Factorization of heavy-to-light form factors in soft-collinear effective theory

M.Beneke (RWTH Aachen); Th.Feldmann (CERN)

REVIEW

Factorization of heavy-to-light form factors in soft-collinear effective theory

M.Beneke (RWTH Aachen), Th.Feldmann (CERN)

High Energy Physics - Phenomenology · Submitted 26 Nov 2003 · 33 references

Pith's verdict: The two-step matching and the λ^8 operator basis are the real contribution; the u-integral convergence is argued by analogy to the toy integral, and the authors say so themselves.

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arXiv IDhep-ph/0311335 (version 1)

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CitationM.Beneke (RWTH Aachen), Th.Feldmann (CERN). "Factorization of heavy-to-light form factors in soft-collinear effective theory." arXiv:hep-ph/0311335 (2003). https://pith.science/paper/UL4SLZG2

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Editor's note

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This is the paper that put the two-step matching for B→π form factors on a proper footing, and it is still worth reading for that. The real contribution is not the slogan 'factorization to all orders' — it is the explicit construction: QCD → SCET(hc,c,s) → SCET(c,s), with the λ^8 operator basis identified (Eq. 83), the power-counting argument in Section 4.3 (Eqs. 101–107) that eliminates everything except the four-quark operator, and the demonstration that only φ_B+ survives because the φ_B− trace vanishes. That last point is load-bearing, not cosmetic: the φ_B− moments do not converge, so its absence matters. The tree-level matching in Section 3 is careful, and the toy integral in Section 2 is the clearest statement in the literature of why hard-collinear and collinear modes must be distinguished. The Section 4.4 critique of [15] is sharp and, as far as I can tell, correct, though it reads a little like a priority claim; that is not a physics problem. The soft spot is where your stress test puts it. Convergence of the u-integral is argued by correspondence: a collinear endpoint divergence should match a soft one, the soft moments (117) converge, therefore the u-integral does. That correspondence is shown for the scalar toy integral and assumed in general. The authors say as much in Section 5 — 'at present it is not clear how to implement the concept of soft-collinear factorization in general and in practice.' So this is an acknowledged gap, not a concealed one, and the definition of ξπ (Eq. 92) is explicitly built to absorb the endpoint-divergent pieces. But the absorption covers ξπ, not T_i, and T_i's convergence is the claim at issue. If a collinear endpoint divergence could arise without a soft partner — a different topology, or u→1 rather than u→0 — the argument fails. I doubt that happens, but it is not proven. Minor: the paper is honest that it does not sum large logarithms and does not pin the parametric size of ξπ beyond λ^3. Anyone doing exclusive B decays or SCET power counting should read this. It is a standard reference and deserves to be.

Referee report

Significance

If correct, this is an important paper: it supplies the first all-orders argument for the conjectured heavy-to-light factorization formula, and it does so within a controlled effective-field-theory framework rather than by one-gluon-exchange inspection. Several results are genuinely new and worth naming. The tree-level matching of Section 3 is explicit and detailed, including the observation that three-particle light-cone distribution amplitudes of the pion and the B meson enter at leading power (Eq. 55 and the surrounding discussion), which was not anticipated in earlier work. The power-counting argument of Section 4.3 (Eqs. 101–107) is a clean, checkable derivation that the hard-collinear gluon operator matches only onto a four-quark operator with one factor each of 1/(in₋∂) and 1/(in₊∂), and the resulting absence of φ_B− from the factorization formula is a non-trivial and physically consequential corollary. The paper is also candid about what it does not do, notably in Section 4.2 remark (iii) and in Section 5, and that candour is itself a strength.

Major objections (3)

[Section 4.3, after Eq. (117)] Point 3 of Section 4 states convergence of the convolution integrals as part of the theorem, but the convergence of the collinear u-integral is argued by analogy rather than proved. The text reads: 'a collinear endpoint divergence should match to a soft endpoint divergence to make the sum of all terms unambiguous. But the soft convolution integral is convergent, so the u-integral must be convergent, too.' The soft-collinear endpoint correspondence on which this rests is demonstrated only for the scalar toy integral of Section 2 (Eqs. 10–15), where the δ-pole of the collinear integral cancels against the δ-pole of the soft integral. The paper does not establish that every collinear endpoint divergence, including those from other diagram topologies or from the u→1 endpoint, has a matching soft divergence. Since the definitional absorption of endpoint divergences into ξ_π (Eq. 92) covers only the ξ_π term and not the hard-scattering term T_i, this is precisely the point at issue. The authors should either qualify the convergence statement in Eq. (73) as conditional on the endpoint correspondence, or supply a direct argument for the u-integral.
[Section 4.3, Eq. (117) and following paragraph] The convergence of the soft moments λ_B^{(n)} is stated to follow provided φ_B+(ω) ∼ ω^a with a > 0 as ω → 0 and ∼ ω^b with b < 0 as ω → ∞, and the text concludes 'It seems safe to conclude that the moments (117) exist.' This is an assumption about non-perturbative input, supported by the Wandzura-Wilczek approximation and by QCD sum rule estimates, not a derivation. Since the convergence claim of point 3 is a stated part of the theorem, the theorem should make explicit that it holds under these endpoint assumptions on φ_B+, in the same way that the paper is explicit elsewhere about what is assumed and what is derived.
[Section 4.1, Eqs. (84)–(87)] The gauge-invariant operators O⁰_j and O¹_j in SCET(hc,c,s) are constructed by what the text calls 'a simple remedy', namely including the transverse position in the definition of x₋ and multipole-expanding soft fields around x₋ = (n₊x)n₋/2 + x⊥, with the gauge transformation of Eq. (84). The text states that gauge invariance in SCET(hc,c,s) with both collinear and hard-collinear modes 'has not been discussed in the literature so far' and that the authors 'do not intend to give a systematic discussion here.' Because Eq. (87), the representation of the QCD currents, is the foundation of the entire factorization proof, a short demonstration that this construction is consistent (for instance, that the modified multipole expansion does not spoil the power counting used in Section 4.3) would materially strengthen the argument. At minimum the status of Eq. (84) as an assumption should be stated where Eq. (87) is introduced.

Minor objections (7)

[Section 4.3, Eq. (100)] The list of objects contains '1in−∂, 1in−∂' twice; the second entry appears to be intended as 1/(in₊∂), which is listed separately in Table 2. The duplication should be corrected.
[Section 3.3, Eq. (70)] The factor 'saitbj' in the coefficient-function argument should presumably read s_i^a t_j^b; the superscripts are lost in the present typesetting.
[Section 4.3, Eq. (117)] The notation λ_B^{(n)} is introduced without a clear statement of how it relates to the λ power-counting parameter used throughout; a one-sentence clarification would avoid confusion between the two uses of λ.
[Section 4.4] The comparison with Ref. [15] is substantive and worth keeping, but several sentences read as a rebuttal rather than an assessment ('in our opinion the factorization arguments of [15] are incomplete or incorrect in three crucial aspects'). A more neutral formulation that states the three points and leaves the reader to judge would serve the same purpose.
[Section 2.1, Eqs. (10)–(15)] The toy integral is instructive, but the text should state explicitly that the δ-pole cancellation between the collinear and soft contributions is a property of this particular integral and is not being offered as a general theorem; otherwise the reader may over-read the example as establishing the correspondence invoked later in Section 4.3.
[Section 3.2.5, Eq. (63)] The step from Eq. (60) to Eq. (63) drops the s- and t-integrations by contour/δ-function arguments that are not spelled out; a brief indication of how the exponentials collapse to the 1/(1−u) and 1/ω factors would help the reader verify the trace structure.
[Table 1] The two-terminology table is genuinely useful and should be retained; a sentence in the caption noting that 'collinear' in this paper corresponds to 'ultracollinear' in Terminology I would make the mapping unambiguous at a glance.

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The paper

The paper claims

The paper argues that the three independent $B$ to $π$ transition form factors at large pion energy all reduce, to all orders in the strong coupling and to leading power in $1/m_b$, to one universal function $ξ_π$ times a short-distance系数,

The reading

The paper asks whether a conjecture about $B$ to $π$ transition form factors at large pion recoil can be lifted from a one-loop check to a theorem valid to all orders in the strong coupling. The conjecture is that the three independent form factors $f_+$, $f_0$ and $f_T$ collapse into a single universal function $ξ_π(q^2)$ multiplied by a short-distance coefficient, plus a hard-scattering term in which a calculable kernel is convolved with the pion and $B$-meson light-cone distribution amplitudes. If that is right, the complicated non-perturbative physics of the soft spectator quark is captured by one number, and everything else is organized by perturbation theory at the scales $m_b$ and $√{m_b Λ}$. The authors prove the formula by matching QCD in two steps onto soft-collinear effective theory, first removing hard modes of virtuality $m_b^2$ and then removing hard-collinear modes of virtuality $m_bΛ$. The price of the proof is that $ξ_π$ is defined as a matrix element that already contains hard-collinear, collinear and soft dynamics, so it is not itself decomposed into separate factors.

What carries the argument

The load-bearing device is a two-step matching: QCD to SCET(hc,c,s), then SCET(hc,c,s) to SCET(c,s). The second step is controlled by a power-counting ledger that assigns each operator building block a $λ$ scaling, a boost scaling and a mass dimension, then solves the boost-invariance and dimension constraints for the number of inverse derivative factors. For the operator $ξ̄_C A_{⊥hc} h_v$ with mass dimension four, the ledger forces $[λ] ≥ 8$ and admits only the solution with no extra gluons, no derivatives and one factor each of $1/(in_-∂)$ and $1/(in_+∂)$. Boost invariance then restricts the hard-collinear coefficient functions to depend on $ω$ only through powers of $ln ω$, so the soft integrals become the moments $λ_B^{(n)} = ∫ dω φ_{B+}(ω) ω ln^n(m_B ω/μ^2)$, which converge under mild endpoint behaviour of $φ_{B+}$. A scalar toy integral, expanded by momentum regions, supplies the interpretive frame: hard-collinear, collinear and soft regions each contribute at leading order, and the collinear and soft endpoint divergences cancel only in their sum.

Extended reading notes

Core claim

The central claim is the formula $f_i(q^2) = C_i(q^2) ξ_π(q^2) + ∫ dω ∫ du T_i(q^2; ω, u) φ_{B+}(ω) φ_π(u)$ for $i = +, 0, T$, valid to all orders in $α_s$ and to leading power in $1/m_b$, with $ξ_π$ independent of $i$ and with convergent convolution integrals. The authors establish it by showing that the leading QCD currents match onto only three soft-collinear operator structures, $ξ̄_{hc}Γ'_j h_v$, $ξ̄_{hc}A_{⊥hc}Γ'_j h_v$ and $ξ̄_c A_{⊥hc}Γ'_j h_v$, and that the last of these, once hard-collinear modes are integrated out, matches onto four-quark operators $(ξ̄_c ξ_c)(ξ̄_s h_v)$ carrying exactly one factor each of $1/(in_+∂)$ and $1/(in_-∂)$ and no additional fields or derivatives. That restricted structure is what makes the matrix element factorize into a pion distribution amplitude times a $B$-meson distribution amplitude, with a hard kernel in between. A by-product is that only $φ_{B+}$ appears and $φ_{B-}$ is absent, which matters because the analogous moments of $φ_{B-}$ are not expected to converge.

Load-bearing premise

The convergence of the pion-side convolution integral is not shown directly; the authors argue that any endpoint divergence there must be matched by a soft endpoint divergence, that the soft moments are convergent, and therefore the pion integral must converge, an argument demonstrated only on the toy scalar integral and then assumed to hold in general.

Editorial extensions

If this is right

The three $B$ to $π$ form factors become one unknown function plus a calculable hard-scattering correction, so a measurement of any one form factor constrains the other two up to perturbative corrections.
The same reduction extends to light vector mesons, where the seven independent form factors collapse to two, $ξ_⊥$ and $ξ_∥$, plus the same type of hard-scattering term.
Three-particle light-cone distribution amplitudes of the pion and the $B$ meson enter at leading power, contrary to what a two-particle treatment would suggest, but they can be absorbed into the definition of $ξ_π$.
Power corrections to the form factors are suppressed by $1/m_b$ rather than $1/√{m_b}$, settling a question the earlier effective-theory treatments left open.
Because $ξ_π$ is defined as a matrix element in the theory that still contains hard-collinear modes, its scale dependence is not simply that of a conventional distribution amplitude, and it cannot be split into soft and collinear factors without confronting endpoint divergences.

Reading between the lines

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

The proof strategy suggests a general template: whenever a conjectured factorization fails at the endpoints, one can define the non-perturbative object in the intermediate theory so that it swallows the divergent region, then prove that only a restricted operator class survives the second matching. That template should transfer to other exclusive processes where endpoint divergences appear.
The paper's own remark that Figure 2 translates to the pion form factor at sub-leading power implies that the same ledger-and-boost-invariance argument could be run for two-collinear-mode processes, which would give a systematic way to classify which endpoint divergences are genuine and which are artifacts of splitting one mode into two.
If the collinear endpoint divergence really is tied to a soft one as argued, then the convergence of the $u$-integral should be checkable independently by computing the kernel $T_i$ at two loops and looking for a $1/u$ or $1/(1-u)$ pole that is not cancelled; a clean two-loop check would either confirm the correspondence or expose it as an accident of the toy model.
Since $ξ_π$ is defined to include hard-collinear effects, its value depends on a factorization scale in a way that is not the standard distribution-amplitude evolution; a reader wanting to connect this to phenomenology would need the anomalous dimension of the defining operator, which the paper explicitly leaves for future work.

Original abstract

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections. In this paper we extend our previous investigation of heavy-to-light currents in soft-collinear effective theory to final states with invariant mass Lambda^2 as is appropriate to exclusive B meson decays. The effective theory contains soft modes and two collinear modes with virtualities of order m_b*Lambda (`hard-collinear') and Lambda^2. Integrating out the hard-collinear modes results in the hard spectator-scattering contributions to exclusive B decays. We discuss the representation of heavy-to-light currents in the effective theory after integrating out the hard-collinear scale, and show that the previously conjectured factorization formula is valid to all orders in perturbation theory. The naive factorization of matrix elements in the effective theory into collinear and soft matrix elements may be invalidated by divergences in convolution integrals. In the factorization proof we circumvent the explicit regularization of endpoint divergences by a definition of the universal form factors that includes hard-collinear, collinear and soft effects.

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The abstract, sentence by sentence

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

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections. In this paper we extend our previous investigation of heavy-to-light currents in soft-collinear effective theory to final states with invariant mass Lambda^2 as is appropriate to exclusive B meson decays. The effective theory contains soft modes and two collinear modes with virtualities of order m_b*Lambda (`hard-collinear') and Lambda^2. Integrating out the hard-collinear modes results in the hard spectator-scattering contributions to exclusive B decays. We discuss the representation of heavy-to-light currents in the effective theory after integrating out the hard-collinear scale, and show that the previously conjectured factorization formula is valid to all orders in perturbation theory. The naive factorization of matrix elements in the effective theory into collinear and soft matrix elements may be invalidated by divergences in convolution integrals. In the factorization proof we circumvent the explicit regularization of endpoint divergences by a definition of the universal form factors that includes hard-collinear, collinear and soft effects.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections.	background 1.00	0.05	0.09	speculative 0.95
s1	In this paper we extend our previous investigation of heavy-to-light currents in soft-collinear effective theory to final states with invariant mass Lambda^2 as is appropriate to exclusive B meson decays.	contribution 0.98	0.84	0.04	asserted 0.89
s2	The effective theory contains soft modes and two collinear modes with virtualities of order m_b*Lambda (`hard-collinear') and Lambda^2.	method 0.74	0.24	0.03	asserted 1.00
s3	Integrating out the hard-collinear modes results in the hard spectator-scattering contributions to exclusive B decays.	method 0.45	0.38	0.04	asserted 1.00
s4	We discuss the representation of heavy-to-light currents in the effective theory after integrating out the hard-collinear scale, and show that the previously conjectured factorization formula is valid to all orders in perturbation theory.	result 0.93	0.94	0.87	asserted 1.00
s5	The naive factorization of matrix elements in the effective theory into collinear and soft matrix elements may be invalidated by divergences in convolution integrals.	background 0.70	0.19	0.03	hedged possibility 1.00
s6	In the factorization proof we circumvent the explicit regularization of endpoint divergences by a definition of the universal form factors that includes hard-collinear, collinear and soft effects.	method 0.90	0.93	0.17	asserted 0.99

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

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sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
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levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Photon vertex correction to B¯ → γlν. Γ denotes the weak b → u decay vertex. We consider the corresponding vertex integral with all lines simplified to scalar propagators and all vertex factors set to 1. The integral I is ultraviolet and infrared finite, but dimensional regularization will be needed to construct the expansion. The +iǫ prescription for the propagators is understood. The external momenta of the vertex subgraph are: a collinear “photon” momentum p ′ = (n+p ′ , p′ ⊥, n−p ′ ) ∼ (1, 0, 0) with p ′ 2 = 0; a soft “light quark” momentum l ∼ (λ 2 , λ2 , λ2 ) with l 2 = m2 ; a…

**Figure 2.** Divergence structure of the integral (3) and its off-shell, massless version when expanded by regions. The arrows indicate the divergences in different regions that are related and cancel each other. The dashed lines mark the various factorization steps. 10

**Figure 3.** Diagrammatic and operator/matrix element representation of the hardcollinear (left column), soft (middle column) and collinear contribution to the diagram of

**Figure 4.** Kinematics of an exclusive heavy-to-light transition in SCET

**Figure 5.** Infinite sets of Feynman graphs with attachments of soft gluons to collinear quarks and vice-versa. Integrating out the intermediate hard-collinear propagators leads to the Wilson lines Y † s and Wc, respectively, see (37). diagrams result in6 ψ¯ c

**Figure 6.** Tree diagrams that contribute to the form factor at lead

Figures reproduced from arXiv:hep-ph/0311335 by the authors.

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The four-loop quark and gluon rapidity anomalous dimensions are obtained and used to resum the energy-energy correlation in the back-to-back limit at N4LL, the first event-shape resummation at fourth logarithmic order.
One-loop matching of QCD currents to power-suppressed two-jet operators
hep-ph 2026-07 unverdicted novelty 7.0 of 10

Computes NLO matching of QCD currents to power-suppressed two- and three-particle two-jet operators in SCET up to O(λ²), with endpoint factorization relations shown.

Show 44 more citing papers

Drell-Yan $q_T$ Resummation of Fiducial Power Corrections at N$^3$LL
hep-ph 2020-06 accept novelty 7.0 of 10

Linear fiducial power corrections in Drell-Yan q_T spectra arise solely from the leptonic tensor, making them computable and resummable to N3LL, which fixes q_T subtractions and the lepton p_T spectrum at the Jacobian peak.
Power-enhanced leading-logarithmic QED corrections to $B_q \to \mu^+\mu^-$
hep-ph 2019-08 conditional novelty 7.0 of 10

A systematic SCET treatment of the power-enhanced QED correction to Bq->mu+mu- is derived, leading QED and QED-QCD logarithms are resummed, and updated branching fractions are presented.
First Subleading Power Resummation for Event Shapes
hep-ph 2018-04 accept novelty 7.0 of 10

Subleading-power logarithms in thrust are resummed to all orders and shown to exponentiate into a Sudakov factor governed by the cusp anomalous dimension, with a new class of theta-function jet and soft operators requ...
Resummation of Transverse Momentum Distributions in Distribution Space
hep-ph 2016-11 conditional novelty 7.0 of 10

A distributional scale-setting prescription allows the QCD renormalization-group evolution of the transverse-momentum spectrum to be carried out entirely in momentum space, resumming the logarithmic plus distributions...
Drell-Yan production at small q_T, transverse parton distributions and the collinear anomaly
hep-ph 2010-07 accept novelty 7.0 of 10

The product of two transverse-position-dependent parton distributions carries an anomalous, exponentiated dependence on the hard scale, and once this is separated the Drell-Yan q_T spectrum resums in closed form, yiel...
Factorization at Subleading Power and Irreducible Uncertainties in $\bar B\to X_s\gamma$ Decay
hep-ph 2010-03 conditional novelty 7.0 of 10

Resolved-photon contributions in B-bar to X_s gamma obey a new factorization formula with extra soft and jet functions, and generate non-local 1/m_b corrections that leave an irreducible about 5 percent theory uncerta...
The Zero-Bin and Mode Factorization in Quantum Field Theory
hep-ph 2006-05 conditional novelty 7.0 of 10

A Lagrangian-level zero-bin subtraction removes double counting between soft, ultrasoft, and collinear fields, converting spurious infrared singularities into ultraviolet divergences that renormalize cleanly.
Soft Collinear Effective Theory for Heavy QCD Axions
hep-ph 2026-08 conditional novelty 6.0 of 10

A new dimension-seven operator is generated by the gluon equation of motion in the presence of the axion-gluon coupling, and its spectator-scattering contribution to B to K a is roughly 25 percent of the soft-overlap ...
SCET sum rules for $B\to D_1(2420)$ and $B\to D_1'(2430)$ form factors at next-to-leading order
hep-ph 2026-07 conditional novelty 6.0 of 10

First O(αs) SCET light-cone sum rules for B o D1 and B o D1' form factors yield R(D1)=0.070+0.028-0.018 and R(D1')=0.159+0.032-0.025.
$B_c \to \eta_c$ form factors at large recoil: SCET analysis and a three-loop consistency check
hep-ph 2026-05 unverdicted novelty 6.0 of 10

SCET factorization confirms the double-logarithmic resummation for B_c to eta_c form factors up to three loops and derives the iterative structure from RG equations of light-cone distribution amplitudes with cutoff re...
Random Reshuffling-Based Distributed Nash Equilibrium Seeking
math.OC 2026-04 unverdicted novelty 6.0 of 10

Random reshuffling yields distributed Nash-seeking algorithms that, under partial decision information, converge linearly to a neighborhood (constant steps) or exactly a.s./in mean square (diminishing steps), outperfo...
SCET sum rules for $\Lambda_b \to \Lambda \ell^+\ell^-$, $\Lambda \gamma$ decays
hep-ph 2025-06 conditional novelty 6.0 of 10

SCET light-cone sum rules give NLO soft form factors and small non-factorizable corrections for Λ_b → Λ decays, with a Λ_b → Λ γ branching fraction consistent with data.
Precision calculations of $B\to K^*$ form factors from SCET sum rules beyond leading-power contributions
hep-ph 2024-12 conditional novelty 6.0 of 10

New subleading-power corrections to B to K* form factors from SCET sum rules yield SM branching fractions BR(Bbar0 to Kbar*0 nu nubar)=8.09(96)e-6 and F_L=0.44(4).
Dispersive Analysis of $B\to K^{(*)}$ and $B_s\to \phi$ Form Factors
hep-ph 2023-05 accept novelty 6.0 of 10

Including below-threshold B_s pi0 and B_s pi0 pi0 branch cuts in a joint dispersive fit of the B to K(*), B_s to phi form factors produces parametrizations whose truncation error is bounded rather than guessed.
Violation of the Kluberg-Stern--Zuber theorem in SCET
hep-ph 2019-07 conditional novelty 6.0 of 10

Equation-of-motion operators do mix into physical operators in SCET at subleading power, so a preferred field basis is required to reproduce QCD infrared singularities.
Subleading Power Factorization with Radiative Functions
hep-ph 2019-05 conditional novelty 6.0 of 10

The paper derives all-orders gauge-invariant bare factorizations for subleading-power soft quark and gluon emission in SCET, defines the resulting radiative functions as multilocal lightcone operator insertions, and c...
Towards Quasi-Transverse Momentum Dependent PDFs Computable on the Lattice
hep-ph 2019-01 conditional novelty 6.0 of 10

At one loop, the natural lattice definition of a quasi-TMDPDF fails an infrared consistency test on the transverse separation b_T, while a modified 'bent' soft operator passes it and yields an explicit matching coefficient.
Subleading Power Rapidity Divergences and Power Corrections for $q_T$
hep-ph 2018-12 accept novelty 6.0 of 10

A new rapidity-only regulator makes subleading-power rapidity divergences tractable, and the resulting one-loop power correction to color-singlet q_T spectra is shown to involve derivatives of parton distribution functions.
Determining the Nonperturbative Collins-Soper Kernel From Lattice QCD
hep-ph 2018-10 conditional novelty 6.0 of 10

A ratio of lattice-computable quasi-TMDPDFs at two hadron momenta isolates the nonperturbative Collins-Soper evolution kernel, with soft contributions canceling between the two momenta.
A Complete Basis of Helicity Operators for Subleading Factorization
hep-ph 2017-03 conditional novelty 6.0 of 10

A complete basis of helicity hard-scattering operators through O(lambda squared) is constructed for two-jet-direction processes in SCET, with tree-level matching coefficients for the operators that contribute to dijet...
A Subleading Operator Basis and Matching for $gg \to H$
hep-ph 2017-03 accept novelty 6.0 of 10

A complete, helicity-organized basis of power-suppressed SCET operators for gluon-fusion Higgs production, with tree-level Wilson coefficients computed for every operator that contributes to the cross section at order...
Perturbative corrections to $B \to D$ form factors in QCD
hep-ph 2017-01 accept novelty 6.0 of 10

One-loop hard and jet functions for the vacuum-to-B correlation function with a D interpolating current are computed under the counting mc ~ sqrt(Lambda m_b), yielding NLL sum rules for B to D form factors.
Building Blocks for Subleading Helicity Operators
hep-ph 2016-01 conditional novelty 6.0 of 10

A complete-in-SCET-I set of scalar helicity building blocks, plus an angular momentum selection rule, lets subleading-power SCET operators be enumerated by simple multiplication instead of Lorentz and Dirac index contraction.
QCD corrections to $B \to \pi$ form factors from light-cone sum rules
hep-ph 2015-06 conditional novelty 6.0 of 10

The authors compute the O(alpha_s) corrections to B to pi form factors in the B-meson distribution-amplitude light-cone sum rule, prove one-loop factorization with the method of regions, and extract |Vub| = 3.05 x 10^-3.
Reassessing the discovery potential of the $B \to K^{*} \ell^+\ell^-$ decays in the large-recoil region: SM challenges and BSM opportunities
hep-ph 2014-12 conditional novelty 6.0 of 10

Allowing power corrections to the heavy-quark limit to float reduces the B to K* mu mu angular anomaly from 3.7 sigma to below 2 sigma, and new muon-to-electron angular ratios are proposed as clean lepton-universality tests.
Soft Theorems from Effective Field Theory
hep-th 2014-12 accept novelty 6.0 of 10

A new one-loop subleading soft theorem for gauge-theory amplitudes, derived from soft-collinear effective theory, replaces the tree-level Low-Burnett-Kroll theorem that breaks down when collinear particles or loops ar...
A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory
hep-ph 2012-02 conditional novelty 6.0 of 10

A new renormalization-group scale and equation (the rapidity renormalization group) lets theorists factorize and resum rapidity logarithms in soft-recoil-sensitive QCD observables, demonstrated on the Sudakov form fac...
B meson distribution amplitude from B --> \gamma l \nu
hep-ph 2011-10 conditional novelty 6.0 of 10

Including NLL radiative and 1/mb power corrections roughly halves the predicted B to gamma lepton neutrino branching fraction, lowering the BABAR-derived lower bound on lambda_B from about 670 MeV to about 310 MeV.
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...
Exclusive processes in position space and the pion distribution amplitude
hep-ph 2007-09 conditional novelty 6.0 of 10

The two-electromagnetic-current correlator between vacuum and pion is computed to NNLO plus twist-4 in position space, enabling a lattice extraction of the pion distribution amplitude that avoids operator renormalization.
Precision calculations of $B \to V$ form factors in QCD
hep-ph 2019-07 conditional novelty 5.0 of 10

Light-cone sum rules with B-meson distribution amplitudes, computed at next-to-leading order and up to twist-six, yield B to vector form factors and give |Vub| = (3.05 +0.67-0.52) x 10^-3 from B to rho l nu and (2.54 ...
QCD calculations of $B \to \pi, K$ form factors with higher-twist corrections
hep-ph 2018-10 conditional novelty 5.0 of 10

Adding twist-five and twist-six B-meson distribution-amplitude corrections shifts the B to pion vector form factor by 20 to 30 percent and gives |Vub| = (3.23 +0.66-0.48) x 10^-3.
Weak Decays of Doubly Heavy Baryons: SU(3) Analysis
hep-ph 2017-07 conditional novelty 5.0 of 10

Flavor SU(3) symmetry yields sum rules connecting weak decay widths and CP asymmetries of doubly heavy baryons, providing testable predictions for future experiments.
Weak Decays of Doubly Heavy Baryons: the $1/2\to 1/2$ case
hep-ph 2017-07 conditional novelty 5.0 of 10

Light-front quark-diquark form factors yield predicted semileptonic and nonleptonic branching ratios, mostly at the 10^-2 to 10^-4 level, for weak decays of doubly charmed, doubly bottom, and bottom-charm baryons into...
Analytic Regularization in Soft-Collinear Effective Theory
hep-ph 2011-12 conditional novelty 5.0 of 10

In massless gauge theories, adding an analytic regulator only to the final-state phase-space measure makes each momentum region of the effective theory well defined while preserving gauge invariance and the eikonal structure.
Theoretical and Phenomenological Constraints on Form Factors for Radiative and Semi-Leptonic B-Meson Decays
hep-ph 2010-04 accept novelty 5.0 of 10

Two-term series expansions, constrained by newly computed NLO tensor-current dispersive bounds, describe B to K, rho, K* and phi form factors across the whole decay region and reproduce lattice points not used in the fits.
Charmless non-leptonic B_s decays to PP, PV and VV final states in the pQCD approach
hep-ph 2007-03 conditional novelty 5.0 of 10

The pQCD approach, at leading order in alpha_s, reproduces the first CDF B_s branching ratios and CP asymmetry within errors and yields a comprehensive table of predictions for B_s to PP, PV, and VV decays.
Radiative B to tensor meson decays at NLO in SCET
hep-ph 2025-05 conditional novelty 4.0 of 10

An NLO SCET calculation predicts B(B to K2*(1430) gamma) = (16.7 ± 6.4 ± 1.2) × 10^-6, close to the experimental average, with much smaller predictions for the a2 and f2 modes.
Subleading-power corrections to the radiative leptonic $B \to \gamma \ell \nu$ decay in QCD
hep-ph 2018-03 conditional novelty 4.0 of 10

The twist-two hadronic photon contribution shifts the leading-power B to gamma form factors by about 30 percent at next-to-leading-logarithmic accuracy, while higher-twist photon effects add only 3 to 5 percent.
The Path forward to N$^3$LO
hep-ph 2022-03 accept novelty 3.0 of 10

N3LO QCD corrections shift key LHC cross sections by a few percent and cut scale uncertainty to the percent level, but widespread use still needs four-loop splitting functions, N3LO PDFs, and better subtraction schemes.
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.
B, D and K decays
hep-ph 2008-01 accept novelty 3.0 of 10

A community review that fixes the pre-LHC status of quark-flavour physics, catalogues the benchmark rare decays most sensitive to new physics, and quantifies the hadronic uncertainties that limit them.
Recent Advances in Transverse-Momentum--Dependent Distributions in Lattice QCD
hep-lat 2026-10 accept novelty 2.0 of 10

A review of lattice QCD progress in computing TMDs via LaMET, covering the Collins-Soper kernel, soft functions, and TMDPDFs.
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

75 later papers cite it

Evidence: substantial. 66 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 13 of the 146 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 of 10 papers with this label · 1 by the same authors

According to the power-counting scheme $ω_M∼ω_s∼𝒪(λ^2)$ and $m_s∼𝒪(λ)$ that employed in Ref. [ref] with the units $λ= Λ_QCD/m_b$, we can derive the scaling rule for leading-power $B→K^*$ form factors, [equation] where $A_12$ and $T_23$ denote the combinations of the form factors corresponding to $m_B+m_V/n ·pA_1-m_B-m_V/m_BA_2$ and $m_B/n ·pT_2-T_3$, respectively.

Precision calculations of $B\to K^*$ form factors from SCET sum rules beyond leading-power contributions (2024) · Higher-twist four-particle contribution · 0.85
In this work, we adopt the analytic regulator [ref] , meaning that we raise the relevant propagators to complex powers $b_i$.

Region analysis of QED massive fermion form factor (2023) · Topology classification · 0.94

By the same authors

Used the non-dimensional regulator from this paper to regularize the endpoint singularity at one-loop order, rendering all integrals well-defined.

Power-enhanced leading-logarithmic QED corrections to $B_q \to \mu^+\mu^-$ (2019) (same authors) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

70 arXiv papers cite this in 145 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 · 89 sentences without a confident label

Contests it 1 paper

The first SCET-based calculation[Earlier attempts of $$-resummation in SCET [ref] missed the effects of rapidity divergences.] was carried out in Becher:2010tm, Becher:2011xn, Becher:2012yn in the so-called collinear anomaly framework using an analytic regulator [ref] .

Resummation of Transverse Momentum Distributions in Distribution Space · Practical implementations · contest 0.77

Uses its method or tool 10 papers

Then, the soft-collinear effective theory (SCET) [ref] and the heavy quark effective theory (HQET) [ref] are used to embody those dynamic modes, thereby calculating the effective amplitudes and the respective differential cross sections.

Projected transverse momentum resummation in top-antitop pair production at LHC · Introduction · 0.96
In our discussion, we will closely follow the detailed treatments of the two-step matching procedure presented in [ref] .

The Simplest B Decay, Precisely · Matching to Soft-Collinear Effective Theory · 0.89
According to the power-counting scheme $ω_M∼ω_s∼𝒪(λ^2)$ and $m_s∼𝒪(λ)$ that employed in Ref. [ref] with the units $λ= Λ_QCD/m_b$, we can derive the scaling rule for leading-power $B→K^*$ form factors, [equation] where $A_12$ and $T_23$ denote the combinations of the form factors corresponding to $m_B+m_V/n ·pA_1-m_B-m_V/m_BA_2$ and $m_B/n ·pT_2-T_3$, respectively.

Precision calculations of $B\to K^*$ form factors from SCET sum rules beyond leading-power contributions · Higher-twist four-particle contribution · 0.85

Compares with it 2 papers

Projecting onto colour singlet states and dropping terms with vanishing quantum numbers for pseudo-scalar mesons yields [equation] After including a global minus sign from the interchange of fermionic field operators in the Fierz relation, identifying $p_⊥= i ∂_⊥$ and $n_-ℓ→-ω$ for an incoming soft momentum of the $B$-meson spectator quark, one recovers the tree-level matching coefficients and SCET operators for two-particle soft and collinear operators obtained in [ref] , and for the case of different quark masses $m_q$ and $m_q'$ in [ref] .

On the gauge-invariance of SCET beyond leading power · An example: the splitting xi Wc -> q xi xi at tree-level · 0.81
A similar phenomenon has been encountered in the context of SCET analyses of the $B→π$ form factor [ref] , where however the factorization breakdown happens at non-perturbative scales.[After submission of this paper, we learned that in this context the term “factorization anomaly” was used in lectures delivered by M.

Drell-Yan production at small q_T, transverse parton distributions and the collinear anomaly · Collinear anomaly and refactorization · 0.91

Cites it as support 4 papers

This can be done, because it can be shown [ref] that these convolutions are convergent to all orders in perturbation theory, which has been confirmed explicitly by one-loop calculations [ref] .

QED factorization of non-leptonic $B$ decays · SCET$_II$ factorization · 0.97
For example, on the theoretical side, a naive collinear factorization breaks down due to these divergences [ref] .

Unregulated Divergences of Feynman Integrals · Introduction · 0.97
The extension to $(λ^2)$ is necessary, since when we square power suppressed jet amplitudes to calculate a cross section, the $(λ)$ power corrections to the cross section tend to vanish [ref] .

A Complete Basis of Helicity Operators for Subleading Factorization · Hard Scattering Operators with Two Collinear Directions · 0.99

Cites it as background 28 papers

The factorisation of resolved contributions introduces a new ingredient, namely an anti-hardcollinear jet function [ref] , typically referred to as a radiative or amplitude-level jet function in collider and flavour applications [ref] .

Refactorisation in subleading $\bar B \to X_s \gamma$ · Introduction · 0.81
Peculiarly, in TMDs this noncommutativity of orders of limits appears naturally, since at intermediate steps of the calculation one must regulate so-called rapidity divergences [ref] , usually by choosing to deviate from lightcone kinematics or by introducing an additional regulator on the lightcone.

Factorization connecting continuum and lattice TMDs · Strategy for proving factorization · 0.94
In the context of SCET, analytical regularization has been used in [ref] .

Analytic Regularization in Soft-Collinear Effective Theory · Analytic regularization · 0.81

Discussion

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