Reparameterization Invariance for Collinear Operators

Aneesh V. Manohar; Dan Pirjol; Iain W. Stewart; Thomas Mehen

REVIEW

Reparameterization Invariance for Collinear Operators

Aneesh V. Manohar, Thomas Mehen, Dan Pirjol, Iain W. Stewart

High Energy Physics - Phenomenology · Submitted 19 Apr 2002 · 15 references

Pith's verdict: The full five-parameter reparameterization argument is correct and closes the loophole in the leading collinear quark Lagrangian; the reader's Eq. (19) scaling worry does not actually threaten the exclusions.

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arXiv IDhep-ph/0204229 (version 1)

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CitationAneesh V. Manohar, Dan Pirjol, Iain W. Stewart, Thomas Mehen. "Reparameterization Invariance for Collinear Operators." arXiv:hep-ph/0204229 (2002). https://pith.science/paper/LXOMH2UC

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Editor's note

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Short version: this is the paper that closes the loophole in the leading collinear quark Lagrangian, and the argument holds up on reading it closely. What is new relative to Chay and Kim (Ref. [15]) is the full five-parameter set of light-cone reparameterizations. Class I was already known. Classes II and III do the real work: class II kills O2 in Eq. (23), class III kills the lambda^2 and lambda^3 operators, and together they force the Lagrangian to be the expansion of Eq. (21) with no free coefficients. The algebra is checkable at each step: Table I, the spinor transformations Eqs. (15)-(17), and the delta(II) computations in Eqs. (26)-(27) and (33)-(34). Nothing is fitted; the coefficients c_a, s_ab, t_ab are driven to zero or to the normalization of the free kinetic term. On the reader's main worry, I think the Eq. (19) scaling assignment is not a soft spot for the claims actually made. Restricting to the subgroup that preserves collinear power counting can only weaken the constraints: an operator that fails under the subgroup fails under the full group too. So the exclusions of O2 and S_ab are safe, and uniqueness under the subgroup gives uniqueness under the full group. The reader scored soundness 7 partly on this; I would put it higher. The stress-test note has this right, and the reader's weakest-assumption flag is misplaced. The genuine soft spots are small. The step from Eq. (35) to Eq. (36) is compressed: the claim that delta(II)O' and the epsilon-perp terms have different Dirac structures glosses over the scalar piece inside iD-slash_perp epsilon-slash_perp, which you have to chase through to see it cannot cancel delta(II)O'. It works out, but the reader has to do it. Same with the 'after some algebra' before Eq. (36). That is Letter-length compression, not a gap. The scope limitation is stated by the authors themselves: footnote 1 and the conclusions say they cannot rule out subleading operators that are reparameterization invariant on their own. Take that at face value. The uniqueness claim is conditional on the expansion starting at lambda^4, and they say so. Who should read it: anyone doing factorization or Sudakov resummation in SCET, and anyone matching QCD currents onto the operator expansion. The L1 and L2 terms in Eq. (22), with no nontrivial Wilson coefficients, are directly usable. Recommendation: accept. A careful referee is worth the time mainly to check the Eq. (35)-(36) step and the S_ab exclusion. The physics is right and the paper is honest about what it does not prove.

Referee report

Significance

If correct, the result is a useful structural statement for SCET: it identifies RI as the remnant of the Lorentz generators broken by the introduction of n and nbar, and turns it into a practical constraint on the operator basis. The exclusion of O2 removes a genuine ambiguity in the leading collinear quark Lagrangian that gauge invariance and power counting alone do not remove, and the identification of the tower of subleading operators tied to the kinetic term (with no independent Wilson coefficients) is directly relevant to factorization proofs and to Sudakov resummation in B and Upsilon decays. The derivation is algebraic and checkable line by line, contains no fitted parameters, and the coefficients c_a, s_ab, t_ab are constrained to zero or to a normalization fixed by the free kinetic term rather than adjusted to any input. The paper is also explicit about the scope of its claim in footnote 1, which is a strength rather than a weakness.

Major objections (2)

[Eqs. (30)-(34)] The exclusion of the S_ab operators, which is load-bearing for the 'complete set' claim, rests on two steps that are asserted rather than shown. First, the statement that the operators appearing in delta(II)S_ab 'cannot be canceled by variations of O_a or T_ab since the latter have a trivial Dirac structure' should be accompanied by the explicit Dirac structure of the variations of O_a and T_ab, so the reader can verify that no gamma-matrix structure is generated. Second, and more importantly, the conclusion s_ab = 0 for all (a,b) except (0,0) requires that the operators N^a iD-slash_perp (1/N^{a+b+1}) iD-slash_perp N^b are linearly independent for distinct (a,b); otherwise different S_ab contributions could cancel among themselves. Please state this independence argument, and also specify the ranges of the sums over a and b in Eq. (30), since the a <-> b symmetrization means only coefficients with a <= b are independent and the powers 1/N^{a+b+1} require a+b+1 >= 1.
[Eqs. (35)-(36)] The final step of the uniqueness proof is compressed. The text states that 'the terms involving delta(II)O' and epsilon-slash_perp must vanish independently since they have different Dirac structures', but iD-slash_perp epsilon-slash_perp contains both a tensor piece and a scalar piece proportional to D_perp · epsilon_perp times the unit Dirac matrix; the scalar piece has the same Dirac structure as delta(II)O' and could in principle cancel against it. Please show explicitly that the tensor piece alone forces O' = n·iD, after which the scalar piece and delta(II)O' both vanish. I expect the argument closes as stated, but as written the last line of the proof of the paper's headline claim is not reproducible from the text.

Minor objections (7)

[Eq. (19) and following paragraph] The scaling assignment Delta_perp ~ lambda, epsilon_perp ~ lambda^0, alpha ~ lambda^0 is justified as the maximum power that leaves the collinear power counting intact. It would help to add the one-line logical remark that restricting to this subgroup can only weaken the constraints, so any operator excluded using the subgroup is also excluded by the full group; this preempts the natural worry that the truncation of the symmetry group makes the resulting constraints non-exhaustive.
[Abstract and footnote 1] The abstract advertises 'the complete set of higher order operators that are related to the collinear quark kinetic term', while footnote 1 concedes that additional terms in the power-suppressed collinear Lagrangians which are reparameterization invariant by themselves are not ruled out. The two statements are compatible but the abstract should be softened slightly, or the footnote cross-referenced there, so the scope of 'complete' is unambiguous.
[Text following Eq. (22)] The sentence that L1 and L2 'have no non-trivial Wilson coefficients' to all orders is established in the body only for the order-lambda^4 basis of Eq. (31); for the lambda^5 and lambda^6 terms the argument relies on footnote 1. A cross-reference to that footnote at this sentence would keep the strength of the claim aligned with what is actually proven.
[Eqs. (30)-(31)] The abbreviation N = nbar·iD is introduced after Eq. (31) but used inside it; move the definition before the operator list. It would also help to state explicitly whether a and b run over non-negative integers and to give the ellipses in Eq. (34) in closed form for the negative-power case, since the pattern of the expansion is used directly in the S_ab argument.
[Table I, Eq. (27), and concluding paragraph] Notation is inconsistent in a few places: epsilon_perp and ǫ_perp are used interchangeably (Table I, Eq. (27)), and the concluding paragraph switches between 'reparameterization' and 'reparametrization'. In the second line of Eq. (33) the factor written as N^a D-slash_perp should presumably read N^a iD-slash_perp. These are cosmetic but worth a pass.
[Eq. (22)] In the second term of L2 the ordering of the two factors of 1/(nbar·iD_c) around nbar·iD_u should be stated as arising from the expansion of 1/(nbar·iD) in Eq. (21), with a reminder of the ordering prescription for the differential operators. A reader cannot otherwise tell whether the displayed ordering is a choice or a consequence.
[Reference [15]] Reference [15] (Chay and Kim) is cited as hep-ph/0201197 with no journal reference. If it has since been published, the entry should be updated; if not, marking it as unpublished would be appropriate given that the comparison with the present L1 result depends on it.

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The paper

The paper claims

The paper argues that requiring SCET to be unchanged under small shifts of the light-cone reference vectors and momentum split uniquely fixes the leading collinear quark Lagrangian and ties an infinite tower of subleading operators to it.

The reading

The paper asks whether the soft-collinear effective theory (SCET) is unchanged when one shifts the split between a collinear particle's large label momentum and its small residual momentum, or when one slightly changes the two light-like reference vectors $n$ and $n$ used to define that split. The authors argue that demanding this reparameterization invariance is not a bookkeeping formality: it is a symmetry that removes operators which gauge invariance alone would allow and relates operators at different orders in the power counting. Their main example is the collinear quark kinetic Lagrangian. They show that the leading operator is unique once gauge invariance, reparameterization invariance, and tree-level matching are imposed, and that an infinite tower of subleading operators related to the kinetic term is fixed by the same symmetry, with no nontrivial Wilson coefficients. They leave open whether there are subleading two-quark operators that are invariant by themselves. If the argument holds, SCET gains a predictive backbone analogous to the reparameterization constraints long used in heavy-quark effective theory.

What carries the argument

The load-bearing device is the set of infinitesimal reparameterization transformations collected in Table I. Type (I) shifts $n$ with $n$ and leaves $n$ fixed; type (II) shifts $n$ and leaves $n$ fixed; type (III) rescales $n$ and $n$. These transformations act on the covariant derivatives, on the collinear spinor $n$, and on the Wilson line $W$ built from $n$. The authors assign $n$, $n$, and $n$, then demand invariance order by order in $n$. The central object carrying the result is the replacement of collinear derivatives by the full covariant derivative $iD=iD_c+iD_u$, which makes the Lagrangian manifestly invariant and generates the tower $L_0,L_1,L_2,...$.

Extended reading notes

Core claim

The central claim is that the ambiguity in decomposing collinear momenta and choosing $n$ and $n$ generates a symmetry of SCET that constrains collinear operators both at a fixed order in $n$ and between different orders. The authors classify the infinitesimal transformations into three types: shifts of $n$ perpendicular to itself, shifts of $n$ perpendicular to itself, and a simultaneous rescaling of $n$ and $n$. Using these, they show that an operator $O_2$ built from two perpendicular covariant derivatives and one factor of $1/(n)$ is collinear-gauge invariant and is not excluded by type (I) alone, but is ruled out by type (II). More generally, they prove that the most general gauge-invariant collinear quark Lagrangian up to order $n$ that respects types (II) and (III) is the manifestly invariant expression of Eq. (21). Expanding that expression in powers of $n$ gives $L_0+L_1+L_2+...$, so the higher-order operators related to the kinetic term are determined by the leading one rather than by independent matching.

Load-bearing premise

The uniqueness claim rests on the power-counting assignment $n$, $n$, and $n$ and on discarding larger transformations that would turn a collinear momentum into a soft one; if that truncation is not the physically correct symmetry, operators excluded by types (II) and (III) could reappear.

Editorial extensions

If this is right

The leading collinear quark kinetic term has its normalization fixed and cannot acquire an anomalous dimension.
The subleading operators $L_1,L_2,...$ obtained by expanding Eq. (21) are tied to $L_0$ and have no independent Wilson coefficients.
The gauge-invariant operator $O_2$ in Eq. (23), which would make the leading kinetic Lagrangian ambiguous order by order, is excluded.
Reparameterization invariance can relate Wilson coefficients of different SCET operators to all orders in perturbation theory, as it does for heavy-quark currents in HQET.
The same symmetry can be used to constrain power-suppressed heavy-to-light and collinear currents, beyond the Lagrangian example treated here.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the symmetry extends cleanly to currents, it would reduce the number of independent Wilson coefficients in factorization theorems for exclusive and semi-inclusive processes, turning some power corrections into symmetry predictions rather than fit parameters.
The most direct stress test is a one-loop calculation of the anomalous dimensions of $L_1$ and $L_2$: the paper's logic predicts zero, so a nonzero result would reveal that the truncation of the transformation group is too restrictive.
The result suggests that SCET's collinear sector may be as rigid as HQET's, with predictive power coming from broken Lorentz symmetry rather than from order-by-order matching.
A natural extension is to ask whether the open question of subleading two-quark operators invariant by themselves can be settled by the same operator basis, which would decide whether the two-quark SCET Lagrangian is fully fixed by symmetry.

Original abstract

We discuss restrictions on operators in the soft-collinear effective theory (SCET) which follow from the ambiguity in the decomposition of collinear momenta and the freedom in the choice of light-like basis vectors $n$ and $\bar n$. Invariance of SCET under small changes in $n$ and/or $\bar n$ implies a symmetry of the effective theory that constrains the form of allowed operators with collinear fields. The restrictions occur at a given order in the power counting as well as between different orders. As an example, we present the complete set of higher order operators that are related to the collinear quark kinetic term.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We discuss restrictions on operators in the soft-collinear effective theory (SCET) which follow from the ambiguity in the decomposition of collinear momenta and the freedom in the choice of light-like basis vectors $n$ and $\bar n$. Invariance of SCET under small changes in $n$ and/or $\bar n$ implies a symmetry of the effective theory that constrains the form of allowed operators with collinear fields. The restrictions occur at a given order in the power counting as well as between different orders. As an example, we present the complete set of higher order operators that are related to the collinear quark kinetic term.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss restrictions on operators in the soft-collinear effective theory (SCET) which follow from the ambiguity in the decomposition of collinear momenta and the freedom in the choice of light-like basis vectors $n$ and $\bar n$.	contribution 0.87	0.70	0.15	asserted 0.97
s1	Invariance of SCET under small changes in $n$ and/or $\bar n$ implies a symmetry of the effective theory that constrains the form of allowed operators with collinear fields.	result 0.59	0.56	0.71	asserted 1.00
s2	The restrictions occur at a given order in the power counting as well as between different orders.	result 0.90	0.58	0.15	asserted 1.00
s3	As an example, we present the complete set of higher order operators that are related to the collinear quark kinetic term.	contribution 0.63	0.93	0.10	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.93.

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References

Works cited (15)

[15]

J. Chay and C. Kim, hep-ph/0201197. 11

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[1]

C. W. Bauer, S. Fleming and M. Luke, Phys. Rev. D 63, 014006 (2001)

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[2]

C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001)

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[3]

C. W. Bauer and I. W. Stewart, Phys. Lett. B 516, 134 (2001)

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[4]

C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 65, 054022 (2002)

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[5]

C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein and I. W . Stewart, hep-ph/0202088. 10

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[6]

C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. 87, 201806 (2001)

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[7]

C. W. Bauer, C. W. Chiang, S. Fleming, A. K. Leibovich and I . Low, Phys. Rev. D 64, 114014 (2001)

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M. E. Luke and A. V. Manohar, Phys. Lett. B 286, 348 (1992)

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M. Neubert, Phys. Lett. B 306, 357 (1993)

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Y. Q. Chen, Phys. Lett. B 317, 421 (1993)

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M. Finkemeier, H. Georgi, and M. McIrvin, Phys. Rev. D 55, 6933 (1997)

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A. V. Manohar, Phys. Rev. D 56, 230 (1997)

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[13]

R. Sundrum, Phys. Rev. D 57, 331 (1998)

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[14]

F. Campanario and T. Mannel, arXiv:hep-ph/0201136

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The Quark Beam Function at NNLL
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The quark beam function is shown to obey the same all-orders renormalization-group equation as the jet function, and its one-loop matching onto quark and gluon PDFs is computed and used to write the NNLL Drell-Yan bea...
Jet Shapes and Jet Algorithms in SCET
hep-ph 2010-01 accept novelty 6.0 of 10

Jet shape distributions in multi-jet events factorize into hard, jet, and soft functions in Soft-Collinear Effective Theory, and their anomalous dimensions cancel at one loop for any jet count, flavor mix, and jet rad...
Factorization Structure of Gauge Theory Amplitudes and Application to Hard Scattering Processes at the LHC
hep-ph 2009-08 accept novelty 6.0 of 10

High-energy gauge-theory scattering amplitudes factor into process-independent one-particle collinear functions times a universal soft function, which lets the authors compute electroweak radiative corrections for all...
Momentum Flow Correlations from Event Shapes: Factorized Soft Gluons and Soft-Collinear Effective Theory
hep-ph 2006-11 accept novelty 6.0 of 10

Boost invariance of lightlike Wilson lines alone implies that leading 1/Q corrections to mean two-jet event shapes are universal, with the universal coefficient given by a Wilson-line matrix element of a single energy...
On the universality of soft emission beyond the next-to-soft order
hep-ph 2026-09 accept novelty 5.0 of 10

The paper extends the soft gluon theorem to all orders using SCET and quantifies the number of non-universal amplitudes for scalar QCD.
Jet Shapes with the Broadening Axis
hep-ph 2014-01 accept novelty 5.0 of 10

Measuring jet angularities about the broadening axis instead of the thrust axis makes them recoil-free and yields one factorization theorem valid for all angularity exponents.
Universality of Nonperturbative Effects in Event Shapes
hep-ph 2006-03 accept novelty 5.0 of 10

Using energy flow operators and boost invariance of eikonal Wilson lines, Lee and Sterman derive the universality of 1/Q power corrections to mean angularities and C-parameter without assuming single-gluon emission.
As Scales Become Separated: Lectures on Effective Field Theory
hep-ph 2019-03 accept novelty 3.0 of 10

A pedagogical set of lecture notes that walks through matching, renormalization group running, and a scalar toy model of Soft Collinear Effective Theory, with every loop integral done explicitly in toy scalar theories.
Introduction to Soft-Collinear Effective Theory
hep-ph 2014-10 accept novelty 2.0 of 10

A self-contained pedagogical construction of Soft-Collinear Effective Theory, from the strategy of regions through the effective Lagrangian to renormalization-group resummation for Drell-Yan and the infrared structure...

How later papers used it

80 later papers cite it

Evidence: some. 58 of the citing papers share no author with it, and 4 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 125 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 3 papers · 1 by the same authors

Used RPI as its main tool.

Factorization and Resummation for Dijet Invariant Mass Spectra (2011) · 1.00 · paraphrased: the licence does not allow quoting
Performed a type-A transformation in this paper's language on the label momentum and a type-IB transformation on the vector n1 itself.

Jet Shapes and Jet Algorithms in SCET (2010) · 0.96 · paraphrased: the licence does not allow quoting

By the same authors

Used reparametrization invariance (RPI-III) from this paper to rescale the jet directions.

Factorization for Jet Radius Logarithms in Jet Mass Spectra at the LHC (2016) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

78 arXiv papers cite this in 121 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 88 sentences without a confident label

Builds on it 1 paper

While these arguments have previously appeared in the literature for four-component spinors [ref] , this is the first derivation that specializes to two-component notation.

Soft-Collinear Supersymmetry · RPI in Two-Components · 0.90

Uses its method or tool 3 papers

The main tool we use is RPI [ref] .

Factorization and Resummation for Dijet Invariant Mass Spectra · All-Order Scale Sensitivity of the Hard Function · 1.00
We can decompose $x$ in $n_1$ light-cone coordinates, so [equation] Performing a type-A transformation (in the language of [ref] ) on the label momentum $p̃_1 = ω_1 n_1/2$, [equation] and a type-IB transformation on the vector $n_1$ itself, [equation] shifts the label momentum on the jet function 1 by $ω_1 n_1/2 →(ω_1 + n̅_1·K) n_1 /2+ K_⊥1$.

Jet Shapes and Jet Algorithms in SCET · Jet Shape Distribution in $e^+ e^-→3 Jets$ · 0.96
This is most easily seen by using reparametrization invariance (RPI-III) [ref] to rescale the jet directions via $n_J →n_J' = n_J β$, $n̅_J →n̅_J' = n_J'/β$ with $β= R/(2 coshη_J)$.

Factorization for Jet Radius Logarithms in Jet Mass Spectra at the LHC · function $S_R$ (regime 2) · 0.98

Compares with it 1 paper

Crucially, since the gauge charges scale with $λ$, a gauge transformation links different orders of $λ$.[This is very similar to $n_±^μ$ reparameterisation (RPI) transformations in QCD SCET [ref] .

Soft-collinear gravity beyond the leading power · QCD vs. gravity · 0.80

Cites it as background 23 papers

Expressions for the leading power Lagrangian can be found in [ref] , and expressions for $L^(1)$, and $L^(2)$ can be found in [ref] (see also [ref] ).

Building Blocks for Subleading Helicity Operators · SCET and Helicity Fields · 0.93
Reparametrization invariance in SCET was first discussed in Ref. [ref] and generalized to the complete set of RPI transformations in Ref. [ref] .

Reparameterization Invariant Collinear Operators · Introduction · 0.89
There has recently been significant progress in understanding the structure of power corrections in the soft and collinear limits [ref] , including the first all order resummation of power suppressed logarithms for collider observables with soft and collinear radiation [ref] and more recently for the case of threshold [ref] .

Helicity Methods for High Multiplicity Subleading Soft and Collinear Limits · Introduction · 0.98

Discussion

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