# Properties of the decay $H\to\gamma\gamma$ using the approximate $\alpha_s^4$-corrections and the principle of maximum conformality

• The Higgs boson decay channel, $H\to\gamma\gamma$, is an important channel for probing the properties of the Higgs boson. In the paper, we analyze its decay width by using the perturbative QCD corrections up to $\alpha_s^4$-order level with the help of the principle of maximum conformality (PMC). The PMC has been suggested in the literature to eliminate the conventional renormalization scheme-and-scale ambiguities. After applying the PMC, we observe that an accurate renormalization scale-independent decay width $\Gamma(H\to\gamma\gamma)$ up to N4LO level can be achieved. Taking the Higgs mass, $M_{\rm H} = 125.09\pm0.21\pm0.11$ GeV, given by the ATLAS and CMS collaborations, we obtain $\Gamma(H\to \gamma\gamma)|_{\rm LHC} = 9.364^{+0.076}_{-0.075}$ KeV.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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## Properties of the decay $H\to\gamma\gamma$ using the approximate $\alpha_s^4$-corrections and the principle of maximum conformality

• 1. Department of Physics, Chongqing University, Chongqing 401331, China
• 2. SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94039, USA
• 3. Department of Physics, Guizhou Minzu University, Guiyang 550025, China
• 4. School of Physics and Electronics, Hunan University, Changsha 410082, China

Abstract: The Higgs boson decay channel, $H\to\gamma\gamma$, is an important channel for probing the properties of the Higgs boson. In the paper, we analyze its decay width by using the perturbative QCD corrections up to $\alpha_s^4$-order level with the help of the principle of maximum conformality (PMC). The PMC has been suggested in the literature to eliminate the conventional renormalization scheme-and-scale ambiguities. After applying the PMC, we observe that an accurate renormalization scale-independent decay width $\Gamma(H\to\gamma\gamma)$ up to N4LO level can be achieved. Taking the Higgs mass, $M_{\rm H} = 125.09\pm0.21\pm0.11$ GeV, given by the ATLAS and CMS collaborations, we obtain $\Gamma(H\to \gamma\gamma)|_{\rm LHC} = 9.364^{+0.076}_{-0.075}$ KeV.

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#### 1.   Introduction

• After the discovery of the Higgs boson at the Large Hadron Collider (LHC) [14], the remaining task is to learn and confirm more of its properties either experimentally or theoretically. Among its various decay modes, the Higgs decays into two photons, $H\to\gamma\gamma$, which could be observed at the LHC or a high luminosity $e^+ e^-$ linear collider, provides a clean platform for studying the Higgs properties and for testing the standard model.

The Higgs boson couples dominantly to the massive particles, the leading-order (LO) term of the total decay width $\Gamma(H\to\gamma\gamma)$ is already at the one-loop level, which inversely makes its high-order pQCD corrections very complicated. At present, the LO, the next-to-leading order (NLO), the N2LO, the approximate N3LO and the N4LO terms for the total decay width $\Gamma(H\to\gamma\gamma)$ have been calculated in Refs.[518]. Especially, the fermionic contributions which form a gauge-invariant subset have been given in the N3LO and N4LO terms [18], as shall be shown below, those state-of-art terms give us the opportunity to achieve a more precise prediction on $\Gamma(H\to\gamma\gamma)$. Due to the complexity of high-order pQCD calculations, it is important to use the known fixed-order terms to achieve the perturbative properties as much as possible and as accurate as possible.

Following the standard renormalization group invariance, a pQCD calculable physical observable, corresponding to an infinite order prediction, should be independent to the choice of renormalization scheme and renormalization scale. However for a fixed-order approximant, one needs to set an optimal scale so as to compare with the data. Conventionally, the renormalization scale is chosen as the typical momentum flow the process or the one to eliminate the large log-terms. Under this simple treatment, the running coupling and its coefficients at the same order cannot be exactly matched, leading to the well-known renormalization scheme-and-scale ambiguities. Due to those ambiguities, the renormalization scale uncertainty is always treated as a key error for theoretical prediction, which is assumed to be decreased when more loop terms have been included. As an example, Ref.[18] shows that when going from the LO level to the approximate N4LO level, the scale dependence of the total decay width $\Gamma(H\to\gamma\gamma)$ does decrease continuously with increasing loop terms. However, such decreasing of scale dependence is caused by compensation of scale dependence among different orders and the exact value for each loop terms cannot be obtained by using the “guessing” scale. There are many other problems for such conventional scale-setting treatment [19, 20]. It is thus important to find a proper scale-setting approach to set the renormalization scale so as to achieve a more accurate fixed-order prediction.

The principle of maximum conformality (PMC) [2125] has been suggested to eliminate the renormalization scheme-and-scale ambiguities. Its key idea is to set the correct momentum flow of the process, whose value is independent to the choice of the renormalization scale, based on the renormalization group equation (RGE) and its prediction thus avoids the conventional renormalization scale ambiguities. When one applies the PMC, all non-conformal terms that govern the $\alpha_s$-running behavior of the pQCD approximant, should be systematically resummed. The PMC prediction satisfies the renormalization group invariance and all the self-consistency conditions of the renormalization group [26]. The PMC resums all of the $\{\beta_i\}$-terms, the divergent renormalon terms which are proportional to $n!\beta_0^n\alpha_s^n$ generally disappear, then a more convergent pQCD series can be naturally achieved. Due to the scheme-independent nature of the conformal series and the commesurate scale relations among different observables [27, 28], the PMC predictions are scheme independent. In this paper we shall adopt the PMC to set the renormalization scale for the decay width $\Gamma(H\to\gamma\gamma)$ up to N4LO level and show that an accurate scale-independent prediction can indeed be achieved. For clarity, we shall adopt the PMC single-scale approach (PMC-s) [29] to do the scale-setting.

• ##### 2.   Calculation technology
• The total decay width of the Higgs decays into two photons at the one-loop level takes the form

 $\Gamma(H \rightarrow \gamma\gamma) = \frac{M_{\rm H}^{3}} {64\pi} \left|A_{W}+\sum\limits_{f} A_{f}\right|^{2},$ (1)

where $M_{\rm H}$ is the Higgs mass, $A_W$ denotes the contribution which arises from the purely bosonic diagrams, and $A_f$ stands for the contribution from the amplitudes with $f = (t, b, c, \tau)$, which corresponds to top quark, bottom quark, charm quark and $\tau$ lepton, respectively.

The higher-order N2LO, N3LO and N4LO expressions have been given in Refs.[17, 18], which are given for the top-quark running mass $(m_t)$. To set the correct momentum flow of the process, only those $\{\beta_i\}$-terms that are pertained to the RGE should be resummed into $\alpha_s$. Thus as has been argued in Ref.[30], we transform those terms into the ones for the top-quark pole mass $(M_t)$ so as to avoid the entanglement of the $\{\beta_i\}$-terms from either the top-quark anomalous dimension or the RGE and then to avoid the ambiguity in applying the PMC. Such transformation of mass can be done by using the relation between $m_t$ and $M_t$, whose explicit expression up to $\alpha_s^4$-order level can be found in Ref.[31].

For convenience, we rewrite the total decay width into two parts,

 $\Gamma(H \rightarrow \gamma\gamma) = \frac{M_{\rm H}^{3}} {64\pi} \bigg( A_{\rm LO}^{2} + A_{\rm EW} {\alpha\over \pi}\bigg) + R(\mu_r),$ (2)

where $\alpha$ is the fine-structure constant.

The LO contribution $A_{\rm LO}$ and the electroweak (EW) correction $A_{\rm EW}$ are [17]

 $A_{\rm LO} = A_{W}^{(0)}+A_{f}^{(0)}+\hat A_t A_{t}^{(0)},$ (3)
 $A_{\rm EW} = 2 A_{\rm LO} A^{(1)}_{\rm EW} ,$ (4)

where $A_W^{(0)}$ is the purely bosonic contribution to the amplitude, $A_f^{(0)}$ is the contribution from the amplitude with $f = (b,c,\tau)$, $\hat{A}_t = 2 Q_t^2 \alpha \sqrt{\sqrt{2} G_{\rm F}}/\pi$, $G_{\rm F}$ is the Fermi constant, and $Q_t$ is the top-quark electric charge. All of them have been calculated in Refs.[5, 6], i.e.

 $\begin{split} A_W^{(0)} = & -\frac{\alpha\sqrt{\sqrt{2}G_{\rm F}}}{2\pi} \left[ 2+\frac{3}{\tau_W}+\frac{3}{\tau_W}\left(2-{1\over\tau_W}\right) f(\tau_W) \right], \\ A_f^{(0)} = & \sum\limits_{f = c,b,\tau} 3\frac{\alpha \sqrt{\sqrt{2} G_{\rm F}}}{\pi \tau_{f}} Q_f^2 \left[1+\bigg(1-{1\over \tau_{f}}\bigg) f(\tau_f)\right], \\ A_t^{(0)} = & 1+{7\over30} \tau_t+{2\over21} \tau_t^2+{26\over525} \tau_t^3+{512\over17325} \tau_t^4 + {1216\over63063} \tau_t^5\\& +{128\over9555} \tau_t^6, \end{split}$

where

 $f(\tau)=\left\{ \begin{array}{ll} {\rm Arcsin}^2(\sqrt{\tau}) &\quad {\rm{for}}\quad \tau\leqslant 1\\ -\displaystyle\frac{1}{4}\left[\ln\displaystyle\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}-i\pi\right]^2& \quad {\rm{for}}\quad \tau>1 \end{array} \right. ,$

$Q_f$ denotes the electric charge for $f = (c,b,\tau)$, $\tau_W = M^2_{\rm H}/(4M_W^2)$, $\tau_t = M_{\rm H}^2/(4 M_t^2)$ and $\tau_{f} = M^2_{\rm H}/(4 M_f^2)$, and the expression for the NLO electroweak term $A^{(1)}_{\rm EW}$ can be read from Refs.[31, 32].

The QCD corrections to the decay width $\Gamma(H \to \gamma\gamma)$ are separately represented by $R(\mu_r)$, whose perturbative series up to $(n+1)$-loop level can be written as

 $R_n(\mu_r) = \sum\limits_{i = 1}^{n} r_{i}(\mu) a_s^{i}(\mu_r),$ (5)

where $a_s = {\alpha_s}/\pi$, $\mu_r$ is the renormalization scale. The perturbative coefficients $r_i$ in the $\overline{\rm MS}$-scheme up to $\alpha_s^4$-order level can be derived from Refs.[17, 18]. To apply the PMC, the $n_f$-power series ($n_f$ being the active flavor number) in the coefficients $r_i$ should be rewritten into conformal terms and non-conformal $\beta_i$-terms [24, 25],

 $r_1 = r_{1,0},$ (6)
 $r_2 = r_{2,0} + r_{2,1} \beta_0,$ (7)
 $r_3 = r_{3,0} + r_{2,1} \beta_1 + 2r_{3,1} \beta_0 + r_{3,2} \beta_0^2,$ (8)
 $\begin{split} r_4 =& r_{4,0} + r_{2,1} \beta_2 + 2r_{3,1} \beta_1 + \frac{5}{2} r_{3,2} \beta_0 \beta_1 \\ & + 3r_{4,1} \beta_0 + 3r_{4,2} \beta_0^2 + r_{4,3} \beta_0^3, \\ &\cdots,\end{split}$ (9)

where the $\beta$-pattern at each order is a superposition of RGE, and all the coefficients $r_{i,j}$ can be fixed from the $n_f$-power series at the same order by using the degeneracy relations among different orders. $r_{i,0}$ are conformal coefficients which are exactly free of $\mu_r$ for the present channel, and $r_{i,j(j\neq 0)}$ are non-conformal coefficients which are functions of $\mu_r$, i.e.,

 $r_{i,j} = \sum\limits_{k = 0}^{j} C_j^k \ln^k(\mu_r^2/M_{\rm H}^2) \hat{r}_{i-k,j-k},$ (10)

where $\hat{r}_{i,j} = r_{i,j}|_{\mu_r = M_H}$. The needed $\{\beta_i\}$-functions under the $\overline{\rm MS}$-scheme are available in Refs.[3341].

Following the standard procedures of the PMC single-scale approach [29], the pQCD corrections to the decay width $\Gamma(H \to \gamma\gamma)$ can be simplified as the following conformal series,

 $R_n(\mu_r)|_{\rm PMC} = \sum\limits_{i = 1}^{n} \hat{r}_{i,0} a_s^i(Q_\star),$ (11)

where $Q_{\star}$ is the PMC scale. Using the known pQCD corrections up to N4LO level, $Q_{\star}$ can be fixed up to next-to-next-to-leading-log (${ \rm N^2LL}$) accuracy, i.e.,

 $\ln\frac{Q_\star^2}{M_{\rm H}^2} = \sum\limits_{i} T_{i} a^{i}_s(M_{\rm H}),$ (12)

whose first three coefficients with $i = (0,1,2)$ can be determined by the known five-loop QCD corrections to the decay width $\Gamma(H \to \gamma\gamma)$, which are

 $T_0 = -{\hat{r}_{2,1}\over \hat{r}_{1,0}},$ (13)
 $T_1 = {2(\hat{r}_{2,0}\hat{r}_{2,1}-\hat{r}_{1,0}\hat{r}_{3,1})\over \hat{r}_{1,0}^2} +{(\hat{r}_{2,1}^2-\hat{r}_{1,0}\hat{r}_{3,2})\over \hat{r}_{1,0}^2}\beta_0,$ (14)
 $\begin{split} T_2 = &{4(\hat{r}_{1,0}\hat{r}_{2,0}\hat{r}_{3,1}-\hat{r}_{2,0}^2\hat{r}_{2,1}) +3(\hat{r}_{1,0}\hat{r}_{2,1}\hat{r}_{3,0}-\hat{r}_{1,0}^2\hat{r}_{4,1})\over \hat{r}_{1,0}^3 }\\ &-{\hat{r}_{2,0}\hat{r}_{2,1}^2 +2(\hat{r}_{2,0}\hat{r}_{2,1}^2-2\hat{r}_{1,0}\hat{r}_{2,1}\hat{r}_{3,1} -\hat{r}_{1,0}\hat{r}_{2,0}\hat{r}_{3,2})\over \hat{r}_{1,0}^3}\beta_0\\ &-{3\hat{r}_{1,0}^2\hat{r}_{4,2}\over \hat{r}_{1,0}^3}\beta_0 +{3(\hat{r}_{2,1}^2-\hat{r}_{1,0}\hat{r}_{3,2})\over 2\hat{r}_{1,0}^2}\beta_1 \\ &+{(\hat{r}_{1,0}\hat{r}_{2,1}\hat{r}_{3,2}-\hat{r}_{1,0}^2\hat{r}_{4,3})+(\hat{r}_{1,0}\hat{r}_{2,1}\hat{r}_{3,2}-\hat{r}_{2,1}^3)\over \hat{r}_{1,0}^3}\beta_0^2. \end{split}$ (15)

It is noted that all the perturbative coefficients $T_i$ are free of $\mu_r$, Eq.(12) then indicates that the PMC scale $Q_\star$ is free of $\mu_r$. Together with the fact that the conformal coefficients $\hat{r}_{i,0}$ are also free of $\mu_r$, the PMC approximant $R_n(\mu_r)|_{\rm PMC}$ shall be exactly scale-independent. Thus the conventional scale ambiguity is eliminated.

As a subtle point, due to the perturbative nature of the PMC scale $Q_\star$, there is residual scale dependence to the pQCD approximant for the unknown higher-order terms in $Q_\star$ perturbative series. However such kind of residual scale dependence is different from conventional renormalization scale ambiguity. In fact this residual scale dependence is usually negligible due to both the $\alpha_s$-suppression and the exponential suppression. This property has been confirmed in many PMC applications done in the literature.

• ##### 3.   Numerical results and discussions
• To do the numerical calculation, we take the following ones as their central values [42]: the $W$-boson mass $M_{W} = 80.379$ GeV, the $\tau$-lepton mass $M_{\tau} = 1.77686$ GeV, the $b$-quark pole mass $M_{b} = 4.78$ GeV, the $c$-quark pole mass $M_{c} = 1.67$ GeV, the $t$-quark pole mass $M_{t} = 173.07$ GeV, and the Higgs mass $M_{H} = 125.9$ GeV. The Fermi constant $G_{\rm F} = 1.1663787\times10^{-5}$ $\rm GeV^{-2}$ and the fine structure constant $\alpha = 1/137.035999139$. To be self-consistent with the N4LO-level pQCD correction to the total decay width $\Gamma(H\to\gamma\gamma)$, we adopt the four-loop $\alpha_s$-running and $\alpha_s(M_{Z} = 91.1876{\rm GeV}) = 0.1181$ to fix the $\alpha_{s}$-running behavior.

As a comparison, we present the total decay width $\Gamma(H\to\gamma\gamma)$ up to N4LO level under conventional and PMC scale-settings in Figs. 1, 2. Agreeing with previous observations done in the literature, Fig. 1 shows that under conventional scale-setting, the scale dependence becomes smaller and smaller when more loop terms have been included. The N4LO total decay width under conventional scale-setting gives

Figure 1.  (color online) Total decay width $\Gamma(H\to\gamma\gamma)$ versus the initial scale $\mu_r$ up to N4LO level under conventional scale-setting.

Figure 2.  (color online) Total decay width $\Gamma(H\to\gamma\gamma)$ versus the initial scale $\mu_r$ up to N4LO level under PMC scale-setting.

 $\Gamma(H\to\gamma\gamma)|_{\rm Conv.} = 9.626^{+0.002}_{+0.002}\; {\rm KeV},$ (16)

where central value is for $\mu_r = M_{\rm H}$, and the renormalization scale error is for $\mu_r\in[M_{\rm H}/2, 2M_{\rm H}]$.

It should be pointed out that the above nearly scale-independence for the N4LO-level total decay width $\Gamma(H\to\gamma\gamma)$ under conventional scale-setting is caused by large cancellations of the scale dependence among different orders. This can be explicitly seen from Table 1, in which the individual decay widths at LO+EW, NLO, N2LO, N3LO and N4LO levels have been presented separately. More explicitly, we define a parameter $\kappa_i$ to measure the scale dependence of the separate decay widths at different orders, i.e.

 $\kappa_i = \frac{\left. \Gamma_{i}\right|_{\mu_{r} = M_{\rm H}/2} -\left. \Gamma_{i}\right|_{\mu_{r} = 2M_{\rm H}}}{\left. \Gamma_{i} \right|_{\mu_{r} = M_{\rm H}}},$ (17)

where the subscript $i$ stands for the individual NLO, N2LO, N3LO and N4LO decay widths, respectively. Under conventional scale-setting, we have

 $\kappa_{\rm NLO} = +20\text{%},$ (18)
 $\kappa_{\rm {N^2LO}} = -1.2\times10^3\text{%},$ (19)
 $\kappa_{\rm {N^3LO}} = +19\text{%},$ (20)
 $\kappa_{\rm {N^4LO}} = -1.6\times10^3\text{%}.$ (21)

Large magnitude of $\kappa_i$ indicates that under conventional scale-setting, there are large scale errors for each order. Due to the cancellation among different orders, the net scale error for the N4LO-level total decay width is small, which is about 0.2%.

On the other hand, as shown by Fig. 2, the PMC prediction is almost scale-independent for each order, and the PMC prediction on $\Gamma(H\to\gamma\gamma)$ quickly approaches its scale-independent “physical” value due to a faster convergence than conventional pQCD series. Because the magnitude of the newly added N3LO and N4LO terms are only about 28% and 4% of that of the N2LO terms whose magnitude is small, our previous N2LO PMC prediction agrees with the present prediction [44] with high precision. Table 1 shows that after applying the PMC, both the separate decay width and the total decay width are unchanged for $\mu_r\in[M_{\rm H}/2, 2M_{\rm H}]$. The N4LO total decay width under PMC scale-setting is

 $\Gamma(H\to\gamma\gamma)|_{\rm PMC} \equiv 9.626\; {\rm KeV}.$ (22)

The four-loop and five-loop fermionic contributions are helpful to set an accurate PMC scale. The effective scale $Q_\star$ can be fixed up to ${\rm N^{2}LL}$-accuracy by using the known five-loop pQCD corrections, i.e.

 $\ln\frac{Q_\star^2}{M_{\rm H}^2} = 1.321-4.271\alpha_s(M_{\rm H})+21.029\alpha^2_s(M_{\rm H}) .$ (23)

Figure 3 shows the perturbative nature of $Q_\star$, e.g. $|Q^{(3)}_\star-Q^{(2)}_\star|<|Q^{(2)}_\star-Q^{(1)}_\star|$, in which $Q^{(1)}_\star$ is at the LL accuracy, $Q^{(2)}_\star$ is at the NLL accuracy and $Q^{(3)}_\star$ is at the ${\rm N^{2}LL}$ accuracy, respectively. To be self-consistent and to ensure the scheme independence of the PMC prediction, in drawing Fig. 3, the ${\rm N^2LO}$ prediction is for $Q^{(1)}_\star$, the ${\rm N^3LO}$ prediction is for $Q^{(2)}_\star$ and the ${\rm N^4LO}$ prediction is for $Q^{(3)}_\star$. The nearly scale-independence for each order under PMC scale-setting is caused by the nearly conformal nature of the pQCD series.

Figure 3.  (color online) The determined effective scale $Q_\star$. $Q^{(1)}_\star$ is at the LL accuracy, $Q^{(2)}_\star$ is at the NLL accuracy and $Q^{(3)}_\star$ is at the ${\rm N^{2}LL}$ accuracy.

It is helpful to be able to estimate the “unknown” higher-order pQCD corrections. The conventional error estimate obtained by varying the scale over a certain range is not reliable, since it only partly estimates the non-conformal contribution but not the conformal one. The Pad$\acute{\rm{e}}$ approximation approach (PAA) provides a practical way for promoting a finite series to an analytic function [4547], which has recently been suggested to give a reliable prediction of uncalculated high-order terms by using the PMC conformal series [48].

As an attempt, following the same approach which has been described in detail in Ref.[48], we give a PAA+PMC prediction for $R_{n}(M_{\rm H})$ by using the preferable $[0/(n-1)]$-type Padé series. The results are presented in Fig. 4, where the “PAA” is the predicted $R_{n}(M_{\rm H})$ by using the known $R_{n-1}(M_{\rm H})$ series and the “EC” is the prediction by directly using the known PMC $R_{n}(M_{\rm H})$ series. Figure 4 shows that the difference between the “EC” and the predicted $R_{n}(M_{\rm H})$ tends to be small as more higher loops have been known, e.g. the difference for $R_{3,4}(M_{\rm H})|_{\rm PAA}$ and $R_{3,4}(M_{\rm H})|_{\rm EC}$ is already less than 1%, thus the “exact” value of $R(M_{\rm H})|_{\rm EC}$ could be directly taken as $R_{5}(M_{\rm H})|_{\rm PAA}$, i.e.

Figure 4.  (color online) Comparison of the exact (“EC”) and the predicted [$0/(n-1)$]-type “PAA” pQCD approximant $R_{n}(M_{\rm H})$ under PMC scale-setting. It shows how the PAA predictions change when more loop-terms are included.

 $R_5(M_{\rm H})|_{\rm PAA} \cong 1.614\times10^{-1}\; {\rm KeV}.$ (24)

Then the total decay width

 $\Gamma_5(H\to\gamma\gamma)|_{\rm PMC} = \left[9.626\pm5.354\times10^{-5}\right]\; {\rm KeV},$ (25)

where the error is the PAA+PMC prediction of uncalculated high-order pQCD contributions, which is negligible.

Total decay width $\Gamma(H\to\gamma\gamma)$ versus the Higgs mass $M_{\rm H}$ is presented in Fig. 5. If taking the Higgs mass as the one given by the ATLAS and CMS collaborations [49, 50], i.e. $M_{\rm H} = 125.09\pm0.21\pm0.11$ GeV, we obtain

Figure 5.  The PMC prediction of the decay width $\Gamma(H\to\gamma\gamma)$ versus the Higgs mass $M_{\rm H}$.

 $\Gamma(H\to \gamma\gamma)|_{\rm LHC} = 9.364^{+0.076}_{-0.075} \; {\rm KeV},$ (26)

As an application, we predict the “fiducial cross section” of the process $pp\to H\to\gamma\gamma$, which has been predicted by the LHC-XS group under the conventional scale-setting [51] and has been measured by ATLAS and CMS collaborations with increasing integrated luminocities [5254]. A PMC prediction has previously been given in Ref.[55] by using $\Gamma(H\to\gamma\gamma)$ up to N2LO level. Taking the same parameters as those of Refs.[51, 55, 56], e.g. $M_{\rm H} = 125$ GeV and $M_t$ = 173.3 GeV, and by using the present $\Gamma(H\to\gamma\gamma)$ up to N4LO level, we obtain $\sigma_{\rm fid}(pp\to H\to \gamma\gamma) = 30.1^{+2.3}_{-2.2}$ fb, $38.3^{+2.9}_{-2.8}$ fb, and $85.8^{+5.7}_{-5.3}$ fb for the proton-proton center-of-mass collision energy $\sqrt{S} = 7$, $8$ and $13$ TeV, respectively. Here the errors are dominated by the error of the Higgs inclusive cross section. A comparison of the recent experimental data is put in Fig. 6. A better agreement with the data at $\sqrt{S} = 7$, $8$ TeV can be achieved by applying the PMC. The ATLAS and CMS measurements at $\sqrt{S} = 13$ TeV are still of large errors and are in disagreement, and the PMC prediction prefers the CMS data.

Figure 6.  (color online) The fiducial cross section $\sigma_{\rm fid}(pp\to H\to \gamma\gamma)$ using the $\Gamma(H\to\gamma\gamma)$ up to N4LO level. The LHC-XS prediction [51], the ATLAS measurements [52, 53] and the CMS measurement [54] are presented as a comparison.

• ##### 4.   Summary
• As a summary, the PMC uses the basic RGE to set the correct $\alpha_s$-running behavior; the resultant conformal series is independent to the initial choice of renormalization scale and renormalization scheme, and thus eliminates conventional scheme-and-scale ambiguities.

Using the pQCD corrections up to the N4LO-level, we can fix the effective PMC scale up to N2LL level and an accurate scheme-and-scale independent prediction for the decay width $\Gamma(H\to\gamma\gamma)$ can be achieved. Because of the elimination of divergent renormalon terms, the pQCD convergence can be naturally improved by applying the PMC. This improvement of pQCD convergence have been found in most of the PMC applications. However as shown by the case of $\gamma\gamma^*\to\eta_c$ form factor [57], in which the magnitude of NNLO-term is still larger than the magnitude of its NLO-term even after applying the PMC, there may have remaining large logarithmic terms in the resultant PMC conformal series, diluting the pQCD convergence. In those special cases, the conventional resummation approach [58] may have some help to further improve the pQCD convergence. A detailed discussion on this point is in progress.

Reference (58)

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