Reanalysis of the Higgs-boson decay H\to gg up to \alpha_{s}^{6}-order level

Reanalysis of the Higgs-boson decay up to -order level

Jun Zeng    Xing-Gang Wu    Shi Bu    Jian-Ming Shen Department of Physics, Chongqing University, Chongqing 401331, P.R. China    Sheng-Quan Wang School of Mechatronics Engineering, Guizhou Minzu University, Guiyang 550025, P.R. China

Using the newly available -order QCD correction to the Higgs decay channel , we make a detailed discussion on the perturbative properties of the decay width by using the principle of maximum conformality (PMC). The PMC provides a systematic and solid way to eliminate the conventional renormalization scheme-and-scale ambiguities, which adopts the renormalization group equation to determine the correct running behavior of the strong coupling constant at each order via a recursive way and well satisfies renormalization group invariance. By using the PMC, a more accurate prediction on , not only for the total decay width but also for the separate decay width of each loop term, can be achieved. Even though there is no ambiguity for setting the renormalization scale, there is residual scale dependence for all the PMC predictions due to unknown even higher-order terms. With the help of the newly -order terms, a somewhat larger residual renormalization scale dependence at the -order level observed in our previous work can be greatly suppressed, which shows KeV, where the first error is caused by the Higgs mass uncertainty GeV, and the second one is the residual scale dependence by varying the initial renormalization scale within the region of .

14.80.Bn, 12.38.Bx, 11.10.Gh

I Introduction

The Higgs boson is an important component of the Standard Model (SM), which arouses people’s great interest either for precision test of the SM or for searching of new physics beyond the SM. Among its decay channels, the decay plays an important role in Higgs phenomenology. At present, the next-to-leading order  Inami:1982xt (); Djouadi:1991tka (); Graudenz:1992pv (); Dawson:1993qf (); Spira:1995rr (); Dawson:1991au (), the next-to-next-to-leading order  Chetyrkin:1997iv (); Chetyrkin:1997un (), the next-to-next-to-next-to-leading order  Baikov:2006ch (), and the next-to-next-to-next-to-next-to-leading order  Herzog:2017dtz () perturbative Quantum Chromodymaics (pQCD) corrections to the Higgs decay width have been given in the literature. Those achievements, especially the newly achieved state-of-the-art -term, provide us great opportunity for achieving precise pQCD predictions.

Because of renormalization group invariance, the physical observable should be independent to the choices of renormalization scheme and renormalization scale. However for a fixed-order prediction of the observable, the mismatching of the QCD strong coupling and the pQCD coefficients at each order leads to the well-known renormalization scheme-and-scale ambiguities. Conventionally, one chooses the typical momentum flow of the process or the one to eliminate the large logs as the renormalization scale with the purpose of minimizing such scale dependence around its central point, hoping to achieve a small scheme-and-scale dependent prediction by finishing higher-and-higher order QCD corrections.

If the pQCD series is well converged, this naive treatment may work well, however I) There are processes which are highly divergent due to the divergent renormalon terms; II) Such possibly small scale-dependence can only be true for global quantity such as total cross-section or total decay width, which is due to cancelations among different orders, while the scale uncertainty for each order could be very large and cannot be cured; III) Even if the prediction of a guessing scale luckily agrees with the data, it cannot explain why this is the case, thus greatly depressing the predictive power of pQCD; IV) The high-order calculation become very complicated, e.g. about eleven years have been passed when people have finished the QCD correction to from the -level to the -level.

Thus it is helpful to have a proper scale-setting approach to fix such kind of problems, and get accurate predictions even at the lower orders. The principle of maximum conformality (PMC) Brodsky:2011ta (); Brodsky:2012rj (); Mojaza:2012mf (); Brodsky:2013vpa () has been suggested for the purpose of eliminating the conventional renormalization scheme-and-scale ambiguities. The PMC has a solid theoretical foundation, satisfying all self-consistency conditions of the renormalization group equation (RGE) or the -function Brodsky:2012ms () and the renormalization group invariance, which has now been applied for various processes, c.f. the reviews Wu:2013ei (); Wu:2014iba (); Wu:2015rga ().

The essential PMC procedure is to identify all contributions which originate from the -terms in a pQCD series; one then shifts the renormalization scale of the QCD running coupling at each order to absorb the -terms. The coefficients of the resulting scale-fixed perturbative series is thus identical to the coefficients of the corresponding scheme-independent “conformal” series. New -terms will occur at each order, so the PMC scale for each order is generally distinct, which reflects the varying virtuality of the amplitude that occurs at each order. One may choose any value (in perturbative region) as the initial renormalization scale, and the determined PMC scale, corresponding to the correct running behavior of the strong coupling at this particular order, shall be highly independent to such choice, thus solving the conventional renormalization scale ambiguity.

The PMC depends on how well we know the scale-running behavior of the strong coupling, which could be determined by using the -function via a superposition way Mojaza:2012mf (); Brodsky:2013vpa (). The conventional -function can be solved recursively, whose solution is known up to five-loop level Brodsky:2011ta (); Kniehl:2006bg (); Baikov:2016tgj (); Tarasov:1980au (); Larin:1993tp (); vanRitbergen:1997va (); Chetyrkin:2004mf (); Czakon:2004bu (); Herzog:2017ohr (). In those solutions, the QCD asymptotic scale can be determined by using the world average of the running coupling at the scale , e.g. , which leads to GeV Patrignani:2016xqp (). The asymptotic scale under different renormalization scheme can be transformed by Celmaster-Gonsalves relation Celmaster:1979km (); Celmaster:1979dm (); Celmaster:1979xr (); Celmaster:1980ji (). Moreover, those -functions are expressed by the -power series, being the active flavor number, thus in dealing with the pQCD approximate, we can inversely transform the usual -power series at each order into the -series and then apply the standard PMC procedures.

There is residual scale dependence in the finite-order PMC predictions, which is caused by the unknown high-order terms of the PMC scale 111The PMC scale is in pQCD series, whose perturbative coefficients are determined by the non-conformal -terms of pQCD series., especially we have no -term information to set the PMC scale of the highest perturbative order. This residual scale dependence is quite different from the above mentioned conventional scale-setting uncertainty, which is purely a guess work. As has been observed in many of PMC applications, such residual uncertainties are highly suppressed, even for low-order predictions, due to the rapid convergence of the conformal pQCD series.

Refs.Wang:2013bla (); Zeng:2015gha () present a PMC analysis on the Higgs decay width up to -order level. It shows a good application of PMC, the residual scale dependence for the total decay width and the decay widths of most of the separate orders are negligibly small. An exception is that the PMC scale for the -terms of the decay width has poor convergence, which leads to a somewhat larger residual scale dependence Zeng:2015gha (). It is interesting to know whether such large residual scale dependence can be suppressed by using the newly achieved -terms, as is the purpose of the present paper. Moreover, since all the PMC scales shall be improved by the newly high-order terms, a more accurate prediction on the decay width can be achieved.

The remaining parts of this paper are organized as follows. We will give the PMC analysis of the decay width up to -order level in Sec.II. Numerical results are given in Sec.III. Sec.IV is reserved for a summary. For convenience, we present the new coefficients emerged at the -order level in the Appendix.

Ii The PMC analysis of the decay width up to -order level

Up to -order level, the decay width of takes the form


where and stands for the arbitrary initial choice of renormalization scale. The perturbative coefficients, , whose expressions under the -scheme can be read from Ref.Herzog:2017dtz (). We can conveniently get their values at any other scale by using the RGE via a recursive way. Before applying the PMC, as suggested by Refs.Zeng:2015gha (); Brodsky:1998kn (); Zheng:2013uja (); Hentschinski:2012kr (); Caporale:2015uva (), it would be better to transform the pQCD series into the minimal momentum space subtraction scheme (mMOM-scheme) such that to avoid the confusion of distributing the -terms into each order. This transformation can be achieved by using the newly available relationship between the -scheme strong coupling and the mMOM-scheme strong coupling up to -level Ruijl:2017eht ().

Following the standard PMC procedures, as described in detail in Ref.Zeng:2015gha (), we can get the optimal behavior of the running coupling at each order up to the present -order. After applying the PMC, the pQCD series (1) can be rewritten as the following scheme-independent conformal series


where are conformal coefficients free of -terms. In addition to previous coefficients for a -order prediction Zeng:2015gha (), we present the new ones in the Appendix. are PMC scales, whose expressions are


To compare with the PMC scales for a -order prediction Zeng:2015gha (), the accuracy of the PMC scales have been improved by the new -terms emerged at the -order, i.e. their highest-order terms are determined by the -order terms. As for which needs the -term information at the uncalculated -order level and is undetermined, our optimal choice for is the last known PMC scale , which ensures the scheme-independence of the resultant PMC pQCD series Mojaza:2012mf (); Brodsky:2013vpa ().

Iii Numerical results

To do the numerical calculation, we take , the top-quark pole mass  ACCPT:2012TOP (), and   Aad:2015zhl ().

iii.1 Perturbative nature of the decay width up to level

Figure 1: Total decay width under conventional scale setting. The dash-dot line, the dotted line with cross symbols, the dotted lines with rhombus symbols, the dashed line and the solid line are for the predictions up to , , , , and levels, respectively.
Figure 2: Total decay width after applying the PMC. The dash-dot line, the dotted with cross symbols, the dotted line with rhombus symbols, the dashed line and the solid line are for the predictions up to , , , , and levels, respectively.

We present the total decay width up to level before and after applying the PMC in Figs.(1, 2). Taking more loop terms into consideration, the conventional scale dependence becomes smaller and smaller. Up to NLO-level, the total decay width under conventional scale-setting is almost flat versus the initial choice of scale. As a comparison, the PMC prediction for the total decay width is scale-independent even for low-order predictions. This shows that if one can determine the correct behavior of the running coupling, one can get the scale-independent prediction at any fixed order.

276.22 101.32 27.33 16.85 3.11 336.48
218.69 116.86 14.48 8.76 3.60 337.67
177.65 118.87 39.78 5.16 1.84 339.63
147.31 114.91 54.58 18.18 3.85 338.83
Table 1: Total and individual decay widths (in unit: KeV) of the decay under conventional scale-setting. stands for the individual decay width at each order with , , , , and , respectively. stands for total decay width. Several typical scales, , , , and , are adopted.
289.84 92.77 32.36 13.47 1.95 338.72
289.85 91.51 31.97 13.47 1.95 337.86
289.93 90.99 31.67 13.47 1.95 337.73
290.07 90.87 31.42 13.47 1.95 338.00
Table 2: Total and individual decay widths (in unit: KeV) of the decay under PMC scale-setting. stands for the individual decay width at each order with , , , , and , respectively. stands for total decay width. Several typical scales, , , , and , are adopted.

We present the total and individual decay widths, and , of the decay under conventional and PMC scale-setting approaches in Tables 1 and 2, where with , , , , and , respectively. For conventional scale-setting,


and for PMC scale-setting


Tables 1 and 2 show that the scale independent predictions for the conventional and PMC scale-setting approaches are quite different. The scale independence for the conventional scale-setting is achieved due to the cancellation of the scale dependence among different orders, i.e. the net scale uncertainty could be negligibly small by including enough high-order terms. It however cannot get precise value for each perturbative order. On the other hand, the scale independence of the PMC prediction is natural, which determines the optimal scale for each order via RGE, thus it shall generally get scale-independent result at each order.

Figure 3: Scale uncertainties of the individual decay width (in unit: ) under conventional and PMC scale-settings. , , , , and , respectively. The central values are for , and the errors are for .

One can define a factor, , which shows the relative importance of the high-order terms to the leading-order terms. Up to level, the factor for under conventional and PMC scale-setting approaches have the following the trends:




If taking , the perturbative nature of is almost unchanged; while the value of changes greatly, particularly for the part of the factor, its magnitude of it changes by for . We put a comparison of the scale uncertainties of the individual decay width under conventional and PMC scale-settings in Fig.(3), in which the error bars are determined by


Here the symbol ‘Max’ stands for the maximum value for . Fig.(3) shows that the separate scale errors for each orders are indeed quite large under conventional scale-setting, which are however negligible under PMC scale-setting.

iii.2 Residual scale dependence and an estimation of unknown high-order contributions after applying the PMC scale-setting

Figure 4: The , , and PMC scales , , and versus the initial choice of scale for up to -order level, which are shown by the dotted line, the dash-dot line, the dashed line, and the solid line, respectively.

Eqs.(3,4,5,6) show the PMC scales are in perturbative series which eliminate all the non-conformal -terms in the pQCD series of the decay width . If setting , the perturbative series of the PMC scales behave in the following way


Fig.(4) presents the PMC scales versus the arbitrary initial choice of renormalization scale . Those PMC scales are optimal and determine the correct behavior of the strong coupling at each order, all of which are unchanged for large values of , e.g. .

There are residual scale dependence due to unknown high-order terms, which however suffer from both the -suppression and the exponential suppression. Thus those residual scale dependence are generally small. Fig.(4) shows the PMC scales are highly independent to the choice of . By varying , the values of those four PMC scales are almost unchanged:

Figure 5: A comparison of the PMC scale from accuracy up to accuracy. The dotted line is the approximate asymptotic limit for , which is for all orders.

In our previous prediction at the -order level Zeng:2015gha (), a somewhat larger residual scale dependence is caused by the accuracy . Fig.(5) shows how changes when more loop terms have been taken into consideration. By using the up to accuracy, the residual scale dependence becomes much smaller. By varying , the variation of will be changed from for accuracy down to for accuracy. As a special case, it is found that the PMC scale such as at the LLO accuracy shall be exactly free of , a strict demonstration has been given in Ref.Brodsky:2013vpa (), which explains why the prototype of PMC, i.e. the well-known Brodsky-Lepage-Mackenzie (BLM) scale-setting approach Brodsky:1982gc (), works so successfully in many of its previous one-loop applications. The PMC provides a underlying background for BLM and extend BLM up to all-orders by systematically and unambiguously identifying all of the terms at each order of perturbation theory using a remarkable “degeneracy” pattern derived from RGE Brodsky:2013vpa (); Mojaza:2012mf ().

Figure 6: Total decay width up to -loop level, (in unit: ), for the decay width together with the predicted unknown high-order contributions (). The central values are for .

As a final remark, it is helpful to estimate the magnitude of ¡®unknown¡¯ higher-order pQCD prediction. We adopt the suggestion raised up by Ref.Wu:2014iba () for such an estimation, i.e. for a -loop pQCD prediction for the decay,


and the unknown high-order contribution is predicted by the following way Wu:2014iba ()


where , and the symbol ‘MAX’ indicates the maximum value. Under conventional scale-setting, ; Under PMC scale-setting, are PMC scales and are conformal coefficients. This way of estimating the unknown high-order pQCD prediction is natural for PMC, since after the PMC scale is set, the pQCD convergence is ensured and the dominant uncertainty is from the last term due to the unfixed PMC scale at this order. The results are presented in Fig.(6). The predicted error bars, as shown by Fig.(6), drop down much quickly when more loop terms are included, being consistent with our previous observations that due to the elimination of divergent renormalon terms, the pQCD series becomes more convergent and the contributions from the unknown high-order terms become reasonably small.

Iv Summary

In the paper, we have made a detailed analysis of the Higgs-boson decay up to -order. After applying the PMC, we obtain


where the first error is caused by the Higgs mass uncertainty GeV, and the second one is residual renormalization scale dependence for . To compare with the residual scale error KeV obtained by a -order prediction Zeng:2015gha (), it is found that the residual scale dependence is greatly suppressed by including the newly calculated -order terms. Similarly, the scale uncertainty under conventional scale-setting shall also be suppressed by including the -order terms, which changes from the -order error KeV to a smaller error KeV.

Tables 1 and 2 show that the scale dependence for the conventional and PMC scale-settings behave quite differently. The relatively smaller net total scale uncertainty for conventional scale-setting is achieved due to the cancellation of the scale dependence among different orders; thus even though the net scale uncertainty could be small by including higher-order terms, one cannot get precise values for each order by using the guessed scale. On the other hand, the net scale independence of the PMC prediction is rightly due to the scale-independence of all separate orders, since the PMC scale for each order is optimal and determined by properly using of RGE.

Even though the unknown high-order terms shall affect the accuracy of the PMC scales and leave the residual scale dependence for the PMC predictions, reliable estimates of the uncertainties associated with the finite-order pQCD predictions can be obtained by applying the PMC. The application of the PMC systematically eliminates a major theoretical systematic error and uncertainty for the pQCD predictions, thus greatly increasing the sensitivity of collider experiments to possible new physics beyond the Standard Model.

Acknowledgement: This work was supported in part by the National Natural Science Foundation of China under Grant No.11625520, No.11547010 and No.11705033; by the Project of Guizhou Provincial Department of Science and Technology under Grant No.2016GZ42963 and the Key Project for Innovation Research Groups of Guizhou Provincial Department of Education under Grant No.KY[2016]028 and No.KY[2017]067.

Appendix: New coefficients needed for a -order analysis

The coefficients with are


where is top-quark pole mass.

The coefficients are


where and are Riemannian Zeta functions.


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