# Gauge dependence of the perturbative QCD predictions under the momentum-space subtraction scheme • The momentum-space subtraction (MOM) scheme is one of the most frequently used renormalization schemes in perturbative QCD (pQCD) theory. In this paper, we discuss in detail the gauge dependence of the pQCD predictions obtained under the MOM scheme. Conventionally, a renormalization scale ambiguity exists for the fixed-order pQCD predictions; this assigns an arbitrary range and error for the fixed-order pQCD prediction and makes the discussions on the issue of the gauge dependence much more involved. The principle of maximum conformality (PMC) adopts the renormalization group equation to determine the magnitude of the coupling constant; hence, it determines the effective momentum flow of the process, which is independent of the choice of renormalization scale. Thus, no renormalization scale ambiguity exists in PMC predictions. To focus our attention on the MOM scheme's gauge dependence, we first apply the PMC to deal with the pQCD series. As an explicit example, we adopt the Higgs boson decay width $\Gamma(H\to gg)$ up to its five-loop QCD contribution, to demonstrate the behavior of the gauge dependence before and after applying the PMC. Interaction vertices are chosen to define five different MOM schemes: mMOM, MOMh, MOMq, MOMg, and MOMgg. Under these MOM schemes, we obtain $\Gamma(H \to gg)|^{\rm{mMOM}}_{\rm{PMC}} =$ $332.8{^{+11.6}_{-3.7}}\pm7.3\; \rm{keV}$ , $\Gamma(H \to gg)|^{\rm{MOMh}}_{\rm{PMC}} = 332.8{^{+27.5}_{-34.6}}\pm7.3\; \rm{keV}$ , $\Gamma(H \to gg)|^{\rm{MOMq}}_{\rm{PMC}} = 332.9{^{+27.4}_{-34.7}}\pm 7.3\; \rm{keV}$ , $\Gamma(H \to gg)|^{\rm{MOMg}}_{\rm{PMC}} = 332.7{^{+27.5}_{-34.6}}\pm7.3\; \rm{keV}$ , and $\Gamma(H \to gg)|^{\rm{MOMgg}}_{\rm{PMC}} = 337.9{^{+1.2}_{-1.7}}\pm 7.7\; \rm{keV}$ ; here, the central values correspond to the Landau gauge with the gauge parameter $\xi^{\rm MOM} = 0$ , the first errors correspond to $\xi^{\rm MOM}\in[-1,1]$ , and the second ones arise through taking $\Delta \alpha_s^{\overline{\rm MS}}(M_Z) = \pm0.0011$ . The uncertainty of the Higgs mass $\Delta M_H = 0.24\; \rm{GeV}$ causes an extra error of $\sim \pm1.7$ (or $\sim\pm1.8$ ) keV for all the aforementioned MOM schemes. It is found that the Higgs decay width $\Gamma (H\to gg)$ depends very weakly on the choice of MOM scheme, which is consistent with renormalization group invariance. It is found that the gauge dependence of $\Gamma(H\to gg)$ under the $\rm{MOMgg}$ scheme is less than ±1%, which is the smallest gauge dependence among all the aforementioned MOM schemes.
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Jun Zeng, Xing-Gang Wu, Xu-Chang Zheng and Jian-Ming Shen. Gauge dependence of the perturbative QCD predictions under the momentum space subtraction scheme[J]. Chinese Physics C. doi: 10.1088/1674-1137/abae4e
Jun Zeng, Xing-Gang Wu, Xu-Chang Zheng and Jian-Ming Shen. Gauge dependence of the perturbative QCD predictions under the momentum space subtraction scheme[J]. Chinese Physics C. Milestone
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## Gauge dependence of the perturbative QCD predictions under the momentum-space subtraction scheme

###### Corresponding author: Jian-Ming Shen, cqusjm@cqu.edu.cn
• 1. Department of Physics, Chongqing University, Chongqing 401331, China
• 2. School of Physics and Electronics, Hunan University, Changsha 410082, China

Abstract: The momentum-space subtraction (MOM) scheme is one of the most frequently used renormalization schemes in perturbative QCD (pQCD) theory. In this paper, we discuss in detail the gauge dependence of the pQCD predictions obtained under the MOM scheme. Conventionally, a renormalization scale ambiguity exists for the fixed-order pQCD predictions; this assigns an arbitrary range and error for the fixed-order pQCD prediction and makes the discussions on the issue of the gauge dependence much more involved. The principle of maximum conformality (PMC) adopts the renormalization group equation to determine the magnitude of the coupling constant; hence, it determines the effective momentum flow of the process, which is independent of the choice of renormalization scale. Thus, no renormalization scale ambiguity exists in PMC predictions. To focus our attention on the MOM scheme's gauge dependence, we first apply the PMC to deal with the pQCD series. As an explicit example, we adopt the Higgs boson decay width $\Gamma(H\to gg)$ up to its five-loop QCD contribution, to demonstrate the behavior of the gauge dependence before and after applying the PMC. Interaction vertices are chosen to define five different MOM schemes: mMOM, MOMh, MOMq, MOMg, and MOMgg. Under these MOM schemes, we obtain $\Gamma(H \to gg)|^{\rm{mMOM}}_{\rm{PMC}} =$ $332.8{^{+11.6}_{-3.7}}\pm7.3\; \rm{keV}$ , $\Gamma(H \to gg)|^{\rm{MOMh}}_{\rm{PMC}} = 332.8{^{+27.5}_{-34.6}}\pm7.3\; \rm{keV}$ , $\Gamma(H \to gg)|^{\rm{MOMq}}_{\rm{PMC}} = 332.9{^{+27.4}_{-34.7}}\pm 7.3\; \rm{keV}$ , $\Gamma(H \to gg)|^{\rm{MOMg}}_{\rm{PMC}} = 332.7{^{+27.5}_{-34.6}}\pm7.3\; \rm{keV}$ , and $\Gamma(H \to gg)|^{\rm{MOMgg}}_{\rm{PMC}} = 337.9{^{+1.2}_{-1.7}}\pm 7.7\; \rm{keV}$ ; here, the central values correspond to the Landau gauge with the gauge parameter $\xi^{\rm MOM} = 0$ , the first errors correspond to $\xi^{\rm MOM}\in[-1,1]$ , and the second ones arise through taking $\Delta \alpha_s^{\overline{\rm MS}}(M_Z) = \pm0.0011$ . The uncertainty of the Higgs mass $\Delta M_H = 0.24\; \rm{GeV}$ causes an extra error of $\sim \pm1.7$ (or $\sim\pm1.8$ ) keV for all the aforementioned MOM schemes. It is found that the Higgs decay width $\Gamma (H\to gg)$ depends very weakly on the choice of MOM scheme, which is consistent with renormalization group invariance. It is found that the gauge dependence of $\Gamma(H\to gg)$ under the $\rm{MOMgg}$ scheme is less than ±1%, which is the smallest gauge dependence among all the aforementioned MOM schemes.

### HTML 1.   Introduction 2.   The momentum-space subtraction schemes 3.   General PMC analysis over the perturbative series 4.   Gauge dependence of the total decay width Γ(H→gg) 5.   Summary Appendix A: The relations between the renormalization constants of the mMOM and $\overline{\rm MS}$ schemes Appendix B: Perturbative transformations of the strong couplings and gauge parameters between the MOM schemes and $\overline{\rm MS}$ scheme Appendix C: The perturbative coefficients of the PMC scale Appendix D: The $\overline{\rm MS}$-scheme coefficients $r_{i,j}(M_H)$ for $\Gamma(H\to gg)$ up to $\alpha_s^6$-order level
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