REVIEW
Two-loop mixing of dimension-five flavor-changing operators
High Energy Physics - Phenomenology · Submitted 30 Sep 1994 · 19 references
Pith's verdict: A careful, standard-reference two-loop calculation whose numbers have held up; the only real soft spot is that three of the four new entries rest on the automated pipeline rather than an independent hand check.
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Referee report
The paper
The paper claims
The two-loop QCD mixing between the photonic and gluonic magnetic moment operators is computed, completing the two-loop anomalous dimension matrix for the dimension-five flavor-changing operators of the Standard Model.
The reading
What carries the argument
The calculation is done off shell in the background-field version of Feynman-'t Hooft gauge, on one-particle-irreducible Green's functions with a single operator insertion, because the divergent parts of those functions are what the renormalization constants are. Two devices carry the argument. First, the authors identify the complete set of operators that can act as counterterms: besides $O_7$ and $O_8$ themselves, only two operators that vanish by the quark equations of motion, $tilde{O}_{DD} = bar{q} D!!!/ D!!!/ q'$ and $tilde{O}_{QD}$, can appear, and they argue this from dimension counting, background gauge invariance, charge conjugation, and the absence of a BRS-exact candidate. Second, to make the two-loop integrals tractable they introduce a common auxiliary mass in every propagator denominator and expand in external momenta before integrating, reducing everything to the scalar two-loop integrals $I_{abc}$ of eq. (10), whose divergent parts are given in closed form by eqs. (11)-(13) and derived in the appendix from $J_{211}$. The auxiliary mass is essential: without it the momentum expansion would generate spurious infrared divergences indistinguishable from ultraviolet ones in dimensional regularization.
Extended reading notes
Core claim
The paper's claim is that the $O(g^4)$ part of the anomalous dimension matrix for the magnetic moment operators is now known, so that the two-loop QCD mixing among all the dimension-five flavor-changing operators is complete. Concretely, the authors give the matrix elements $a$, $b$, $c$, $d$ of eqs. (16)-(19) in the basis $(O_7, O_8)$ and, after subtracting the quark mass anomalous dimension $6 C_F$ to convert to the normalization used in $b$ to $s$ photon analyses, the two-by-two matrix of eq. (25): $4688/27$ and $4063/27$ on the diagonal, zero in the upper-right corner, and $-2192/81$ in the lower-left. They are explicit that this does not finish the next-to-leading calculation: the three-loop mixing of four-quark operators into the magnetic moment operators contributes at the same order, and they say so plainly, so no phenomenological conclusion can be drawn until those pieces exist. What the paper does deliver is the last two-loop ingredient, plus a numerical estimate that this piece by itself lowers the predicted rate by about five percent for $a_s(M_Z) = 0.12$, $m_t = 174$ GeV and $mu = 5$ GeV, and lowers the coefficient of the gluonic operator by about 2.4 percent.
Load-bearing premise
The whole numerical result rests on the claim that adding a common auxiliary mass to every propagator and expanding in external momenta before integrating gives the same divergent parts as a direct massless calculation without expansion; the authors justify this with Weinberg's theorem and the polynomiality of pole parts after subdivergence subtraction, and they cross-check the photonic self-mixing by hand without auxiliary masses, but the other entries depend on it.
Editorial extensions
If this is right
- The two-loop anomalous dimension matrix for the photonic and gluonic magnetic moment operators is fixed in the MS scheme, so the next-to-leading renormalization group evolution of their Wilson coefficients can be written down once the remaining ingredients exist.
- Inserted into the existing next-to-leading analysis, the new matrix elements shift the effective coefficient of the photonic operator downward, giving roughly a five percent reduction of the predicted $B$ to $X_s$ photon branching ratio from this contribution alone.
- The coefficient of the gluonic operator receives a comparable correction of about 2.4 percent, which the authors expect to have no measurable consequence.
- Because the additional diagrams for two identical quark flavors contain a trace over an odd number of Dirac matrices and vanish for massless quarks, the same results apply directly to flavor-conserving magnetic moment operators.
- The mixing is insensitive to the scheme used for $gamma_5$ in dimensional regularization, since chirality conservation removes $gamma_5$ from the calculation entirely, unlike the case of four-quark operators.
Reading between the lines
- The remaining obstacle to a complete next-to-leading $B$ to $X_s$ photon rate is now sharply located: it is the three-loop mixing of four-quark operators into the magnetic moment operators, and the automated auxiliary-mass pipeline described here looks reusable for that harder problem.
- A five percent shift from a single two-loop entry is of the same order as the precision the experimental measurements were approaching, which is why the scheme-independence of the result matters as much as its value.
- The treatment of equation-of-motion-vanishing operators as bookkeeping devices that must be tracked through intermediate stages, even though their on-shell matrix elements vanish, is a template for any off-shell multi-loop mixing computation where such operators appear as subdivergence counterterms.
Original abstract
We calculate the two-loop QCD mixing of the dimension-five flavor-changing operators that arise in the Standard Model after integrating out the $W$, $Z$ and Higgs bosons and the top-quark, i.e.~the mixing among gluonic and photonic magnetic moment operators. Our calculation completes the two-loop anomalous dimension matrix of operators that govern low energy flavor-changing processes. It is an important ingredient of the next-to-leading calculation of the $B \rightarrow X_s \gamma$ decay rate.
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
We calculate the two-loop QCD mixing of the dimension-five flavor-changing operators that arise in the Standard Model after integrating out the $W$, $Z$ and Higgs bosons and the top-quark, i.e.~the mixing among gluonic and photonic magnetic moment operators. Our calculation completes the two-loop anomalous dimension matrix of operators that govern low energy flavor-changing processes. It is an important ingredient of the next-to-leading calculation of the $B \rightarrow X_s \gamma$ decay rate.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We calculate the two-loop QCD mixing of the dimension-five flavor-changing operators that arise in the Standard Model after integrating out the $W$, $Z$ and Higgs bosons and the top-quark, i.e.~the mixing among gluonic and photonic magnetic moment operators. | contribution 1.00 | 0.95 | 0.24 | asserted 1.00 |
| s1 | Our calculation completes the two-loop anomalous dimension matrix of operators that govern low energy flavor-changing processes. | result 0.33 | 0.89 | 0.59 | asserted 1.00 |
| s2 | It is an important ingredient of the next-to-leading calculation of the $B \rightarrow X_s \gamma$ decay rate. | interpretation 0.97 | 0.26 | 0.17 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v3
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Figures
Figures reproduced from arXiv:hep-ph/9409454 by the authors.
References
How later papers used it
68 later papers cite it
3 later papers by other authors are listed below: 1 used its method or tool and 2 compared their own results with it.
Uses its method or tool 1 of 36 papers with this label
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Applied the method proposed in this paper, sometimes called IR-rearrangement, which allows all masses and external momenta to be set to zero.
NNLO vertex corrections in charmless hadronic B decays: Real part (2009) · 1.00 · paraphrased: the licence does not allow quoting
Compares with it 2 papers
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We verified that our results are consistent with the results of Ref. [ref] .
Electric dipole moment constraints on CP-violating heavy-quark Yukawas at next-to-leading order (2018) · Two-loop anomalous dimensions · 0.99
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Found that its own result for L̇ agrees with this paper, while the remaining terms with L are new to the best of its knowledge.
Renormalization-group equations of the LEFT at two loops: dimension-five effects (2024) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
66 arXiv papers cite this in 86 sentences
Papers citing it · 38 sentences without a confident label
Uses its method or tool 34 papers
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In our case, the required three-loop calculations were conducted with a custom version of the framework [ref] , employing infrared rearrangement via a mass parameter [ref] to isolate UV from IR poles.
General Quartic $\beta$-Function at Three Loops · Computation · 1.00
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Feynman integrals are evaluated by the method from Refs. [ref] and our own implementation of the Laporta's algorithm [ref] in the form of the MATHEMATICA package BAMBA with the master integrals from Ref. [ref] .
Higgs self-coupling beta-function in the Standard Model at three loops · Introduction · 0.99
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For evaluation of Feynman integrals we used the method from Refs. [ref] and our own implementation of the Laporta's algorithm [ref] in the form of the MATHEMATICA package BAMBA with the master integrals from Ref. [ref] .
Four loop anomalous dimension of the second moment of the non-singlet twist-2 operator in QCD · Introduction · 1.00
Uses its data or values 1 paper
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The only exceptions are those which describe the selfmixing of $O_8$ at one-loop and the mixing of $O_8$ into $O_7$ up to two-loops; they can be extracted from [ref] .
Fermionic corrections to the interference of the electro- and chromomagnetic dipole operators in anti-B --> X(s) gamma at O(alpha(s)**2) · Fermionic corrections · 0.81
Compares with it 2 papers
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We verified that our results are consistent with the results of Ref. [ref] .
Electric dipole moment constraints on CP-violating heavy-quark Yukawas at next-to-leading order · Two-loop anomalous dimensions · 0.99
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For the RGEs we obtain [equation] The result for $L̇$ agrees with Ref. [ref] , the remaining terms with $L$ are new to the best of our knowledge.
Renormalization-group equations of the LEFT at two loops: dimension-five effects · Single-flavor QCD with $CP$-even dipole · 1.00
Cites it as background 2 papers
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The LEFT has been studied in detail as the low-energy EFT of the SM [ref] .
Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR · Introduction · 0.97
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In QCD and flavour physics, higher loop order anomalous dimensions (ADs) have been calculated for dimension-five and dimension-six operators [ref] , going up to four loops in the last reference.
Renormalization and non-renormalization of scalar EFTs at higher orders · Introduction · 0.95
Discussion
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