REVIEW
Two-loop corrections to Higgs boson production
High Energy Physics - Phenomenology · Submitted 27 Aug 2004 · 40 references
Pith's verdict: A solid, honest two-loop calculation: the pseudo-scalar Higgs vertex is genuinely new, and the single-pole observation is real but is an observation, not a theorem.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
The two-loop Higgs production vertex is computed for general colour factors, and the single-pole term in both the quark and gluon form factors is traced to one shared function.
The reading
What carries the argument
The load-bearing device is the logarithmic form factor identity inherited from the Collins–Sen line of work, Eq. (4.1): the logarithmic derivative of the form factor with respect to $ln(-q^2/μ^2)$ is written as an integral of the cusp anomalous dimension $γ_{K,ii}$ plus a kernel $G_{ii}$ plus a term $K_{ii}$ that collects all collinear and infrared poles. Integrating this once in scale and once in $ln(-q^2/μ^2)$, and converting renormalized to bare coupling, produces a template (Eq. 4.10) in which every pole coefficient is expressed through $γ_{K,ii}$, $γ_{ii}$, the $\beta$-function coefficients, and two vertex-dependent constants $f^{(0)}_{i,1}$ and $f^{(1)}_{i,2}$. The one-to-one correspondence between $ln(-q^2/μ^2)$ terms and $1/ε$ poles is what lets the authors read the single-pole coefficient off the template rather than guess it. The non-conserved operators add the constants $z_1 = 2β_0$ and $z_2$ from the operator renormalization constants $Z_O$ and $Z_{11}$, which enter the pole terms but, the authors note, not the logarithms.
Extended reading notes
Core claim
The paper's central claim is that the coefficient of the second-order single pole in a two-loop vertex correction is not arbitrary. Written as $1/2 γ^{(1)}_{ii} + f^{(1)}_{i,2}$, it splits into a piece fixed by the two-loop anomalous dimension $γ^{(1)}_{ii}$ and a remainder $f^{(1)}_{i,2}$ that, up to the overall colour factor $C_i$, takes the same value for the quark ($C_F$) and the gluon ($C_A$): $f^{(1)}_{i,2} = C_i[-β_0 ζ(2) + C_A(404/27 - 14ζ(3)) + n_f T_f(-112/27)]$. The authors reach this by computing, in the heavy-top effective theory, the complete two-loop vertex for scalar Higgs production (Eq. 3.3) and for pseudo-scalar Higgs production (Eq. 3.6) with general colour factors, taking the logarithm of each (Eqs. 4.17 and 4.19), and comparing against the electromagnetic vertex correction already in the literature. The $1/ε^4$, $1/ε^3$ and $1/ε^2$ terms reproduce the structure known from electromagnetism; the single pole is the term the paper extends, and it is presented as a general formula covering conserved and non-conserved operators alike.
Load-bearing premise
The claim that the leftover single-pole function is universal rests on agreement between exactly two computed cases, the electromagnetic quark vertex and the gluon Higgs vertex, so the pattern is observed rather than proved to hold for other processes.
Editorial extensions
If this is right
- The pseudo-scalar Higgs two-loop vertex is available explicitly for the first time, so the $δ(1-z)$ piece of the pseudo-scalar coefficient function can be written for general colour factors rather than only for $C_A = 3$, $C_F = 4/3$.
- The single-pole structure of any two-loop vertex correction can be predicted from the two-loop anomalous dimension plus one universal function, without a fresh Feynman-diagram calculation.
- Combined with the known three-loop anomalous dimensions, the same template predicts the three-loop vertex function and three-loop scattering amplitudes up to the double pole.
- The authors' rewriting of the $H^{(2)}_i$ functions in two independent amplitude calculations (Eq. 4.21) puts those results into a common form, making cross-checks between them mechanical.
Reading between the lines
- If the universality of $f^{(1)}_{i,2}$ survives at three loops in the same shape, the pole structure of multi-leg amplitudes becomes largely a bookkeeping exercise in anomalous dimensions, which would shift effort in higher-order QCD from pole extraction to finite remainders.
- The paper's comparison set is small — one quark vertex and one gluon vertex. A natural test would be a third two-loop vertex with a different operator or a different external state; agreement there would turn an observed pattern into a working rule.
- Because the argument leans on the correspondence between $ln(-q^2/μ^2)$ and $1/ε$, it implicitly assumes the regularization preserves that correspondence; a scheme where the Levi-Civita and $γ_5$ treatment interferes differently could in principle disturb the single-pole identification.
Original abstract
In this paper we present the complete two-loop vertex corrections to scalar and pseudo-scalar Higgs boson production for general colour factors for the gauge group ${\rm SU(N)}$ in the limit where the top quark mass gets infinite. We derive a general formula for the vertex correction which holds for conserved and non conserved operators. For the conserved operator we take the electromagnetic vertex correction as an example whereas for the non conserved operators we take the two vertex corrections above. Our observations for the structure of the pole terms $1/\epsilon^4$, $1/\epsilon^3$ and $1/\epsilon^2$ in two loop order are the same as made earlier in the literature for electromagnetism. However we also elucidate the origin of the second order single pole term which is equal to the second order singular part of the anomalous dimension plus a universal function which is the same for the quark and the gluon. [3mm]
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
In this paper we present the complete two-loop vertex corrections to scalar and pseudo-scalar Higgs boson production for general colour factors for the gauge group ${\rm SU(N)}$ in the limit where the top quark mass gets infinite. We derive a general formula for the vertex correction which holds for conserved and non conserved operators. For the conserved operator we take the electromagnetic vertex correction as an example whereas for the non conserved operators we take the two vertex corrections above. Our observations for the structure of the pole terms $1/\epsilon^4$, $1/\epsilon^3$ and $1/\epsilon^2$ in two loop order are the same as made earlier in the literature for electromagnetism. However we also elucidate the origin of the second order single pole term which is equal to the second order singular part of the anomalous dimension plus a universal function which is the same for the quark and the gluon. [3mm]
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we present the complete two-loop vertex corrections to scalar and pseudo-scalar Higgs boson production for general colour factors for the gauge group ${\rm SU(N)}$ in the limit where the top quark mass gets infinite. | contribution 0.99 | 0.95 | 0.63 | asserted 1.00 |
| s1 | We derive a general formula for the vertex correction which holds for conserved and non conserved operators. | contribution 0.66 | 0.96 | 0.22 | asserted 1.00 |
| s2 | For the conserved operator we take the electromagnetic vertex correction as an example whereas for the non conserved operators we take the two vertex corrections above. | method 0.81 | 0.64 | 0.05 | asserted 0.95 |
| s3 | Our observations for the structure of the pole terms $1/\epsilon^4$, $1/\epsilon^3$ and $1/\epsilon^2$ in two loop order are the same as made earlier in the literature for electromagnetism. | result 0.78 | 0.55 | 0.10 | asserted 1.00 |
| s4 | However we also elucidate the origin of the second order single pole term which is equal to the second order singular part of the anomalous dimension plus a universal function which is the same for the quark and the gluon. [3mm] | result 0.89 | 0.90 | 0.31 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-ph/0408315 by the authors.
References
How the literature uses it
52 arXiv papers cite this in 108 sentences
Papers citing it · 75 sentences without a confident label
Uses its data or values 9 papers
-
The $f_i^q$ for $i=1,2$ can be found in [ref] and in [ref] for $i=3$.
Higgs Rapidity Distribution in $b {\bar b}$ Annihilation at Threshold in N$^{3}$LO QCD · Threshold Corrections Beyond NLO · 0.97
-
Here, $q$ denotes the independence of the quantities on the quark flavors and absence of $I$ represents the independence of the quantities on the nature of colorless particles. $f^H_bb̅$ can be found in [ref] , $A^H_bb̅$ in [ref] and $B^H_bb̅$ in [ref] up to three loop level.
QCD Radiative Corrections to Higgs Physics · The Form Factor · 0.80
-
Since, $A^i$ [ref] ,$B^i$ [ref] and $f^i$ [ref] are known up to three loop level, we can use the solution to K-G equation to determine the renormalisation constants $Z_IJ$.
Three loop form factors of a massive spin-2 particle with nonuniversal coupling · Infrared structure · 0.88
Compares with it 5 papers
-
To make contact with the “universal” quantities $f_q(a_s)$ appearing in [ref] and defined in [ref] , we can put [equation] with $G̃$ as defined in [ref] , such that [equation] We note that $ΔG(a_s)$ is also proportional to the beta function, i.e. has the structure [equation] where $κ(a_s)$ is a power series with polynomial dependence on the group theory factors.
Constant terms in threshold resummation and the quark form factor · Threshold resummation of the physical anomalous dimension (DY case) · 0.87
-
An important aspect of this derivation is an explicit check of the prediction [ref] for the infrared poles of the renormalized $Hbb̅$ form factor using recent results for the four-loop cusp [ref] and quark collinear [ref] anomalous dimensions.
The $Hb\bar{b}$ vertex at four loops and hard matching coefficients in SCET for various currents · Introduction · 0.81
-
One and two loop results completely agree with the existing literature [ref] .
QCD Radiative Corrections to Higgs Physics · Results · 0.99
Cites it as support 1 paper
-
However a generalized Casimir scaling holds for the cusp, and this is inherited by the soft anomalous dimension, since they exhibit the same maximally non-abelian structure [ref] .
The Four Loop QCD Rapidity Anomalous Dimension · Summary of Perturbative Data for Anomalous Dimensions · 0.97
Cites it as background 15 papers
-
For example, in the context of hard scattering processes such as Drell-Yan [ref] and the Higgs boson production in gluon fusion [ref] , the form factors corresponding to the vector current operator $ψγ_μψ$ and the gluonic operator $G^a_μν G^a,μν$ contribute, respectively.
QCD Radiative Corrections to Higgs Physics · Prologue · 0.89
-
After the pioneering work more than 20 years ago [ref] where the quark form factor has been computed to two-loop order the corresponding quantity for the Higgs-gluon coupling has been evaluated by Harlander in Ref. [ref] (see also [ref] ).
Quark and gluon form factors to three loops · Introduction · 0.80
-
$f_g^A$'s were introduced for the first time in the article [ref] .
Radiative corrections and threshold resummed predictions to pseudoscalar Higgs boson production in QCD · The Form Factor · 0.83
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.