# 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 the paper, we make a detailed discussion on the gauge dependence of the pQCD prediction under the MOM scheme. Conventionally, there is renormalization scale ambiguity for the fixed-order pQCD predictions, which assigns an arbitrary range and an arbitrary 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 and hence determines an effective momentum flow of the process, which is independent to the choice of renormalization scale. There is thus no renormalization scale ambiguity in PMC predictions. To concentrate our attention on the MOM gauge dependence, we first apply the PMC to deal with the pQCD series. We adopt the Higgs boson decay width, $\Gamma(H\to gg)$ , up to five-loop QCD contributions as an explicit example to show how the gauge dependence behaves before and after applying the PMC. Different interaction vertices have been chosen for defining the MOM-schemes such as the mMOM, the MOMh, the MOMq, the MOMg, and the MOMgg schemes. Under those MOM schemes, we obtain $\Gamma(H \to gg)|^{\rm{mMOM}}_{\rm{PMC}} = 332.8{^{+11.6}_{-3.7}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMh}}_{\rm{PMC}} = 332.8{^{+27.5}_{-34.6}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMq}}_{\rm{PMC}} = 332.9{^{+27.4}_{-34.7}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMg}}_{\rm{PMC}} = 332.7{^{+27.5}_{-34.6}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMgg}}_{\rm{PMC}} = 337.9{^{+1.2}_{-1.7}}\pm 7.7\; \rm{KeV}$ , where the central values are for the Landau gauge with the gauge parameter $\xi^{\rm MOM} = 0$ , and the first errors are for $\xi^{\rm MOM}\in[-1,1]$ , the second ones are caused by taking $\Delta \alpha_s^{\overline{\rm MS}}(M_Z) = \pm0.0011$ . The uncertainty of the Higgs mass $\Delta M_H = 0.24\; \rm{GeV}$ will cause an extra error $\sim \pm1.7$ (or $\sim\pm1.8$ ) KeV for all the mentioned MOM schemes. It is found that the Higgs decay width $\Gamma (H\to gg)$ depends very weakly on the choices of the MOM schemes, being consistent with the 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 mentioned 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.
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, P. R. China
• 2. School of Physics and Electronics, Hunan University, Changsha 410082, P. R. China

Abstract: The momentum space subtraction (MOM) scheme is one of the most frequently used renormalization schemes in perturbative QCD (pQCD) theory. In the paper, we make a detailed discussion on the gauge dependence of the pQCD prediction under the MOM scheme. Conventionally, there is renormalization scale ambiguity for the fixed-order pQCD predictions, which assigns an arbitrary range and an arbitrary 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 and hence determines an effective momentum flow of the process, which is independent to the choice of renormalization scale. There is thus no renormalization scale ambiguity in PMC predictions. To concentrate our attention on the MOM gauge dependence, we first apply the PMC to deal with the pQCD series. We adopt the Higgs boson decay width, $\Gamma(H\to gg)$ , up to five-loop QCD contributions as an explicit example to show how the gauge dependence behaves before and after applying the PMC. Different interaction vertices have been chosen for defining the MOM-schemes such as the mMOM, the MOMh, the MOMq, the MOMg, and the MOMgg schemes. Under those MOM schemes, we obtain $\Gamma(H \to gg)|^{\rm{mMOM}}_{\rm{PMC}} = 332.8{^{+11.6}_{-3.7}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMh}}_{\rm{PMC}} = 332.8{^{+27.5}_{-34.6}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMq}}_{\rm{PMC}} = 332.9{^{+27.4}_{-34.7}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMg}}_{\rm{PMC}} = 332.7{^{+27.5}_{-34.6}}\pm7.3\; \rm{KeV}$ , $\Gamma(H \to gg)|^{\rm{MOMgg}}_{\rm{PMC}} = 337.9{^{+1.2}_{-1.7}}\pm 7.7\; \rm{KeV}$ , where the central values are for the Landau gauge with the gauge parameter $\xi^{\rm MOM} = 0$ , and the first errors are for $\xi^{\rm MOM}\in[-1,1]$ , the second ones are caused by taking $\Delta \alpha_s^{\overline{\rm MS}}(M_Z) = \pm0.0011$ . The uncertainty of the Higgs mass $\Delta M_H = 0.24\; \rm{GeV}$ will cause an extra error $\sim \pm1.7$ (or $\sim\pm1.8$ ) KeV for all the mentioned MOM schemes. It is found that the Higgs decay width $\Gamma (H\to gg)$ depends very weakly on the choices of the MOM schemes, being consistent with the 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 mentioned 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 relation of the renormalization constants under the mMOM and $\overline{\rm MS}$ schemes Appendix B: Perturbative transformations of the strong couplings and gauge parameters among the MOM schemes and the $\overline{\rm MS}$ 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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