Soft-Collinear EFT

  1. 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.

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  1. 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.

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  1. 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.

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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.


  1. 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 $\slash\!\!\! n\, u_n=0$.

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  1. 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 $(n\cdot A_n,\bar n\cdot A_n,A_n^\perp) \sim (\lambda^2,\lambda^0,\lambda)$.

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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.


  1. 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 $W_n[\bar n\cdot A_n]$.

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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.


  1. 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.

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  1. 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.

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  1. 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.

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  1. 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.

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  1. 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.

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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.


  1. 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.

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Insert Pset #4, part #3

Insert Pset #4, part #4

Chap.#11

  1. 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${}_{\rm II}$.

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  1. 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.

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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.


  1. 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${}_{\rm II}$.

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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.


  1. 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.

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  1. 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.

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  1. 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.

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  1. 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).

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  1. 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.

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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

  1. 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.

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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.


  1. 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.

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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.


  1. 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>.

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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.


  1. 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.

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  1. 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.

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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

  1. 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.

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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

  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.

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  1. 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.

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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

  1. 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.

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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.


  1. 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 $\mu$.

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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).


  1. 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.

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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.


  1. 78) The Cusp anomalous dimension (L21)

Pre Text: In this component we discuss why the cusp anomalous dimension has this name.

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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.


  1. 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.

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Post Box Question: none


  1. 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.

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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.


  1. 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.

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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.


  1. 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.

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Post Box Question: none


  1. 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.

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Post Box Question: none


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  1. 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.

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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 $p^+ \sim M^2/Q$.


  1. 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.

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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.


  1. 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.

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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

$$ F(k) = \delta(k) - \Omega_1\: \delta'(k) + \frac{1}{2!} \Omega_2 \delta''(k) + \ldots = \sum_{j=0}^\infty \frac{(-1)^j}{j!}\: \Omega_j\: \delta^{(j)}(k) $$


  1. 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.

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Post Box Question: none


  1. 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.

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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.


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Chap.#14

Chap.#15