing.
Video
Post Box Question: What are some other examples of neutral current processes that connect fermions from amongst the three different families present in the Standard Model?
Answer: Another example is given by the leptonic analog, $\mu \to e\gamma$. This process is lepton number violating, and would require an oscillation between muon and electron neutrinos inside a one-loop diagram. Hence it is highly suppressed in the Standard Model. Other examples with quarks include $b\to d\gamma$ and $t\to c\gamma$. The rates for both of these are smaller than for $b\to s\gamma$ in the Standard Model. Other neutral current processes replace the photon by a lepton pair, such as $b\to s \mu^+\mu^-$.
35)HQET Power Corrections Directly using QCD (L9)
Pre Text: Having studied perturbative corrections to HQET, in this component we now turn our attention to power corrections. We will start with a top-down approach by using the relations between fields in QCD and HQET to perform the tree level matching. This will allow us to construct the HQET Lagrangian for the first power corrections which appear at ${\cal O}(1/m_Q)$.
Video
Post Box Question (distinct vertical): The path integral for integrating out $B_v$ is quadratic, so it can be solved explicitly. Nevertheless, our analysis was only valid at tree-level. Which parts of the calculation would be made more complicated by including loop corrections?
Post Box Answer: If we include loop corrections then one addition would be including the fermionic determinant in the path integral, to account for closed fermion loops with antiquarks. In addition, when we expand the denominator obtained from integrating out $B_v$ we assume $iv\cdot D \ll 2 m_Q$, but this expansion is not valid for all loop corrections. In full QCD some loops can have momentum $iv\cdot \partial\sim m_Q$ which invalidate this expansion. For these loops we must integrate first, and then carry out the expansion $iv\cdot \partial m_Q$ only for external momenta. If gauge fields carry hard momentum then they will also invalidate the expansion by having $v\cdot A \sim m_Q$.
36)HQET Power Corrections from Symmetry & Reparameterization Invariance (L9)
Pre Text: In this component we consider HQET power corrections from the bottom-up. To construct higher order operators in the $1/m_Q$ expansion we simply need to find operators of higher dimensions, and impose symmetries of the low energy theory. These operators will break the heavy quark spin-flavor symmetry, but we retain gauge symmetry, discrete symmetries, and a version of Lorentz invariance. The heavy quark velocity four vector provides a preferred frame, so part of the Lorentz symmetry is realized as a reparameterization invariance (RPI) of the theory to variations in the choice of this four vector. This RPI acts between different orders in the $1/m_Q$ expansion, and therefore can relate the Wilson coefficients of operators that appear at different orders in the power expansion to all orders in the strong coupling. In this component we will discuss in some detail the transformations needed to implement RPI.
Video
Post Box Comment: The RPI is a symmetry of HQET. As with all symmetries we must think carefully about what happens once loops are included, since the process of regulating and renormalizing the theory may break symmetries. A regulator like dimensional regularization with the $\overline{\rm MS}$ preserves the RPI symmetry of HQET. On the other hand, the use of a Wilsonian cutoff regulator, which are widely used in heavy quark physics, will in general break the RPI symmetry. Although more Wilson coefficients receive loop corrections in such schemes, they can still be used, and the consequences of Lorentz symmetry will still be encoded in relations between observables where cutoff dependence cancels out.
37) Simple Phenomenology with HQET (Hadron Masses) (L10)
Pre Text: As a simple application of the operators obtained from the HQET power expansion, in this component we derive non-trivial results for the $1/m_Q$ expansion of the hadron masses of B and D mesons.
Video
Post Box Question: none
38) Operator Product Expansion for HQET (L10)
Pre Text: A powerful application of the HQET power expansion occurs for inclusive semileptonic decays of heavy hadrons, where for example a $B$ meson decays to any set of hadrons involving one charm quark ($D$, $D\pi\pi$, $D^*\pi$, $\ldots$) plus a lepton and a neutrino. Here an operator product expansion (OPE) allows the inclusive decay rate for these hadrons to be systematically related to the decay rate for the underlying quarks, including both perturbative corrections from the strong coupling and power corrections. In this component a brief discussion of the key aspects of this OPE are given.
Video
Post Box Question (distinct vertical): From the point of view of making predictions for a decay rate, what is the key difference between an exclusive semileptonic $B$ decay to a particular final state, and an inclusive semileptonic $B$ decay where we sum over possible final states?
Post Box Answer: For exclusive decays a non-trivial hadronic form factor can enter. Since this form factor involves physics of gluons with momenta $k^\mu \sim \Lambda_{\rm QCD}$ it can not be computed in perturbation theory. For inclusive decays the use of completeness of the sum over hadronic states allows us to related the hadronic and parton level decay rates at lowest order in the $1/m_Q$ expansion. Even though we can not perturbatively calculate the exclusive form factors, we can still impose the symmetries of HQET to reduce them to a minimal basis and derive properties that follow from symmetry considerations. For the $B\to D \ell\bar\nu_\ell$ and $B\to D^*\ell\bar\nu_\ell$ decays the symmetry is strong enough such that there is only a single form factor, known as the Isgur-Wise function, which is also normalized at the point where there is zero recoil between the $B$ and $D^{(*)}$ systems.
Insert Pset #3, part #2 (Reparameterization Invariance Problem)
Chap.#7
Chap.#8
29)HQET (static sources) (L7)
Pre Text: What happens if we have a heavy particle whose dynamics we wish to explore at energies much less than that particles mass? The relevant energy scales are below the particles mass, but obviously we can not integrate it out. Instead the heavy particle becomes a static source for whatever quantum numbers it carries. A classic example of this is the physics of a heavy quark in QCD, where the relevant effective theory is Heavy Quark Effective theory (HQET). In this component and the following components we will discuss HQET in detail. Included in this chapter will be a discussion of symmetries and perturbative corrections. In the next chapter we will discuss HQET corrections that are higher order in the power counting, so-called power corrections, that allow our static source to wiggle. The leading Lagrangian we will construct here is very general, and can be applied whether the static source is a heavy quark or some other static object.
Video
Post Box Question (distinct vertical): At lowest order a static heavy quark simply sits at rest, acting as a color source, and describing low energy interactions of the heavy quark with other particles. Can we apply HQET for a heavy quark moving at 9/10 of the speed of light? If so, what dynamics would HQET describe in this case?
Post Box Answer: To describe a heavy quark of mass $m_Q$ moving at speed $v_0$ along the z-axis we can simply use a boosted four vector $v^\mu = \Big(\frac{1}{\sqrt{1-v_0^2}}, 0,0, \frac{v_0}{\sqrt{1-v_0^2}}\Big)$, which still has $v^2=1$. HQET always describes the interaction of the heavy quark with soft modes of four-momentum $p$ where $p^2 \ll m_Q^2$. In the rest frame of the heavy quark, these soft particles have small energy $p^0 \ll m_Q$. When we boost to the moving frame their energy grows, and can even be larger than $m_Q$ as $v_0$ approaches $1$. Nevertheless, these boosted soft particles still have $p^2 \ll m_Q^2$ and the dynamics of how they couple to the heavy quark is still described by HQET. Physically the cloud of soft particles interacting with the heavy quark in its rest frame has the same dynamics as the cloud of comoving particles interacting with the heavy quark in the boosted frame.
30)HQET from QCD (L8)
Pre Text: In this component we derive the HQET Lagrangian directly from the QCD Lagrangian by using a tree level relation between the fields in the two theories. The key idea here is that we must shift our origin for the low energy expansion. We are expanding about the heavy particle, so low energy for the heavy quark means energy that is very close to its mass.
Video
Post Box Question: What is the interpretation of the $2 m_Q$ term in the $\bar B_v (iv\cdot D + 2 m_Q) B_v$ piece of the Lagrangian?
Post Box Answer: By pulling out the phase $\exp(-i m_Q v\cdot x)$ we have shifted our zero point for energy so that it is centered on the heavy quark. After the shift a heavy quark at rest has $E=0$ rather than $E=m_Q$. The $B_v$ field describes antiquark, and the energy gap between the quark and the anti-quark is $2 m_Q$, which explains why the explicit $2 m_Q$ shows up in quadratic $B_v$ piece of the Lagrangian.
31)HQET Properties (HQS, label velocity, p.c.) (L8)
Pre Text: In this component we discuss the physics encoded by the HQET Lagrangian, including the heavy quark flavor and spin symmetries. We also discuss the conservation of the heavy quark's velocity, and the power counting in the heavy quark mass which motivates using a different normalization convention for hadronic states containing a heavy quark.
Video
Post Box Question:
32)Covariant Fields for Symmetries (L8)
Pre Text: The heavy quark spin-flavor symmetry yields interesting information about the dynamics of processes involving heavy quarks. Since the gluon couplings to the heavy quark in HQET are nonperturbative, and lead to heavy meson and heavy baryon bound states, we need a formalism to derive the implications of the symmetry without using perturbation theory. In this component we construct covariant fields which encode the symmetry properties for the bound states and show how they are used to derive symmetry predictions for matrix elements.
Video
Post Box Discussion (distinct vertical): In addition to encoding symmetry properties for HQET matrix elements, the covariant fields $H_v^{(Q)}$ discussed here also turn out to be useful if we want to formulate heavy meson fields as sources for goldstone bosons in Chiral Perturbation Theory. We can use $H_v^{(Q)}$ to construct a chiral Lagrangian that is invariant under both chiral symmetry and heavy quark symmetry. A single pion vertex will cause a $B$-meson to fluctuate into a $B^*$, and the coupling $g_\pi$ for this process is related by heavy quark symmetry to the pion coupling that connects a $B^*$ to another $B^*$. Furthermore, since $B$ and $D$ mesons are related in the heavy quark limit, the same $g_\pi$ coupling describes the interaction of a pion with $D$ and $D^*$ mesons. Chiral symmetry also yields vertices which couple more than one pion, again with the coupling $g_\pi$. The study of matter fields in chiral perturbation theory (which could be heavy mesons, heavy baryons, or light baryons) is beyond the scope of topics we will cover in the main course sequence, but could make a good topic for the end of term video project.
33)Loop Corrections with a Four Vector Label (impact of v labels, symmetries, pure dim.reg. trick) (L8)
Pre Text: In the operators and Feynman rules of HQET the velocity four vector plays an important role. It serves as an auxiliary vector labeling the operators and hence can appear in Wilson coefficients and anomalous dimensions when we consider loop corrections. This shows up in a nontrivial way when we consider a transition between two heavy quarks, such as in a weak decay, where there is an external transfer of momentum that causes a change in the velocity vector between the initial and final heavy quarks. These ideas are discussed in this component.
Video
Post Box Question (distinct vertical): We have seen that the anomalous dimension for the current $\bar Q_{v'} \Gamma Q_v$ is a nontrivial function $\gamma(w)$ of the dot product of velocities $w=v\cdot v'$. Taking the one loop result discussed in lecture, what do you get for the limit $\gamma(w\to 1)$? Do you think that you would get the same result using the two loop expression for $\gamma(w)$?
Post Box Answer: The $w\to 1$ limit takes $v' \to v$, and gives $\gamma(w\to 1) = 0$ from our one-loop expression. In this limit our current is $\bar Q_v \Gamma Q_v$. Since the HQET Feynman rules do not modify the spin structure, the anomalous dimension is independent of $\Gamma$ at any order in $\alpha_s$. For the case $\Gamma= \gamma^\mu$ we have a conserved current, with $\mu=0$ we have $\bar Q_v \gamma^0 Q_v$ which corresponds with the charge that counts the number of heavy quarks with velocity $v$. Thus the vanishing of $\gamma(w\to 1)$ is simply related to current conservation, and will remain true as a property of $\gamma(w)$ to all orders in $\alpha_s$.
34)HQET Matching (L9)
Pre Text: In this component we consider the steps needed to carry out a one-loop matching calculation between QCD and HQET. As before, the matching results are obtained by comparing renormalized S-matrix elements in the two theories. Here particular attention is paid to the role of the (finite) residue factors that appear when we carry out wavefunction renormalization in the ${\overline {\rm MS}}$ scheme. In addition we discuss how to exploit the use of "pure dimensional regularization" which makes the HQET loop integrals scaleless. This setup provides a useful trick that can often be applied to efficiently carry out EFT matching calculations.
Video
Post Box Question (distinct vertical): To test your understanding of the content in this module, answer the following questions. If pure dimensional regularization is used for the QCD and HQET loop diagrams, then:
1)The result for the one-loop matching coefficients is just the QCD result dropping $1/\epsilon_{\rm IR}$ divergences. TRUE or FALSE?
2) The HQET diagrams are scaleless, and hence do not require UV counterterms. TRUE or FALSE?
3) In order to extract the Wilson coefficients purely from the full theory graphs, the HQET renormalization scheme has to be chosen as $\overline{\rm MS}$, TRUE or FALSE?
4) It is important that both the QCD and HQET calculations are using the same IR regulator. TRUE or FALSE?
5) The fact that both QCD and HQET treat UV divergences and renormalization in the $\overline{\rm MS}$ scheme, is an attribute that is special to the setup here. TRUE or FALSE?
Post Box Answer: 1) is TRUE, 2) is FALSE, and 3) is TRUE. The HQET graphs are scaleless, so adding the $\overline {\rm MS}$ HQET counterterms, the renormalized graphs involve $1/\epsion_{\rm IR}$s, which, when subtracted will exactly cancel these same divergences in the QCD graphs. Hence it suffices to simply drop the $1/\epsion_{\rm IR}$ poles in the QCD graphs to obtain the answer. 4) is TRUE in general for any matching calculation. 5) is also TRUE, more generally we can choose to use different renormalization schemes to define parameters in QCD and HQET.
Insert Pset #3, part #1 (HQET Renormalization Problem)
Chap.#2
Chap.#9
39)Renormalons (Pole Mass) (L10)
Pre Text: With HQET, we have seen an explicit example of an effective theory that has both a coupling expansion and a power expansion, so it is reasonable to ask whether there are connections between them. An obvious connection is that Wilson coefficients and renormalized operators have to be defined in a definite renormalization scheme. In general this actually provides a freedom to move contributions back and forth between coefficients and operators, which is the analog of varying the hard cutoff that would distinguish these the high energy coefficients from the low energy operators in a Wilsonian EFT. Given this freedom, are their constraints we can impose on the choice of scheme we should use? In this component we learn that even within the class of schemes that preserve the EFT symmetries, the answer to this question is yes. A poor scheme choice can lead to poor convergence for the coupling expansion in Wilson coefficients, with a corresponding instability in the value of nonperturbative matrix elements of the operators. The most important instabilities are related to the concept of renormalons. In this (somewhat longer) lecture component we in the first half motivate the concept of renormalons phenomenologically, then in the second half define mathematically the concepts of asymptotic series, Borel transforms, and renormalon poles. We will explore these ideas in detail in this and the following components.
Video
Post Box Question (distinct vertical): This question requires some calculations, but the results will be useful for your understanding of material throughout this chapter. Consider the Incomplete Gamma Functions $$\Gamma(a, x) = \int_x^\infty\! dz\: z^{a-1} e^{-z}$$. Find a series for the function $e^{-1/y} Gamma(0,-1/y)$ about $y=0$, and think of this like a coupling expansion in $y$. Find the Borel transform of this series and determine whether it has a renormalon pole. Can you think of a function that diverges so severely that it does not have an asymptotic expansion? Can you think of two functions which have the same asymptotic series?
Post Box Answer: The series expansion gives the asymptotic series $$e^{-1/y} Gamma(0,-1/y) = - \sum_{n=0}^\infty \Gamma(1+n) y^{n+1} \,. $$ To take the Borel transform from the variable $y$ to the variable $b$ we replace $y^{n+1} \to b^n /\Gamma(n+1)$ giving $$\sum_{n=0}^\infty b^n = \frac{1}{1-b}$$. When we consider the inverse transform integral $$f(y) = \int_0^\infty \! db\ e^{-u/b}\: \frac{1}{1-b} $$ we see that there is a renormalon pole on the integration contour at $b=1$. Note that the normalization choice for the variable $y$ does impact the location of the renormalon pole. If we took $y\to 2y'$, considered the series $y'$, and did the tranform $y'\to b$, then the pole would be at $b=1/2$. As an example of a function which does not have an asymptotic expansion, consider $e^{1/y}$ about $y=0$, where all the series coefficients are zero. As an example of two functions with the same asymptotic series about $y=0$ consider $e^{-1/y} Gamma(0,-1/y) $ and $e^{-1/y} Gamma(0,-1/y) + e^{1/y}$. For a field theory like QCD, ambiguities like that obtained from $e^{1/y}$ are related to nonperturbative effects which are not seen in the coupling expansion.
40)Renormalons Part 2 (when does MSbar lead to trouble, b-mass example, math, Borel transform) (L11)
Pre Text: As we saw numerically in the previous lecture component, a nice example of a renormalon occurs for the heavy quark pole mass. To find the renormalon pole we need to take the Borel transform of an infinite series in the coupling constant. Here we carry out this calculation by making use of a simple and unique subclass of Feynman diagrams, known as the fermion bubble chain. The residue of the renormalon pole for the pole mass is proportional to $\Lambda_{\rm QCD}$, providing a quantitative estimate for the parametric size of the infrared renormalon ambiguity in the pole mass.
Video
Post Box Question: In this lecture we used the fermion bubble chain to compute the renormalon ambiguity in the relation between the heavy quark pole and $\overline{\rm MS}$ masses. Lets ask, what would happen if we in addition included other diagrams? Imagine you had computed analytically another unique subclass of diagrams that also had a leading renormalon pole, how do you think it would modify the answer we found from the fermion bubble chain?
Post Box Answer: A leading pole for heavy quark masses will always give an ambiguity $\propto \Lambda_{\rm QCD}$, so we will get an answer with a pole at $u=1/2$ in the Borel plane. The thing that will be modified from this additional set of diagrams is the residue of the pole, and thus the magnitude of the ambiguity.
41) Renormalon Implications (L11)
Pre Text: In this component we discuss general observations about renormalon ambiguities building on the results of our pole mass calculation. We discuss several examples of heavy quark mass schemes that are free of the renormalon ambiguity, but which differ because of the scale that is introduced for each of these schemes when the renormalon is removed.
Video
Post Box Question: none
42) All Order LambdaQCD result (L11)
Pre Text: This is a brief aside about nonabelian quantum field theory. In this component we derive an all orders expression for $\Lambda_{\rm QCD}$ in terms of the strong coupling constant and beta function coefficients, a result that is useful in various contexts. Here we will use the resulting formula for our analysis of renormalons using the renormalization group.
Video
Post Box Question: none
43) Renormalons and the Renormalization Group (MSR mass example) (L11)
Pre Text: In this component we study in detail the dependence of the quantum field theory on the scale introduced to remove the renormalon ambiguity, which we will call $R$. This scale acts like a cutoff between infrared and ultraviolet contributions and hence can be studied with the renormalization group. Unlike the standard analysis of EFT Wilson coefficients, here we have power law dependence on the cutoff parameter which leads to an interesting solution for the corresponding leading log renormalization group equation.
Video
Post Box Question (distinct vertical): The MSR mass, $m(R)$, depends on a scale $R$ which is the dimensionful prefactor that is introduced when we remove the renormalon ambiguity in the heavy quark pole mass by subtracting an appropriate perturbative series (here the $\overline {\rm MS}$ series). If a physical observable is written in terms of $m(R)$ then what cancels the $R$ dependence?
Post Box Answer: To answer this question, consider first calculating the observable in terms of the heavy quark pole mass, and expressing the perturbative corrections as a series in $\alpha_s$. One might imagine that at lowest order the observable depends on a power of $m_{\rm pole}$, like the inclusive semileptonic decay rate. This series will have a renormalon ambiguity since the observable has none, but the pole mass is ambiguous. Now use the relation between the pole mass and MSR mass to write the observable in terms of $m(R)$, and a new series that is obtained by combining the original $\alpha_s$ series for the observable, and the $\alpha_s$ series in the mass relation. The new series will be renormalon free, and will have a dependence on $R$ at each order in $\alpha_s$ that ensures the observable is independent of $R$ at that order. So the answer to the posed question is, the series in $\alpha_s$ that appears for this physical observable.
44) Renormalon Renormalization Group at Higher Orders (L12)
Pre Text: In this component we study our solution to the renormalon RGE in more detail. We extend our solution to all orders in the strong coupling in terms of coefficients of the perturbative anomalous dimension, and discuss how the renormalon cancels out in the anomalous dimension coefficients.
Video
Post Box Question: none
45) RGE as a Renormalon Probe (L12)
Pre Text: In this component we show that the solution of our renormalon RGE can also be used as a probe for renormalons. We analytically continue our RG solution past the QCD Landau pole in order to connect the renormalon free mass definition to the pole mass which carries the ambguity. Thus we can see analytically the connection between the Landau pole in $\mu$-space and the renormalon pole in Borel space, and derive a sum rule for the residue of the renormalon pole.
Video
Post Box Question: Physically, why does it make sense that the renormalon pole is connected to the QCD Landau pole?
Post Box Answer: In QCD the existence of the single dimensionful scale $\Lambda_{\rm QCD}$ is related to the Landau pole in the (perturbative $\overline{\rm MS}$) running coupling. In the heavy quark pole mass, the renormalon ambiguity is related to the existence of nonperturbative physics, and hence also to $\Lambda_{\rm QCD}$.
46) Renormalons in OPEs (Wilsonian versus MSbar OPE) (L12)
Pre Text: In this component we consider the general structure of an Operator Product Expansion (OPE) obtained from expanding in $\Lambda_{\rm QCD}\ll Q$. Using a toy example we contrast the structure of the OPE obtained in the $\overline{\rm MS}$ scheme, Wilsonian scheme, and then in an ``MSR scheme'' that is defined in an analogous manner to what we did to define the MSR mass. Wilson coefficients for the OPE in the MSR scheme have power law terms in the evolution parameter, and the form of the resulting solution of the RGE is very similar to what we obtained for the MSR mass.
Video
Post Box Question: none
47) Mass Splitting OPE example (L12)
Pre Text: In this component we carry out an explicit OPE example using a ratio of differences of heavy meson masses that can be computed with perturbative QCD at leading order in the power expansion. We show how the OPE works out in the $\overline{\rm MS}$ and MSR schemes using known 3-loop results for the perturbative series. Interestingly, in the MSR scheme we can use scale variation to estimate higher order perturbative uncertainties (as usual), and we can use scale variation to numerically estimate the size of power corrections since they are connected to the leading power result by an explicit cutoff parameter.
Video
Post Box Question: none
Insert Pset #3, part #3 (R-RGE Problem)
48)EFT with a Fine Tuning (two nucleon or cold atom EFT) (L13)
Pre Text: In this chapter we will study an example of an effective field theory where the simplest dimensional expansion does not lead to a useful power counting. This happens because certain parameters are fine tuned from the perspective of this dimensional analysis. To obtain the desired power counting, a dimensionally irrelevant operator must be promoted to a relevant term. Our study will be carried out for a nonrelativistic system with two heavy particles, so we will also have the opportunity to discuss some of the features of nonrelativistic field theories. In this first component we introduce the nonrelativistic field theory with two-body contact interactions and setup the corresponding operators.
Video
Post Box Question (distinct vertical): In lecture we discussed the example of nonrelativistic nucleons, which carry spin and isospin degrees of freedom, interacting through contact interactions. The same theory is also relevant for describing atomic systems, such as ultracold atoms. Since the atoms are nonrelativistic, they will have the same kinetic term
$$\psi^\dagger \big[i\partial_t + {\nabla^2}/(2M)\big] \psi$$ whether they are fermions or bosons. A fermion degree of freedom could also have just one relevant spin state. This happens because the hyperfine interaction splits the energy levels for the spin states, and this small splitting can actually be a relatively large energy scale for our very cold atoms. Compared to lecture the quantum numbers change, so some of the contact interactions that were allowed for nucleons may now be forbidden. For fermionic atoms with a single spin state, can you think of an example of a contact interaction that is no longer an allowed operator? Can you think of a contact interaction that will not appear for bosonic atoms?
Post Box Answer: For fermions with a single spin, $\psi\psi $ vanishes since the fermion fields anticommute. Thus there is no contact interaction operator ${\cal O}_0$ without derivatives, and thus no $S$-wave scattering. For bosons the ${\cal O}_0$ operator is allowed, but $\psi \overleftrightarrow\grad \psi$ vanishes since the nonrelativistic bosonic fields $\psi$ commute, but the $\overleftrightarrow\grad$ yields a minus sign when we swap the fields. Therefore there is no $P$-wave scattering operator $O_2= (\psi \overleftrightarrow\grad_i \psi)^\dagger (\psi \overleftrightarrow\grad_i \psi) $ for bosons.
49)Loops and Power Counting (L13)
Pre Text: When two heavy particles scatter, whether by contact interactions or Coulomb interactions, the kinetic energy is a relevant operator. The importance of the kinetic energy is familiar from nonrelativistic quantum systems like the hyrdrogen atom, which you will recall has two heavy particles from the point of view of the relevant energy scales. In this component we will discuss the relevance of the kinetic energy from the point of view of loops in the field theory. We then work out the dimensional power counting for local contact interactions, being careful to separately count factors of the heavy particles mass $M$ and the scale $\Lambda$ associated to the potential that is encoded by the contact interactions. Finally we compute all the non-relativistic two-body scattering diagrams in this EFT, and use the result to derive a theorem in quantum mechanics about short range potentials.
Video
Post Box Question (distinct vertical): In lecture we only considered short range interactions. We could also extend our power counting analysis to consider long range interactions. Consider a long range potential in coordinate space,$V(\vec x -\vec x^{\,\prime})$, encoded by the Lagrangian
$${\cal L}_{\rm LR} = -\frac12 \int \!\! dt\, d^3x\, d^3x'\ (\psi^\dagger\psi)(x) V(\vec x -\vec x^{\, \prime})
(\psi^\dagger\psi)(x')$$
where $x$ and $x'$ both have time coordinate $t$. Counting the scaling of objects with factors of momenta $p$ and mass $M$, what is the power counting relevant for a Coulomb interaction? What is the power counting for a $V=C_6/r^6$ van der Waals interaction, which is the relevant long distance potential for neutral atoms?
Post Box Answer: Since the nonrelativistic fermion or boson field has dimension $3/2$, using the discussion in lecture we have $\psi \sim M^0 p^{3/2}$. The measure gives $dt d^3x d^3x' \sim M p^{-8}$. So letting the scaling of the potential $V$ be denoted by $[V]$, the scaling of the operator is $M p^{-2} [V]$. For a Coulomb potential $V(r) \propto \alpha/r \sim \alpha p$, so each factor of the Coulomb potential gives a factor of $\alpha M/p = \alpha/v$ by power counting. Here $v=p/M$ is the relative velocity of the heavy particles (not the HQET four-velocity). For a van der Waals force we have $V(r) = C_6/ r^6 \sim p^6$, so each insertion of this potential is $\sim M C_6 p^4$. Associating a scale $\Lambda_{VW}$ to the van der Waals force, we see that $C_6\sim 1/(M \Lambda_{VW}^4)$. Since the long range van der Waals force does not enter until $\sim p^4$ in the power counting, short range contact interactions will dominate for cold atoms.
50) Matching and Fine Tuning (power counting, enhanced coupling, beta function & RGE, other
enhanced couplings) (L13)
Pre Text: In this lecture component we relate the couplings in our EFT with parameters in the effective range expansion, and discuss the implications for systems of two nucleons which have large scattering lengths $a$ (systems with a large scattering length also appear in many atomic physics examples). The fine tuning of the EFT as $a\to \infty$ is connected to renormalization group flow from a linear power law divergence in the loop graphs. Switching to a scheme that tracks power divergences, we see that RG flow can be used to follow the transition from dimensional power counting to the fine-tuned power counting relevant for a large scattering.
Video
Post Box Question (distinct vertical): The classic example of fine tuning discussed in quantum field theory is that of the Higgs boson mass, which is sensitive to quadratic divergences in loop graphs. Using matching, can you give an example which expresses this statement of fine tuning in terms of physical parameters rather than cutoffs?
Post Box Answer: Imagine that we extend the Standard Model with a new physics model that has the SM particle plus additional heavy particles with a mass $M_{NP}$ much larger than the electroweak scale. Since generically these new particles will couple with the Higgs boson, upon integrating them out of the theory they will generate matching corrections for the low energy Higgs mass that go as $m_H^2 \sim M_{NP}^2$. We will then have to fine tune these threshold corrections against the initial value of $m_H^2$ to achieve a fine tuned cancellation that gives the many order of magnitude smaller value for the physical Higgs mass.
51) Power Divergence Subtraction Scheme (L13)
Pre Text: In the previous lecture we used the offshell momentum subtraction scheme, which allowed us to see power divergences in the EFT $\beta$-function. In this component we discuss how the same result can be achieved with an $\overline {\rm MS}$ type dimensional regularization scheme, which is called Power Divergence Subtraction. This scheme makes finite subtractions that are associated with $1/\epsilon$ poles that occur for dimensions other than $d=4$. We also use RGE equations to determine the the fine-tuning enhancement that occurs for the coefficients of other contact operators in the theory.
Video
Post Box Question: none
52) Nonrelativistic Conformal invariance, SU(4) invariance (L14)
Pre Text: In this lecture component we discuss the non-relativistic conformal invariance for the field theory of nucleons at infinite scattering length $a\to \infty$. We also discuss the enhancement of spin and isospin into Wigner's SU(4) symmetry for $a\to \infty$.
Video
Post Box Question: none
53) Bound States in QFT, the Deuteron (E.M. form factor as example, LSZ for bound state) (L14)
Pre Text: In this lecture component we use our non-relativistic EFT to demonstrate how calculations involving bound states work in a quantum field theory.
Video
Post Box Question (distinct vertical): Unlike in our analysis from this lecture, in a relativistic quantum field theory we most often can not analytically compute the infinite number of Feynman diagrams that are responsible for generating a bound state. Nevertheless we can still write down matrix elements involving a bound state as vacuum matrix elements with interpolating fields, in the same manner that we discussed. How does the normalization of the interpolating field drop out of the calculation of a physical observable like a form factor?
Post Box Answer: The more general statement about interpolating field choice is this: As long as our interpolating field has the same quantum numbers as our bound state, and has a large overlap with that state (for example, is the leading order operator by power counting), then the result for the physical matrix element will be independent of the choice of the interpolating field. This is true for both non-relativistic and relativistic field theories. The truncation of the matrix element by the bound state two-point functions, and the inclusion of the bound state LSZ factor $Z$, play a crucial role in ensuring that this is the case. For example if we scale our interpolating field for the deuteron by a constant factor of two, then the $3$-point function, $2$-point functions, and $Z$ factor each scale like this interpolating field squared, and the factors of two will cancel in the product that gave the physical form factor $Z G_{\rm 3-pt} G^{-1}_{\rm 2-pt} G^{-1}_{\rm 2-pt}$. Expressions of this type with interpolating fields are used to compute properties of relativistic QCD bound states in lattice QCD.
54) Coupling to a Charge: Axions in the Sun (L14)
Pre Text: As a final lesson from our non-relativistic EFT, we consider the implications of spin symmetry and SU(4) invariance for the coupling of light axions to non-relativistic nucleons in the Sun. The axions and nucleons have a similar energy, but the momenta of the axions are much less than the nucleons. This forces us to carry out a multipole expansion in order to have a systematic power counting, leading to some interesting consequences.
Video
Post Box Comment: In this lecture component I did not prove that setting $\vec x=0$ on the axion part of the operator is equivalent to dropping the axion $3$-momentum relative to the nucleon $3$-momentum. We will have the occasion to demonstrate this explicitly later on, see <a href="../../../courseware/Ch11/seq_35/2/" target="_blank">Chapter 11, Lecture Module 62</a>.
Insert Pset #4, part #1
Insert Pset #4, part #2
Chap.#10
Soft-Collinear EFT
55)Introduction to SCET (L14)
Pre Text: In this chapter we start our discussion of the Soft-Collinear Effective Theory (SCET), which is an effective field theory whose Wilson coefficients encode the hard interactions of energetic particles, and whose operators and Lagrangian describe the dynamics of energetic jets and energetic hadrons as well as soft radiation. In this lecture component we motivate the study of SCET based on the type of processes and physical effects that one can treat. We also discuss some of the complications SCET involves compared to other EFTs that we have studied earlier in the lecture sequence. Finally, we will introduce light cone basis vectors for decomposing four-momenta and other tensors.
Video
Post Box Question: none
56)Degrees of freedom for SCET2, momentum planes (L15)
Pre Text: In this lecture component we continue our discussion of SCET degrees of freedom, defining collinear scaling for momenta, and discussing the physics of soft and collinear particles in the $B\to D\pi$ process. It is useful to picture the modes for this process in a $p^+$-$p^-$ momentum space plane. The degrees of freedom for this process correspond to what is known as a SCET$_{\rm II}$ theory.
Video
Post Box Question: none
57)Degrees of freedom for SCET1 (L15)
Pre Text: In this lecture component we discuss the SCET degrees of freedom for typical measurements of $e^+e^-\to $ two-jets. This process involves $n$-collinear, $\bar n$-collinear, and ultrasoft modes, which corresponds to a SCET$_{\rm I}$ theory. Some general comments about methods to determine the correct degrees of freedom are also given.
Video
Post Box Question (distinct vertical): a) Consider the production process $e^+e^-\to \pi^+\pi^-$ in the CM frame. What degrees of freedom would you need to describe this process? b) As an advanced example, what SCET degrees of freedom would you need to consider for $pp \to$ two-jets in the center of momentum (CM) frame? Here you can assume that the jets go off at a large angle compared to the incoming protons, and that you measure jet-mass type variables to ensure there are only two jets in the final state.
Post Box Answer: a) The process $e^+e^-\to \pi^+\pi^-$ is like the example from lecture where we produced two-jets, except now each of the jets is replaced by a pion. In the CM frame these energetic pions will be described by collinear degrees of freedom, much like the jets, where $p^\mu \sim Q(\lambda^2,1,\lambda)$ for the degrees of freedom in one pion, and $p^\mu\sim Q(1,\lambda^2,\lambda)$ for those in the other. The difference is that now $\lambda \sim \Lambda_{\rm QCD}/Q$, and these collinear modes are nonperturbative. Since we know there are just two pions in the final state we do not have any other type of modes that can cause real emissions. We do not have to consider virtual ultrasoft modes since $p^\mu \sim Q\lambda^2 \sim \Lambda_{\rm QCD}^2/Q$, and any modes with such small virtuality are propagating at distance scales that are outside the hadrons. Due to confinement in QCD at zero temperature we do not have colored modes that propagate in this manner (which we can refer to as saying there are no "hyper-confining" modes). We should consider whether virtual soft modes with momenta $p^\mu\sim Q(\lambda,\lambda,\lambda)$ can contribute. It turns out that they will not contribute to the physical process at leading power. At subleading power these soft modes could contribute through vacuum matrix elements (which people sometimes refer to as condensates), but I am not aware of anyone that has demonstrated explicitly whether or not this happens. Note that the theory for the process $e^+e^-\to \pi^+\pi^-$ is like an SCET${}_{\rm II}$.
b) For $pp\to $ two-jets, we now have energetic incoming protons as well as energetic outgoing jets. So we need collinear modes for the incoming protons with back-to-back light-like vectors $n_a$ and $\bar n_a$. We also need collinear modes for the outgoing jets with directions $n_1$ and $n_2$. The collinear modes for the protons are just like those for the pions in part a, they are nonperturbative with $p^2 \sim Q^2 \lambda^2\sim \Lambda_{\rm QCD}^2$. (Depending on precisely how we make the measurement of the final state particles, they may also have a perturbative component.) The collinear modes for the jets are exactly like those for $e^+e^-\to $ two-jets, as we discussed in lecture, with $p^2 \sim Q^2 \lambda^{\prime\, 2}\gg \Lambda_{\rm QCD}^2$. Since we are measuring jet masses we can also have ultrasoft modes with $p^2 \sim Q^2 \lambda^{\prime\, 4}$, just like we did for $e^+e^-\to $ dijets. Due to the proton scales, we can also consider soft modes with $p^2 \sim Q^2 \lambda^2$. There are also offshell virtual modes referred to as Glauber gluons with $p^\mu \sim Q(\lambda^a, \lambda^b,\lambda)$, where $a+b>1$, and the same with $\lambda\to \lambda'$. These modes need to be considered for $pp$ scattering processes. The list of modes to consider here is quite large, and should makes it clear why we choose to focus on simpler processes than this one in the remainder of the SCET lectures.
58) Collinear Spinors (L15)
Pre Text: Here we consider full QCD spinors, expand them in the collinear limit $p^+\ll p_\perp \ll p^-$, and demonstrate that both the collinear quark spinors and collinear field should obey $\slash\!\!\! n\, \xi_n=0$. Note that the act of expanding the spinors in this fashion assumes that we have fixed the reference frame, such as using the center-of-momentum frame for the hard interaction. Thus we can say that the hard interaction preferentially produces quarks whose spinors obey $\slash\!\!\! n\, u_n=0$.
Video
Post Box Question: none
59) Collinear Propagators and Power Counting (L15)
Pre Text: In this lecture component we discuss the collinear quark propagator, and derive the power counting for the collinear quark field $\xi_n\sim \lambda$, and for the collinear gluon field $(n\cdot A_n,\bar n\cdot A_n,A_n^\perp) \sim (\lambda^2,\lambda^0,\lambda)$.
Video
Post Box Question: Is the power counting for the collinear gluon field gauge invariant?
Post Box Answer: Yes it is. Although we derived the power counting using a general covariant gauge gluon propagator, the scaling of the $n$-collinear gluon field is the same in any gauge. If we make a specific gauge choice, such as Feynman gauge or the light-cone gauge ($\bar n\cdot A_n=0$), then this gauge alone may still allow redundancies in the power counting. But the power counting we assigned will always be a valid solution. One way to see this is to look at the gauge invariant $G_{\mu\nu} G^{\mu\nu}$ term in the gluon action, and note that if we just have interactions amongst collinear gluon fields alone then the Lagrangian must be the same as QCD (it's just QCD in a boosted frame). Therefore no terms can be dropped. If we consider the type of terms that enter the $GG$ operator, we have $\partial^\mu A^\nu \partial^\alpha A^\beta$, $\partial^\mu A^\nu A^\alpha A^\beta$, and $A^\mu A^\nu A^\alpha A^\beta$ with various contractions of these indices. Since a $\partial^\mu$ acts on a collinear field, it has collinear power counting. If we have an $A_\mu$ dotted into this $\partial^\mu$, then $A_\mu$ must have collinear scaling too, otherwise some term in the dot product would be larger than another term, and we would be forced to drop some term relative to another. Another way to make the same argument is write $G_{\mu\nu}$ as a commutator of covariant derivatives, and then note that the two terms in the covariant derivative must have the same power counting. This leads to the same conclusion. We discussed in lecture the fact that gauge invariance requires $i\partial^\mu$ and $A_n^\mu$ to have the same scaling in order to be able to combine them into a $D^\mu$ that is covariant in all of its components.
60) Wilson Line W from offshell propagators (L16)
Pre Text: One component of an $n$-collinear gluon field is not suppressed by power counting, $\bar n\cdot A_n\sim \lambda^0$. This implies that any number of $\bar n\cdot A_n$ fields can appear in a leading power SCET operator. Recall that SCET can be derived explicitly from QCD in a top-down manner, by integrating out offshell quarks and gluons. In this lecture component we use QCD calculations to explicitly construct the SCET operators involving $\bar n\cdot A_n$ fields for a current that produces a collinear quark. We demonstrate that the sum of all tree level diagrams with offshell propagators produces a collinear Wilson line $W_n[\bar n\cdot A_n]$.
Video
Post Box Comment: We will explore the collinear Wilson line $W$ in more detail in the homework.
Post Box Question: If we had an operator constructed out of collinear fields in the $n$ and $\bar n$ directions, then what type of collinear Wilson lines might we expect to appear?
Post Box Answer: In this case we would expect to have $W_n[\bar n\cdot A_n]$ as well as $W_{\bar n}[n\cdot A_{\bar n}]$. We will discuss how these Wilson lines will appear in later lectures.
61) SCET Lagrangian, initial steps (L16)
Pre Text: If we consider isolated interactions which are purely between $n$-collinear particles, or purely between ultrasoft particles, then the appropriate Lagrangians for each of these sectors is just full QCD. For ultrasoft particles the momenta are homogeneous, so there is nothing from QCD to expand. For collinear particles we can boost everything to a frame such that $p_n^\mu \sim (\lambda^2,1,\lambda) \to (\lambda,\lambda,\lambda)$, so again physically there is nothing to expand. These special cases provide a constraint on the form of our SCET Lagrangians. What SCET does is describe the interactions {\it between} different sectors. In particular, it handles interactions between ultrasoft and collinear particles and between collinear particles in the presence of a hard interaction where the boost argument can no longer be used.
In this lecture component we start our derivation of the leading order SCET$_{\rm I}$ Lagrangian for collinear quarks. Motivated by the fact that hard interactions produce collinear quarks obeying $\slash\!\!\! n\,\xi_n=0$, we first rewrite the QCD Lagrangian in terms of such a field $\xi_n$, plus a field $\varphi_{\bar n}$ obeying $\slash\!\!\! {\bar n} \varphi_{\bar n}=0$. This gives a Lagrangian that is equivalent to standard QCD, but written in a non-standard form.
Video
Post Box Question: none
62) Expansion of Gauge Field and Momenta with a Multipole Expansion (L16)
Pre Text: In this lecture component we expand the QCD Lagrangian by noting that in certain components the ultrasoft gauge fields are power suppressed relative to the collinear gauge fields, such as $A_\perp^{us} \ll A_n^\perp$. Likewise, certain components of an ultrasoft momentum are suppressed relative to their collinear counterparts. This momentum expansion is carried out with a multipole expansion. Here we start by showing how one carries out a multipole expansion in position space, and then motivate why in SCET it is often more convenient to carry out the multipole expansion in momentum space.
Video
Post Box Question: none
63) Multipole Expansion in Momentum Space, Label Operators (L17)
Pre Text: In this lecture we setup some notation to make it easy to carry out a momentum space multipole expansion, in particular by introducing collinear momentum operators called Label operators.
Video
Post Box Question: none
64) Antiparticles and Momentum Labels (L17)
Pre Text: In this lecture component we discuss how the particle and antiparticle operators are treated within the same collinear fields.
Video
Post Box Question: none
65) Leading Order SCET Lagrangian (L17)
Pre Text: In this lecture we assemble the results from the past few lecture components, and construct the leading order SCET quark Lagrangian. We then discuss some of its properties. The success of our leading order formulation is noted by confirming that in different situations we immediately obtain Feynman rules and propagators that do not require any further expansions.
Video
Post Box Question (distinct vertical): How does our SCET Lagrangian give back full QCD for the situation where we are treating isolated collinear particles?
Post Box Answer: We mentioned earlier that, despite its strange form, the Lagrangian obtained by dropping the ultrasoft field $n\cdot A_{us}$ is just QCD. To see this even more explicitly, we can consider integrating back in an auxilliary field $\varphi_{\bar n}$ that satisfies $(\slash\!\!\! \bar n \: \slash\!\!\! n / 4) \varphi_{\bar n} = \varphi_{\bar n}$. This does not take us back to our starting point because we carried out the multipole expansion. What it gives is an equivalent form for the Lagrangian written in terms of the field $\psi_n = \xi_n + \varphi_{\bar n}$, namely
$$ {\cal L}_{n\xi}^{(0)} = \bar\psi_n \Big( i \slash\!\!\! D + \frac{\slash \!\!\! \bar n}{2} n\cdot A_{us} \Big) \psi_n $$.
From this form of the Lagrangian it is completely obvious that we just get back the Dirac Lagrangian when we drop $n\cdot A_{us}$. The reason to prefer the form of the collinear Lagrangian discussed in lecture is simply that the equations of motion imply that this auxiliary field has the power counting $\varphi_{\bn} \sim \lambda^2$, whereas $\xi_n\sim \lambda$, so the object $\psi_n$ is not homogeneous in its $\lambda$ power counting.
66) Wilson line identities, Collinear Gluon Lagrangian (L17)
Pre Text: In this lecture component we discuss operator identities that allow the $\bar n\cdot A_n$ component of the collinear gluon field to be completely traded for the collinear Wilson line $W_n$. We also discuss the collinear gluon Lagrangian.
Video
Post Box Question: none
Insert Pset #4, part #3
Insert Pset #4, part #4
Chap.#11
89) Interactions in SCET2 (offshell S-C interactions, S-C factorization, obtaining SCET2 from SCET1, Theorem about matching) (L24)
Pre Text: In this final chapter we will discuss various aspects of the theory SCET${}_{\rm II}$. We begin in this component by discussing soft-collinear factorization, and develop an efficient mechanism for deriving SCET${}_{\rm II}$ by first passing through and SCET${}_{\rm I}$ theory: QCD $\to$ SCET${}_{\rm I}$ $\to$ SCET${}_{\rm II}$.
Video
Post Box Question: none
90) SCET1 and SCET2 power counting formula (L24)
Pre Text: In this lecture component we briefly discuss general power counting formulas for SCET${}_{\rm I}$ and SCET${}_{\rm II}$. The power counting is determined entirely by the $\lambda$ scaling of operators. These results demonstrate that the SCET power counting works the same way in any gauge, since it is entirely determined by gauge invariant operators.
Video
Post Box Exercise (no solution given): Consider some examples of SCET Feynman diagrams at 1-loop order, and verify that the power counting formula gives you the same answer that you would find by the "brute force" method of assigning \(\lambda\) power counting to the explicit momentum factors that occur from propagators, loop measures, and vertices.
91) gamma-Pion form factor example (L24)
Pre Text: In this lecture component we carry out an example of factorization involving energetic hadrons, and using SCET${}_{\rm II}$.
Video
Post Box Question: For this example we had a purely collinear theory at leading power, the soft fields cancel out of our purely $n$-collinear operators and do not play any role. In what manner could soft contributions show up in the photon-pion form factor at subleading power? How would the analysis done in the lecture change at leading power if we had instead analyzed the factorization using the pion rest frame?
Post Box Answer: At any order in the power counting we can always consider the pion to be generated by a purely collinear interpolating field. The choice of interpolating field does not matter as long as we make a choice with a large overlap, so there is no reason to consider power corrections to the interpolating field. In particular we should not consider an interpolating field with one soft and one collinear field, since the invariant mass of this combination is too large to hadronize into a pion. So, since the final state pion is purely collinear, and the initial state only involves photons, the only type of soft matrix elements that can appear at subleading power are condensates, $\langle 0 | \text{operator} | 0\rangle$.
If we had instead considered this process in the pion rest frame, then the pion would be described by soft fields. The boost that takes us to this frame increases the energy of the onshell photon so that it is $\sim Q^2/m_\pi$. The offshell photon has $q^\mu = (m_\pi/2) n^\mu - (Q^2/2m_\pi) \bar n^\mu$, so it has a minus-momentum that is much smaller than its plus-momentum. In this situation it is natural to consider the offshell and onshell photon as hard-collinear photons with a hard scale $\sim Q^2/m_\pi$. The intermediate propagator in our tree level diagram is also hard-collinear with the same scaling, and the process takes place through a time-ordered product of two currents that couple these hard-collinear modes and the soft quarks. This is effectively an SCET${}_{\rm I}$ type theory. Here we would factorize the process by factorizing the product of two currents into a hard-collinear matrix element (analog of the jet function) convoluted with a soft matrix element that gives precisely the light-cone pion distribution. Here this is ultrasoft-collinear factorization. It leads to precisely the same factorization theorem, but with a different description of the degrees of freedom that led to this result. This example illustrates an interesting connection between hard-collinear and ultrasoft-collinear factorization.
92) B to D pi example (L25)
Pre Text: In this lecture component we carry out another example of factorization involving energetic hadrons. This time we will also see hadrons whose dynamics is dominated by soft momenta.
Video
Post Box Question: none
93) SCET2 and Rapidity divergences (massive Sudakov form factor, various examples) (L25)
Pre Text: In this lecture component we consider the factorization of modes in SCET${}_{\rm II}$ in greater detail. In general, in order to properly distinguish soft and collinear modes we must make use of rapidity variables since these modes live at the same invariant mass. In some processes this factorization in rapidity space leads to logarithmic singularities, known as rapidity divergences. As an example of this situation, we consider the massive Sudakov form factor, and discuss the an appropriate renormalization regularization procedure for this SCET${}_{\rm II}$ process.
Vide
Post Box Question: none
94) Rapidity Renormalization Group (L26)
Pre Text: In this lecture component we consider renormalization group equations for the massive Sudakov form factor. Due to the presence of both invariant mass and rapidity logarithms, the corresponding RG evolution takes place with two cutoff variables or renormalization group parameters.
Video
Post Box Question: none
95) Factorization with Rapidity Divergences (L26)
Pre Text: In this lecture component we briefly discuss another physical example of a SCET${}_{\rm II}$ processes that has rapidity singularities, namely $p_T$ resummation for Higgs production from gluon fusion. Despite our extensive use of SCET${}_{\rm I}$ when discussion $e^+e^-\to$ dijets, it is also worth noting that observables can be chosen for this process which put us into SCET${}_{\rm II}$ (the classic example is described here, and is called broadening).
Video
Post Box Question: none
96) Beam Functions (L26)
Pre Text: In this lecture component we turn to a final type of perturbative function, beam functions, which often show up when we consider processes with hadrons in the initial state. The beam functions contain both perturbative initial state radiation, as well as the nonperturbative parton distribution functions, and obey a factorization theorem that cleanly separates these two components.
Video
Post Box Comment: Note that in our discussion of Drell-Yan we have implicitly assumed that there are no other degrees of freedom that have to be considered in the EFT. It is actually known that this is not the case for processes involving \( pp\) collisions, one must consider Glauber degrees of freedom that have momenta \( p^\mu \sim Q(\lambda^a, \lambda^b,\lambda)\), where \(a+b>1\). In the case of inclusive Drell-Yan it is known from the literature (the proof of factorization by Collins-Soper-Sterman = CSS) that there are actually no Glauber gluons that modify the leading power amplitude. The CSS proof is non-trivial. The same level of rigor has not yet been applied to less inclusive processes, like the Isolated Drell-Yan example that we treated, because the nature of Glauber gluons for these processes has not been worked out yet in the literature. Since in general the data on less inclusive \(pp\) scattering processes does agree with factorization predictions, it is expected that if Glauber gluons do play a role, that the severity varies process by process, and presumably may be often treated as a correction rather than something that spoils the leading order factorized results. Glauber gluons are also known to play a role when SCET is applied to scattering in medium, such as the "liquid" produced in heavy ion collisions. Here the Glauber gluon acts as a background source field that allows the active quarks and gluons to communicate with the medium.
Chap.#12
SCET beyond tree level, Symmetries & Factorization
67) Any Spin symmetry? Two-component SCET (L18)
Pre Text: In this chapter we continue our discussion of SCET, focussing on properties of the theory that are true to all orders in the strong coupling constant. In the first few lecture components we will talk about symmetries of SCET, including gauge symmetry and reparameterization invariance. Then we will turn to properties related to interactions between different sectors, including ultrasoft-collinear factorization and hard-collinear factorization. In this lecture component we review the SCET Lagrangian, and then discuss the nature of spin symmetry for collinear fermions. This also gives us an opportunity to write the collinear quark Lagrangian with 2-component fields.
Video
Post Box Discussion: We saw that the spin symmetry of the collinear Lagrangian in SCET is just the same spin symmetry of tree level massless QCD, namely a chiral symmetry or helicity symmetry. The only difference for the collinear Lagrangian is that we already have a preferred direction in $n^\mu=(1,\hat n)$, and therefore we talk about the perturbative helicity symmetry about this $\hat n$ direction. If collinear quarks become nonperturbative and/or we include quark masses, then the fate of the chiral symmetries here is the same as that in QCD.
68) Gauge Symmetry in SCET (L18)
Pre Text: In this lecture component we discuss the gauge symmetry of SCET${}_{\rm I}$. Since we have two types of gluon fields, ultrasoft and collinear, it may not come as a surprise that these gauge fields are connected to two types of gauge transformations, which are also called ultrasoft and collinear.
Video
Post Box Question: What would happen if we were to make a gauge transformation on the EFT fields that was not in the collinear or ultrasoft classes that we considered, for example by having larger momenta?
Post Box Answer: In a typical EFT one does not have to restrict the momentum of gauge transformations in the way we did in our lecture. For example, if we make a gauge transformation carrying large momentum (short range curvature) to a field in HQET, then as long as we continue to transform the EFT fields in the same manner, this gauge transformation will also cancel out of the HQET action. The EFT remains gauge invariant when it is used to consider fluctuations outside the low momentum region. The same is true of SCET, but the description of how this happens is a bit more tricky. The important point of our discussion of ultrasoft and collinear gauge transformations is not having an upper bound on the momenta, but rather that there are two classes of transformations that allow us to have a meaningful simultaneous definition of collinear and ultrasoft gauge fields. The key point is that we have a "collinear" class of transformations where the collinear gluons transform as a gauge field and the ultrasofts do not transform, and an "ultrasoft" class where the collinears transform as a background adjoint field and the ultrasofts as a gauge field. If we consider a gauge transformation with larger momenta than the collinear scaling for some components, then we can put this transformation in our "collinear" class, and transforming the fields in the manner appropriate to this class, we then see that SCET remains gauge invariant for these large momentum transformations, much like HQET. Although the actions are gauge invariant under such transformations, they are not needed to constraint the form of operators in the EFT. For that purpose our ultrasoft and collinear transformations suffice.
69) Reparameterization Invariance in SCET, part 1 (intro) (L18)
Pre Text: In this lecture component we discuss reparameterization invariance in SCET. One part of the reparameterization freedom is related to our choice of the collinear basis vectors $n$ and $\bar n$, while another part is related to the freedom inherent in how we split momenta into large label components and small residual components. You may wish to first review the discussion of reparameterization invariance in HQET, which is done in <a href="../../../courseware/Ch11/seq_22/" target="_blank">Chapter 8, Lecture Module 36</a>.
Video
Post Box Question: How does RPI in SCET differ from a passive infinitesimal Lorentz transformation that we could make on our coordinate system?
Post Box Answer: With a single type of collinear field, the RPI transformations we discussed are quite similar to infinitesimal Lorentz transformations, or combinations of them. To see this explicitly, consider RPI-I is related to a rotation plus a boost. An infinitesimal rotation will modify both $n$ and $\bar n$, and the boost is needed to restore $\bar n$ to its original form. For example, consider the canonical $n=(1,0,0,1)$ and $\bar n = (1,0,0,-1)$. Rotating a small component $\delta$ in the $x$-direction gives: $(1,\delta,0,1)$ and $(1,-\delta,0,-1)$ where we have dropped ${\cal O}(\delta^2)$ terms in the $z$-components. Including a boost along $-\hat x$ by the same amount $\delta$ we get $(1,2\delta,0,1)$ and $(1,0,0,-1)$, again dropping second order terms. This has the form of the RPI-I transformation.
One difference between RPI and the Lorentz transformation is the power counting restriction $\Delta_\perp\sim \lambda$. Even with repeated RPI-I transformations we are not allowed to build up a transformation of our $n$ vector that woud take it outside the collinear cone, which we can certainly do with a Lorentz transformation. We will comment on another more dramatic difference between RPI and Lorentz after Lecture Module 71 below.
70) RPI part 2 (L19)
Pre Text: Here we continue our discussion of SCET reparameterization invariance. Using the full power of reparameterization invariance and gauge invariance we show that the collinear covariant derivatives $i D_{n\perp}^mu$ and $\bar n\cdot D_n$ are always connected with the presence of power suppressed ultrasoft covariant derivatives.
Video
Post Box Question: none
71) Extension to multiple collinear fields (L19)
Pre Text: In this lecture component we extend our description of $n$-collinear fields to include multiple independent collinear directions $n_1$, $n_2$, $\ldots$. A majority of the results we derived with a single collinear direction immediately carry over to this more general situation.
Video
Post Box Comment: Note, as remarked in lecture, that once we have multiple collinear fields with a $\{n_i,\bar n_i\}$ basis, that we have corresponding RPI transformations for each one. Here there is a clear difference between RPI transformations and Lorentz transformations since an RPI transformation acts only on a single set of basis vectors, eg. just $n_1\to n_1+\Delta_{n_1}^\perp$, whereas an infinitesimal Lorentz transformation would necessarily transform all basis vectors. There are now more generators of the RPI transformations than there are for Lorentz transformations.
Factorization
72)Study L^(0): Usoft-Collinear factorization (L19)
Pre Text: In this lecture component we come back to our SCET${}_{\rm I}$ Lagrangian, and investigate the interactions between collinear particles and ultrasoft gluons in more detail. In particular, we will see that the leading ultrasoft interactions with collinear particles also generate Wilson lines, but now involving the ultrasoft gauge field rather than the collinear one. The general factorized structure of these ultrasoft Wilson lines is derived by exploiting a field redefinition.
Video
Post Box Question: When we consider Wilson lines in the fundamental representation $Y_n$ and adjoint representation ${\cal Y}_n$, a useful set of identities is
$$ Y_n^\dagger T^A Y_n = {\cal Y}_n^{AB} T^B \,,\qquad\quad Y_n T^A Y_n^\dagger = {\cal Y}_n^{BA} T^B $$.
How would you prove these?
Post Box Answer: Part of this relation just has to do with color algebra, rather than the Wilson lines in particular, so lets start with that. The object $Y_n^\dagger T^A Y_n$ is a product of matrices in color space and hence can always be decomposed in terms of components along the basis vectors $\id$ and $T^B$. Since its traceless there is no $\id$ component, and we can now think of $ Y_n^\dagger T^A Y_n = {\cal Y}_n^{AB} T^B$ as the general basis decomposition. It remains to prove that ${\cal Y}$ is an orthogonal matrix:
$$ {\cal Y}_n^{AB} {\cal Y}_n^{CB} = \frac{1}{T_F} {\rm tr} \big[ {\cal Y}_n^{AB} T^B {\cal Y}_n^{CB'} T^{B'}\big] = \frac{1}{T_F} \big[ Y_n^\dagger T^A Y_n Y_n^\dagger T^C Y_n \big] = \delta^{AC} $$
Each step also applies for the transposes, so ${\cal Y}_n^{AB}$ is an orthogonal matrix. The second identity follows from the first by using the unitarity of $Y_n$ and orthogonality of ${\cal Y}_n$ to swap each to the other side.
To prove that the orthogonal matrix is an adjoint Wilson line along the same direction as the fundamental Wilson line, we can just consider the termwise expansion using the following matrix identity:
$$ e^{X} Y e^{-X} = Y + [X,Y] + \frac{1}{2!} [X,[X,Y]] + \frac{1}{3!} [X,[X,[X,Y]]] +\ldots$$
Each commutator gives an $i f^{ABC}$ and the coefficients lead to the path-ordered exponential which is precisely that of the Wilson line.
Insert Pset #5, part #1
73) Wilson Coefficients (L19)
Pre Text: Having discussed in detail the dynamics of SCET itself, we now turn to the issue of determining the most general form for the hard physics we are integrating out. This physics is encoded in Wilson coefficients for our SCET operators, which can be functions of the large momentum component of collinear building blocks.
Video
Post Box Question: none
74) Hard-Collin Fact & Building Blocks (L20)
Pre Text: By exploiting operator relations for the Wilson line $W$, we show that the collinear covariant derivative $i\bar n\cdot D_n$ can always be traded for the Wilson line $W$. Using this result we generalize the discussion of the previous lecture component to show that the most general SCET operators can be built from a few basic building blocks. The most general form for the corresponding Wilson coefficient is a function of the large momentum components of the collinear building blocks appearing in the operator.
Video
Post Box Question: In lecture we discussed the fact that the collinear components of any hard scattering operator just involves the building blocks $\{ \chi_n , {\cal B}_{n\perp}^\mu, {\cal P}_\perp \}$. Can you argue physically that this is not unexpected?
Post Box Question: Physically there are two spins for a quark which are encoded in the fields $P_L \chi_n$ and $P_R\chi_n$, and there are two polarizations for a gluon which are encoded by the choices $\mu=1,2$ for ${\cal B}_{n\perp}^\mu$. So, the number of field building blocks is equal to the number of physical spins that we have to describe. Since we can simplify things with the equations of motion (i.e. work onshell) it is not surprising that all other components can be eliminated. In particular momenta can be described by just the $\perp$-momenta, picked out by ${\cal P}_{n\perp}^\mu$, as well as the large momenta $\bar n\cdot p$ that are either hidden inside the operator definitions or encoded in generic Wilson coefficients. Thus, the result can be expected.
Chap.#13
75)Loops and Matching (collinear loops, label sums & 0-bin) [ long! 1 hour ] (L20)
Pre Text: In this chapter we will consider various interesting results that follow from considering SCET at one-loop order. In particular we will study operator anomalous dimensions that allow us to sum Sudakov logarithms, and operator anomalous dimensions that yield the DGLAP evolution for parton distributions. To start, the following lecture component has a comparison of one-loop QCD and one-loop SCET, including a discussion of the matching of IR divergences and of the zero-bin subtractions that are part of evaluating collinear loop integrals. In order to be self contained, this lecture component is longer than usual.
Video
Post Box Comment: In this lecture we demonstrated the matching of IR divergences between QCD and SCET for $b\to s\gamma$. Along the way we discussed the various loop diagrams in SCET, including a careful analysis of the UV and IR divergences in the collinear diagrams (accounting for 0-bin subtractions). Generally we can quite easily see how to quickly jump to the final collinear loop integrands with continuous loop integration variables. Since the 0-bin subtraction integrands are often (though not always) scaleless, we can also check this and often ignore this complication. Having been through these complications once, future loop calculations in SCET can be carried out more quickly.
76) Summing Sudakov Logs in SCET1 (LL RGE, solution) (L21)
Pre Text: In this lecture component we carry out the renormalization for the one-loop SCET Feynman graphs for the process $b\to s\gamma$. We derive a leading logarithmic RGE equation for the Wilson coefficient, and demonstrate that its solution (with and without a runnng coupling) sums Sudakov double logarithms into a Sudakov form factor. Physically these logarithms are related to the kinematic restrictions on the type of radiation that can be emitted by our collinear operators. The Sudakov form factor encodes the probability of having no emissions between a hard scale $\omega$ and a low energy scale $\mu$.
Video
Post Box Question: Which diagrams in SCET contributed to the terms needed to find the LL anomalous dimension?
Post Box Answer: The one-loop diagrams that gave $1/\epsilon^2$ poles are the ones that contribute to the LL anomalous dimension. In our Feynman gauge calculation these are the collinear loop graph with the contraction involving the Wilson line (giving a $1/\bar n\cdot k$ propagator) and the ultrasoft loop graph which also involves eikonal propagators (up to the presence of our IR regulator).
77) Sudakov Logs at higher orders (L21)
Pre Text: In general we can consider our anomalous dimension equation at higher orders in perturbation theory, and thus extend the resummation of logarithms from leading-log (LL), to next-to-leading-log (NLL), then next-to-next-to-leading-log (NNLL), etc. Due to the presence of both double and single logarithmic series, the organization of the terms in our anomalous dimension equations here are different than what we considered earlier in our earlier discussion of the RGE obtained from integrating massive particles (in Chapter 5). The counting and contributions needed to carry out higher order Sudakov resummation are discussed in this lecture component.
Video
Post Box Question: Previously, when integrating out the massive W-boson for the electroweak Hamiltonian we discussed what type of logarithmic terms are resummed by considering the NLL anomalous dimension equation, <a href="../../../courseware/Ch05/seq_13/1/ target="_blank">link</a>. Since it is more common to have single logarithmic series, I will call this the standard resummation. At what order in the Sudakov resummation, (LL, NLL, ...) would our resummed series contain the terms that are NLL in the standard resummation?
Post Box Answer: The LL${}_{std}$ series from the electroweak Hamiltonian corresponds to single logarithmic terms of the form $(\alpha_s \ln)^k$. The NLL${}_{std}$ series corresponds to terms of the form $\alpha_s (\alpha_s \ln)^k$. To determine where these terms show up in the Sudakov resummation, we first have to deal with the fact that the Sudakov logarithms exponentiate. If we first answer this question for $\ln C(\mu,\omega)$ then we see that there is an offset by one, the LL${}_{std}$ series is only determined when we have the NLL${}_{sudakov}$ series, and the NLL${}_{std}$ series only appears once we have the NNLL${}_{sudakov}$ result. If we instead consider $C(\mu,\omega)$ itself, then we have to expand the resulting exponential, and the correspondence becomes more complicated.
78) The Cusp anomalous dimension (L21)
Pre Text: In this component we discuss why the cusp anomalous dimension has this name.
Video
Post Box Comment: The cusp anomalous dimension plays a role in many calculations involving energetic light quarks and gluons. The fact that it is universal, related to kinks from Wilson lines rather than full quark and gluon fields, is very useful since it means that once we know how the cusp enters at one-loop we can immediately infer the coefficients of these terms in the anomalous dimension at higher orders. Currently in QCD the cusp anomalous dimension is known at 3-loop order.
79) When are labels fixed in SCET operators? (L21)
Pre Text: In the example treated so far, the large momentum $\omega$ in our Wilson coefficient was fixed by external kinematics. In this lecture component we discuss the conditions under which this statement is no longer the case. When the momentum is no longer fixed, various equations that were previously simple products of functions, now involve integrations over large momentum variables. Following the standard terminology, we will refer to these integrals as convolutions.
Video
Post Box Question: none
80) DIS Introduction and Operators (L21)
Pre Text: As an explicit application of a couple aspects of SCET, we will consider the separation of short and long distance contributions in deep inelastic scattering (DIS). In this lecture component we establish notation and setup the leading order basis of SCET operators.
Video
Post Box Question: In this lecture we setup the analysis for DIS, including the leading ${\cal O}(\lambda^2)$ operators. Can you guess what type of convolutions will occur in the final DIS factorization theorem?
Post Box Answer: The DIS operators have two collinear building blocks in the same direction. There will be large collinear momenta of these building blocks that is not fixed by the kinematics in the matrix element, so we might guess that the final factorization theorem will have a convolution. We will derive the precise form of this convolution next.
81) DIS factorization (L22)
Pre Text: Using general properties that we have learned about the factorization of hard-collinear modes, it becomes easy to establish the general factorized structure of short and long distance contributions in DIS. In this lecture component we derive the factorization theorem for DIS which involves a convolution of a hard coefficient function for the short distance scattering with parton distribution functions describing the long distance dynamics.
Video
Post Box Question: How do we know that only the combination $\xi/x$ appears in the DIS hard function?
Post Box Question: The Wilson coefficient in SCET depends on $\omega_+$, and our proton matrix element sets $\omega_+ = \xi \bar n \cdot p = (\xi/x) Q$, where the last equality comes from the kinematic identity $\bar n \cdot p = Q/x$ that we derived in the previous lecture. The same result could be derived by dimensional analysis and RPI-III: The Wilson coefficient can depend on hard momenta, $\omega_+$, $\omega_-$ (which is equal to zero), and $n\cdot q = \bar n\cdot q = Q$. Since it is dimensionless and has to be RPI-III invariant, it can only depend on the combination $\omega_+/Q$. The parton distribution is also dimensionless and hence can only depend on $\omega_+ / \bar n\cdot p$. Calling this $\xi$, the Hard function depends on $\xi \bar n\cdot p/Q=\xi/x$. Note that the dependence on the proton $\bar n\cdot p$ enters because of the choice to make a change of variable, and that there is actually no dependence on the hadron variables in this Wilson coefficient. The equality that gives the $x$ dependence requires the use of the kinematic relations.
82) One Loop RGE for PDF with operators (L22)
Pre Text: The operators for DIS provide an example where the large momenta are not completely fixed by momentum conservation. In this lecture component we carry out the one-loop renormalization of these operators, and also show that the RGE equation that is valid to all orders in perturbation theory involves a convolution. Indeed the anomalous dimension we obtain at one-loop is just the leading order DGLAP result with the standard quark splitting kernel.
Video
Post Box Question: none
83) When do we get convolutions? SCET1 versus SCET2 processes (L22)
Pre Text: Lets take for granted that the hard, collinear, and ultrasoft (or soft) components of a process can be factorized into distinct objects or functions. We can then ask the general question, when will there be convolutions between the objects in different momentum sectors? That question is answered here.
Video
Post Box Question: none
Insert Pset #5, part #2
84) e+e- to dijets: Modes & Expansions (L23)
Pre Text: In this chapter we discuss another very useful example, that of dijet production from $e^+e^-$ annihilation. This will give us an opportunity to discuss a nontrivial example of ultrasoft-collinear factorization, as well as introducing the concepts of jet and soft functions. In this first lecture component we setup the required modes and power counting, and discuss in detail a few nontrivial things that we can already learn about dijet production just from these considerations.
Video
Post Box Question: The observable we focussed on measuring was the hemisphere invariant mass $M$. There are two obvious scales in the problem, $Q$ and $M$, but we also found nontrivial effects coming from the ultrasoft scale $M^2/Q$. Why did this happen?
Post Box Answer: The appearance of the ultrasoft scale in the ratio combination $M^2/Q$ is what is sometimes called the SCET${}_{\rm I}$ see-saw (named after the famous see-saw mechanism that generates small left-handed neutrino masses by the ratio of the weak scale over a heavy right handed neutrino mass scale). The appearance of this scale is related to the fact that $p^2 = p^+p^- + p_\perp^2$. For the case at hand the $\perp$-momentum is not playing a role in the measurements, so we have $p^2 = p^+ p^-$. The jet mass measures a $p^2$ and the CM energy fixes a large $p^-=Q$, therefore the scaling represented by the dispersion relation induces dependence on the ultrasoft scale $p^+ \sim M^2/Q$.
85) Observable Factorization & Factorized Cross Section (L23)
Pre Text: In this lecture component we derive a factorized cross section for the dijet hemisphere mass distribution. Beyond just factorizing the corresponding SCET operators, we also discuss the factorization of the measurement. The final cross section result involves hard functions, jet functions, and soft functions.
Video
Post Box Comment: A nice feature of SCET is that one can derive a factorization theorem in a class room lecture. Many of the subtleties about factorization are handled at the level of determining the appropriate fields for the process and constructing the appropriate operators and Lagrangians. Then one must consider an observable that can be suitably factorized. With these things in hand the remaining steps to put things together typically require some mathematical effort, but not further deep thought.
86) Scales & Regions for Thrust (L23)
Pre Text: In this lecture component we discuss the scales in our factorized dijet cross section, and how large logarithms that can be summed by solving the appropriate RGE equations. We also carry out a further factorization of the soft function into perturbative and nonperturbative components. The importance of summing large logarithms, and of accounting for nonperturbative corrections depends on the value of the measurement variable. Using the thrust event shape variable, we illustrate the generic regions that occur. This includes a {\em peak region} where summing logs is important but nonperturbative corrections give leading contributions, {\em a tail region} where the resummation is important and nonperturbative corrections are suppressed, and {\em a fixed order region} where it is important not to carry out resummation.
Video
Post Box Question: The function $F(k)$ describes nonperturbative physics and is related to the hadronization of particles. It varies on scales of $k\sim \Lambda_{\rm QCD}$ and we can assume that $0\le k < \infty$, that $F(k)\ge 0$. Let the j'th moment of $F(k)$ be called $\Omega_j$, so $\Omega_j = \int_0^\infty \!\! dk\: k^j \: F(k)$. $F(k)$ is normalized so $\Omega_0=1$, while the other moments carry non-trivial information. Do we know that $F(k)$ has an exponentially suppressed tail for large $k$? Assuming $F(k)$ has an exponentially suppressed tail, can you construct a formula for it using its moments?
Post Box Answer: Since $$\Omega_j = \int_0^\infty \!\! dk\: k^j \: F(k)$$ are all finite we know that $F(k)$ must fall to zero faster than any polynomial. A criteria for a function to be uniquely reconstructable from its moments is that its moment generating function $$M(t) = \int_{-\infty}^{+\infty} \!\! dk \: e^{t k }\: F(k)$$
exists in an interval including $t=0$. The Taylor series of $M(t)$ about $t=0$ has coefficients which are the moments of $F(k)$. From this criteria we see that if $F(k)$ has (at least) an exponentially suppressed tail, then it is uniquely specified by its moments. The answer to the posed question is actually no, because there exist functions with non-exponential tails, but whose moments all still exist. Generically these functions have moments $\Omega_j$, such that the dimensionless ratios $\Omega_j/(\Omega_1)^j$ increase with increasing $j$. An example is the log-normal distribution $$ F(k) = \frac{\Lambda}{\sqrt{2\pi} k} \exp\Big( -\frac12 \ln^2(k/\Lambda) \Big)\, ,$$ which is an example of a function that is actually not uniquely specified by its moments $\Omega_j = e^{j^2/2}$. There exist different functions that have these same moments. For these moments the dimensionless ratio mentioned above grows quite quickly with increasing $j$. So in order to ensure that $F(k)$ has an exponentially falling tail we have to also add the physically reasonable criteria that the ratio of moments does not grow with increasing $j$, which makes sense given our expectations about the nature of confinement. The moments can actually be specified as matrix elements of operators of increasing dimension, which due to confinement we believe should simply be $\sim \Lambda_{\rm QCD}^j$ without a large prefactor depending on $j$.
If the tail falls exponentially with increasing momentum $k$, then we can write down a distribution valued formula for $F(k)$ via
$$ F(k) = \delta(k) - \Omega_1\: \delta'(k) + \frac{1}{2!} \Omega_2 \delta''(k) + \ldots = \sum_{j=0}^\infty \frac{(-1)^j}{j!}\: \Omega_j\: \delta^{(j)}(k) $$
87) e+e- to dijets: Perturbative Hard, Jet, and Soft Functions (L23)
Pre Text: In this lecture component we consider the form of the one-loop results for the hard, jet, and soft functions, as well as the form of RGE equations for each of these functions.
Video
Post Box Question: none
88) RGE, Cusp anomalous dimension with +-function (L24)
Pre Text: In this lecture component we consider the solution of the RGE equations for dijet production, making use of the form of these equations in Fourier space. We also discuss consistency equations that relate the anomalous dimensions of hard, jet, and soft functions. These consistency equations arise from the general fact that RG evolution yields equivalent results whether we run coefficients or operators.
Video
Post Box Comment: The same result is obtained for the resummation of large logs regardless of the common scale \(\mu\) that we choose for the endpoint of the evolution, where here we run the hard function \(H\) from \(\mu_H\to \mu \), the jet function from \(\mu_J\to \mu\), and the soft function from \(\mu_S\to \mu\). Since we can pick $\mu$ equal to one of \(\mu_{H,J,S}\), it is possible to set things up so that we do not have to evolve one of the functions. As we discussed in lecture, this freedom necessarily implies a relation between the anomalous dimensions of the hard, jet, and soft functions. We wrote down the relation in lecture for the non-cusp part of the anomalous dimensions, \( \gamma_J + \gamma_S = -\gamma_H/2\). There is also a relation for the cusp-part of these anomalous dimensions, and this relation is even more interesting because each of the hard, jet, and soft functions has a different logarithm multiplying the cusp anomalous dimension \(\Gamma^{\rm cusp}[\alpha_s]\). Consistency requires a relation among the variables in these logarithms. It also requires that we can not have more than a single logarithm in the anomalous dimensions (any higher polynomial power of a logarithm can not possibly satisfy the consistency condition). Thus the existence of a factorization theorem for thrust provides one means of proving that the anomalous dimension has at most a single logarithm.
Insert Pset #5, part #3
Chap.#14
Chap.#15