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Three-loop QCD matching of the flavor-changing scalar current involving the heavy charm and bottom quark
High Energy Physics - Phenomenology · Submitted 31 Dec 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The three-loop QCD correction turns out to be significantly large.
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The abstract, sentence by sentence
We compute the matching coefficient between the quantum chromodynamics (QCD) and the non-relativistic QCD ( NRQCD) for the flavor-changing scalar current involving the heavy charm and bottom quark, up to the three-loop order within the NRQCD factorization. For the first time, we obtain the analytical expressions for the three-loop renormalization constant $\tilde{Z}_s(x,R_f)$ and the corresponding anomalous dimension $\tilde{\gamma}_s(x,R_f)$ for the NRQCD scalar current with the two heavy bottom and charm quark. We present the precise numerical results for those relevant coefficients $(C_{FF}(x_0), \cdots, C_{FBB}(x_0))$ with an accuracy of about thirty digits. The three-loop QCD correction turns out to be significantly large. The obtained matching coefficient $C_s(\mu_f,\mu,m_b,m_c)$ is helpful to analyze the threshold behaviours when two different heavy quarks are close to each other and form the double heavy $B_c$ mesons.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compute the matching coefficient between the quantum chromodynamics (QCD) and the non-relativistic QCD ( NRQCD) for the flavor-changing scalar current involving the heavy charm and bottom quark, up to the three-loop order within the NRQCD factorization. | contribution 0.99 | 0.97 | 0.24 | asserted 1.00 |
| s1 | For the first time, we obtain the analytical expressions for the three-loop renormalization constant $\tilde{Z}_s(x,R_f)$ and the corresponding anomalous dimension $\tilde{\gamma}_s(x,R_f)$ for the NRQCD scalar current with the two heavy bottom and charm quark. | contribution 0.56 | 0.97 | 0.22 | asserted 1.00 |
| s2 | We present the precise numerical results for those relevant coefficients $(C_{FF}(x_0), \cdots, C_{FBB}(x_0))$ with an accuracy of about thirty digits. | result 0.95 | 0.96 | 0.10 | asserted 1.00 |
| s3 | The three-loop QCD correction turns out to be significantly large. | result 0.83 | 0.93 | 0.26 | asserted 1.00 |
| s4 | The obtained matching coefficient $C_s(\mu_f,\mu,m_b,m_c)$ is helpful to analyze the threshold behaviours when two different heavy quarks are close to each other and form the double heavy $B_c$ mesons. | interpretation 1.00 | 0.77 | 0.21 | asserted 0.87 |
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The paper
Original abstract
We compute the matching coefficient between the quantum chromodynamics (QCD) and the non-relativistic QCD ( NRQCD) for the flavor-changing scalar current involving the heavy charm and bottom quark, up to the three-loop order within the NRQCD factorization. For the first time, we obtain the analytical expressions for the three-loop renormalization constant $\tilde{Z}_s(x,R_f)$ and the corresponding anomalous dimension $\tilde{\gamma}_s(x,R_f)$ for the NRQCD scalar current with the two heavy bottom and charm quark. We present the precise numerical results for those relevant coefficients $(C_{FF}(x_0), \cdots, C_{FBB}(x_0))$ with an accuracy of about thirty digits. The three-loop QCD correction turns out to be significantly large. The obtained matching coefficient $C_s(\mu_f,\mu,m_b,m_c)$ is helpful to analyze the threshold behaviours when two different heavy quarks are close to each other and form the double heavy $B_c$ mesons.
Human review
How the literature uses it
4 arXiv papers cite this in 20 sentences
Papers citing it · 12 sentences without a confident label
Builds on it 1 paper
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Building upon our earlier works [ref] , we complete the three-loop matching between QCD and NRQCD for the heavy flavor-changing temporal vector $(v,0)$, spatial-spatial tensor $(t,ij)$ and spatial-temporal axial-tensor $(t5,i0)$ currents, which can couple to (can produce/annihilate) the $P$-wave scalar meson $B_c0^*$ and axial-vector meson $B_c1$.
Decay constants of $c \bar b$ mesons involving the ten heavy flavor-changing currents at N$^3$LO QCD · Summary · 0.99
Uses its method or tool 2 papers
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To match QCD with NRQCD, we employ both the coupling running [ref] and the decoupling relation [ref] in $D=4-2ϵ$ for the mutual conversion between $α_s^(n_f)(μ)$, $α_s^(n_l)(μ_f)$ and $α_s^(n_l)(μ)$, where $n_f=n_l+n_b+n_c$ is the total number of flavors.
Three-loop matching of heavy flavor-changing (axial-)tensor currents · NRQCD current renormalization constants · 0.91
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For matching between QCD and NRQCD with the same perturbative expansion parameter $α_s$, we employ both the coupling running [ref] and the decoupling relation [ref] in $D=4-2ϵ$ for the interconversion among $α_s^(n_f)(μ)$, $α_s^(n_l)(μ_f)$ and $α_s^(n_l)(μ)$, where $n_f=n_l+n_b+n_c$ is the total number of flavors, $n_l$ is the number of massless flavors, $n_b(c)$ is the number of bottom (charm) flavors with mass $m_b(c)$, $μ$ is the QCD renormalization scale and $μ_f$ is the NRQCD factorization scale.
Decay constants of $c \bar b$ mesons involving the ten heavy flavor-changing currents at N$^3$LO QCD · NRQCD current renormalization constant · 0.87
Compares with it 1 paper
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To further check our three-loop calculations, we have computed the three-loop matching coefficients at five rational numerical points [ref] : $x=150/475$, $475/150$, $204/498$, $498/204$, and $1$.
Decay constants of $c \bar b$ mesons involving the ten heavy flavor-changing currents at N$^3$LO QCD · Matching coefficient · 0.87
Cites it as support 1 paper
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From Eq. [ref], Fig. [ref] and Table [ref], one can find the higher order QCD corrections have larger values, especially, the $O(α_s^3)$ correction looks quite sizable, which confirms the nonconvergence behaviors of the matching coefficients investigated in previous literature [ref] .
Three-loop matching coefficients for heavy flavor-changing currents and the phenomenological applications · Matching coefficients · 0.97
Cites it as background 1 paper
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Spectroscopic parameters (such as mass and decay constant) of $B_c$ mesons are studied with using various models such as relativistic quark potential model [ref] , nonrelativistic quark model [ref] , covariant light-front quark model [ref] , shifted large-N expansion [ref] , perturbative QCD [ref] , nonrelativistic renormalization group [ref] , lattice QCD [ref] , QCD sum rules (QCDSR) [ref] , light-cone QCDSR [ref] , heavy quark effective theory [ref] , Bethe-Salpeter equation [ref] and field correlator method [ref] .
Bottom-charmed meson states in inverse problem of QCD · Motivation · 0.97
What it cites · 23 sentences without a confident label
Uses the method or tool of 4 papers
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Besides, we can evolve the strong coupling from the scale $μ_f$ to the scale $μ$ with renormalization group running equation [ref] in $D=4-2 ϵ$ dimensions as following [equation] To calculate the values of the strong coupling, we also use the renormalization group running equation [ref] in $D=4$ dimensions as [equation] where $L_Λ=ln(μ^2/Λ_QCD^(n_l)^2)$ and $b_i=β_i^(n_l)/β_0^(n_l)$.
Two-loop form factors for pseudo-scalar quarkonium production and decay · Matching formula · 0.98
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Namely, before generating amplitudes, $m_b$ and $m_c$ are chosen to be particular rational number values [ref] .
Heavy-quark pair production at lepton colliders at NNNLO in QCD · Calculation procedure · 0.91
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Namely, before generating amplitudes, $m_b$ and $m_c$ are chosen to be particular rational number values [ref] .
Complete two-loop electroweak corrections to $e^+e^-\rightarrow HZ$ · Calculation procedure · 0.91
Compares with 3 papers
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To confirm our calculation, we have also applied the same calculation procedure to the evaluation of the three-loop matching coefficients $𝒞_v$, $𝒞_p$, $𝒞_(a,0)$ for the flavor-changing heavy quark vector current, the flavor-changing heavy quark pseudoscalar current, the zeroth component of the axial-vector current with the flavor-changing heavy quarks, respectively, where our results verify $𝒞_p≡𝒞_(a,0)$ and our results $𝒞_v$ and $𝒞_(a,0)$ agree with the known results in previous literature [ref] .
Next-to-next-to-leading order matching of beauty-charmed meson $B_{c}$ and $B^*_{c}$ decay constants · Results · 0.99
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To confirm our calculation, we have also applied the same calculation procedure to the evaluation of the three-loop matching coefficients $𝒞_v$, $𝒞_p$, $𝒞_(a,0)$ for the flavor-changing heavy quark vector current, the flavor-changing heavy quark pseudoscalar current, the zeroth component of the axial-vector current with the flavor-changing heavy quarks, respectively, where our results verify $𝒞_p≡𝒞_(a,0)$ and our results $𝒞_v$ and $𝒞_(a,0)$ agree with the known results in previous literature [ref] .
Three-loop QCD corrections to the decay constant of $B_c$ · Results · 0.99
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To confirm our calculation, we have also applied the same calculation procedure to the evaluation of the three-loop matching coefficients $𝒞_v$, $𝒞_p$, $𝒞_(a,0)$ for the flavor-changing heavy quark vector current, the flavor-changing heavy quark pseudoscalar current, the zeroth component of the axial-vector current with the flavor-changing heavy quarks, respectively, where our results verify $𝒞_p≡𝒞_(a,0)$ and our results $𝒞_v$ and $𝒞_(a,0)$ agree with the known results in previous literature [ref] .
2210.02979 · Results · 0.99
Cites as background 4 papers
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After that, the three-loop matching coefficient of the vector current was also achieved in Ref. [ref] .
2210.02979 · Introduction · 0.80
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This year, the next-to-next-to-next-to-leading order (N$^3$LO) QCD correction for the matching coefficient of the axial-vector current was first numerically calculated in Ref. [ref] .
Three-loop QCD corrections to the decay constant of $B_c$ · Introduction · 0.85
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For more higher-order calculations about matching coefficients for doubly heavy quark systems, one can see the following literature [ref] .
Complete three-loop QCD corrections to leptonic width of vector quarkonium · Introduction · 0.99
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