many of the difficulties of higher - order calculations in qft can be traced back to the treatment , in the framework of dimensional regularization ( dr )  , of the infinities arising in the intermediate steps of the computation .
ultraviolet ( uv ) , infrared ( ir ) , and collinear ( cl ) divergences are first dimensionally regulated , and then renormalized away  in the uv case  or canceled by combining virtual and real contributions , or reabsorbed in the collinear behavior of the initial state parton densities . in order to attack the problem numerically , it is often necessary to subtract and add back approximations of the ir / cl singular structures . at one loop ,
several well - tested subtraction procedures have been introduced in the last two decades  [ 28 ] . at two loops and beyond , the situation is more involved , but progress is under way  [ 914 ]
. the first obvious ingredient , which may lead to a significant simplification in the above picture , is a computational procedure in which all parts of the calculation can be directly treated in four dimensions . as for the virtual contribution ,
the fdr approach has been recently introduced in reference  , which allows a subtraction of the uv divergencesat the level of the integrand , leaving a four - dimensional integration over the loop momenta . in the same work , the use of fdr as an ir regulator in qed with massive fermions is also suggested . 
  in this paper ,
i present the first application of the fdr ideas in the context of fully massless qcd , where the issues related to gauge invariance are much more subtle than in the qed case .
i concentrate , in particular , on the calculation of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{o}(\alpha _ s)$$\end{document}o(s ) gluonic corrections to the \documentclass[12pt]{minimal } 
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				 \begin{document}$$h \rightarrow gg$$\end{document}hgg decay in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_\mathrm{top } \rightarrow \infty $ $ \end{document}mtop limit , and re - derive the well - known fully inclusive result  [ 16 , 17]1\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \vargamma ( h \rightarrow gg ) = \vargamma ^{(0)}(\alpha _ s(m_h^2 ) ) \left [ 1+\frac{95}{4}\,\frac{\alpha _ s}{\pi } \right ] , \end{aligned}$$\end{document}(hgg)=(0)(s(mh2))1 + 954s,where2\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \vargamma ^{(0)}(\alpha _ s(m_h^2))= \frac{g_f \alpha _
s^2(m_h^2)}{36 \sqrt{2 } \pi ^3 } m^3_h \end{aligned}$$\end{document}(0)(s(mh2))=gfs2(mh2)3623mh3is the lowest order contribution , with \documentclass[12pt]{minimal } 
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				 \begin{document}$$n_f= 0$$\end{document}nf=0 in \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha _ s(m_h^2)$$\end{document}s(mh2 ) , since only gluons are considered . despite its simplicity , all key ingredients of massless qcd are present in this process , such as the simultaneous occurrence of ir / cl divergences and uv renormalization .
the fact that the correct expression is reproduced shows that fdr is a valid and consistent approach in massless qfts , and it gives confidence in its potential to simplify multi - leg / loop computations .
section  2 provides the set - up of the calculation . in sect . 
3 , i review the fdr treatment of the uv divergences and discuss its interplay with the ir and cl infinities .
section  4 presents the fdr computation of the virtual part , while sect . 
5 deals with the real contribution and its merging with the one - loop piece .
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				 \begin{document}$$h$$\end{document}h with two , three and four gluons  mediated by an infinitely heavy top loop  is described by the lagrangian  [ 18 , 19]3\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \mathcal{l}_\mathrm{eff } = -\frac{1}{4 } a h g^{a}_{\mu \nu } g^{a,\mu \nu } , \end{aligned}$$\end{document}leff=-14ahgaga,,where4\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } a = \frac{\alpha _
s}{3\pi v}\left ( 1+\frac{11}{4}\frac{\alpha _ s}{\pi } \right ) \end{aligned}$$\end{document}a=s3v1 + 114sand \documentclass[12pt]{minimal } 
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				 \begin{document}$$v$$\end{document}v is the vacuum expectation value , \documentclass[12pt]{minimal } 
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				 \begin{document}$$v^2= ( g_f \sqrt{2})^{-1}$$\end{document}v2=(gf2)-1 .
the corresponding feynman rules are given in  , and the diagrams for the decay rate \documentclass[12pt]{minimal } 
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				 \begin{document}$$\vargamma ( h \rightarrow gg)$$\end{document}(hgg ) are drawn in fig . 
1virtual and real diagrams contributing to \documentclass[12pt]{minimal } 
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				 \begin{document}$$h \rightarrow g g ( g)$$\end{document}hgg(g ) at \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{o}(\alpha _ s^3)$$\end{document}o(s3 ) .
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				 \begin{document}$$v_6$$\end{document}v6 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$v_7$$\end{document}v7 represent gluon wave - function corrections and the dashed line stands for the higgs field . \documentclass[12pt]{minimal } 
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				 \begin{document}$$r_1(p_i , p_j , p_k)$$\end{document}r1(pi , pj , pk ) corresponds to three diagrams with permuted gluons virtual and real diagrams contributing to \documentclass[12pt]{minimal } 
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				 \begin{document}$$h \rightarrow g g ( g)$$\end{document}hgg(g ) at \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathcal{o}(\alpha _ s^3)$$\end{document}o(s3 ) .
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				 \begin{document}$$v_6$$\end{document}v6 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$v_7$$\end{document}v7 represent gluon wave - function corrections and the dashed line stands for the higgs field . \documentclass[12pt]{minimal } 
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				 \begin{document}$$r_1(p_i , p_j , p_k)$$\end{document}r1(pi , pj , pk ) corresponds to three diagrams with permuted gluons there are five graphs contributing to the virtual part \documentclass[12pt]{minimal } 
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				 \begin{document}$$\vargamma _ { v}$$\end{document}v  without counting gluon wave - function corrections  and four diagrams for the real radiation \documentclass[12pt]{minimal } 
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				 \begin{document}$$\vargamma _ { r}$$\end{document}r . in the following ,
i separately compute , in fdr , the two pieces , showing how ir / cl divergences drop in the sum5\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \vargamma _ { v}(h \rightarrow gg ) + \vargamma _ { r}(h \rightarrow ggg ) .
consider the one - loop quadratically divergent rank - two tensor6\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \int d^4q \frac{q_\alpha q_\beta } { d_0 d_1 } , \end{aligned}$$\end{document}d4qqqd0d1,with7\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } d_i ~=~ q^2-d_i,~~~d_i ~=~ m^2_i - p^2_i- 2 ( q\cdot p_i),~~~p_0=0.\nonumber \\ \end{aligned}$$\end{document}di = q2-di , di = mi2-pi2 - 2(qpi),p0=0.its uv convergence can be improved by first deforming the propagators by a vanishing amount \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu ^2$$\end{document}218\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } d_i \rightarrow \bar{d}_i~=~d_i-\mu ^2 , \end{aligned}$$\end{document}didi = di-2,and then by repeatedly using the identity9\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \frac{1}{\bar{d}_i } = \frac{1}{\bar{q}^2 } \bigg ( 1+\frac{d_i}{\bar{d}_i } \bigg ) , \end{aligned}$$\end{document}1di=1q2(1+didi),where10\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \bar{q}^2~=~q^2-\mu ^2 . \end{aligned}$$\end{document}q2=q2-2.note that the prescription in eq . 
( 6 ) can then be rewritten as11\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \frac{q_\alpha q_\beta } { \bar{d}_0 \bar{d}_1}&= q_\alpha q_\beta \left ( \left [ \frac{1}{\bar{q}^4 } \right ] + \left [ \frac{d_0+d_1}{\bar{q}^6 } \right ] + \left [ \frac{d_1 ^ 2}{\bar{q}^8 } \right ] + \frac{d_1 ^ 3}{\bar{q}^8 \bar{d}_1 } \right .
+ \frac{d_0 d_1}{\bar{q}^6 \bar{d}_1 } + \frac{d_0 ^ 2}{\bar{q}^4 \bar{d}_0\bar{d}_1 } \right ) , \end{aligned}$$\end{document}qqd0d1=qq1q4+d0+d1q6+d12q8+d13q8d1+d0d1q6d1+d02q4d0d1,where the terms in square brackets are uv divergent but depend only on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu ^2$$\end{document}2 . the fdr definition of the integral in eq . 
( 6 ) is obtained by integrating the expansion in eq .  ( 11 ) , after dropping the divergent pieces , and taking the physical limit \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu \rightarrow 0$$\end{document}0:12\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&b_{\alpha \beta } ( p_1 ^ 2,m_0 ^ 2,m_1 ^ 2 ) = \int [ d^4q ] \frac{q_\alpha q_\beta } { \bar{d}_0 \bar{d}_1 } \nonumber \\&\quad \equiv \lim _ { \mu \rightarrow 0 } \int d^4q \ , q_\alpha q_\beta \left ( \frac{d_1 ^ 3}{\bar{q}^8 \bar{d}_1 } + \frac{d_0 d_1}{\bar{q}^6 \bar{d}_1 } + \frac{d_0 ^ 2}{\bar{q}^4 \bar{d}_0\bar{d}_1 } \right ) \!.\nonumber \\ \end{aligned}$$\end{document}b(p12,m02,m12)=[d4q]qqd0d1lim0d4qqqd13q8d1+d0d1q6d1+d02q4d0d1.the r.h.s . of eq . 
( 12 ) corresponds to a well - defined four - dimensional integral , in which all uv divergences are explicitly subtracted .
the gauge - invariance properties of this definition are discussed in detail in  [ 15 , 21 ] . in the rest of this section ,
i mostly concentrate on cl and ir infinities , and , in particular , on the matching between virtual and real contributions . a convenient starting point to study the cl singularities
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				 \begin{document}$$\begin{aligned}&b_{\alpha \beta } ( 0,0,0 ) = \lim _ { \mu \rightarrow 0 } \int d^4q \ , \frac{q_\alpha q_\beta d_1 ^ 3}{\bar{q}^8 \bar{d}_1 } \nonumber \\&\quad = -8 p_1^\rho p_1^\sigma p_1^\tau \lim _ { \mu \rightarrow 0 } \int d^4q \frac{q_\alpha q_\beta q_\rho q_\sigma q_\tau } { \bar{q}^8 \bar{d}_1 } = 0 , \end{aligned}$$\end{document}b(0,0,0)=lim0d4qqqd13q8d1=-8p1p1p1lim0d4qqqqqqq8d1=0,which vanishes , after tensor decomposition , since \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_1 ^ 2= 0$$\end{document}p12=0 .
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				 \begin{document}$$\begin{aligned}&b_{\alpha } ( 0,0,0 ) = \int [ d^4q ] \frac{q_\alpha } { \bar{d}_0 \bar{d}_1 } = 0 , \nonumber \\&b(0,0,0 ) = \int [ d^4q ] \frac{1}{\bar{d}_0 \bar{d}_1 } = 0 .
\end{aligned}$$\end{document}b(0,0,0)=[d4q]qd0d1=0,b(0,0,0)=[d4q]1d0d1=0.those results coincide with dr  in which scale - less integrals are zero  and are due to a cancelation between two \documentclass[12pt]{minimal } 
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				 \begin{document}$$\ln ( \mu ^2)$$\end{document}ln(2 ) of cl and uv origin , respectively .
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				 \begin{document}$$\begin{aligned } b(p^2,0,0)&= -i \pi ^2 \lim _ { \mu \rightarrow 0}\int _ 0 ^ 1 \mathrm{d}x\ , \nonumber \\&\times \,\,\left [ \ln ( \mu ^2 -p^2 x ( 1-x ) ) -\ln ( \mu ^2 ) \right ] \ ! , \end{aligned}$$\end{document}b(p2,0,0)=-i2lim001dxln(2-p2x(1-x))-ln(2),where the first logarithm develops a cl singularity in the limit \documentclass[12pt]{minimal } 
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				 \begin{document}$$p^2 \rightarrow 0$$\end{document}p20 .
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				 \begin{document}$$1 \rightarrow 2$$\end{document}12 splittings of massless particles , are naturally regulated by the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu ^2$$\end{document}2-deformed propagators inside the loop , while the external momenta remain massless , as illustrated in fig .  2 a and  c. the real counterpart of this procedure
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				 \begin{document}$$\mu $ $ \end{document} and the internal ones stay massless .
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				 \begin{document}$$\begin{aligned } \frac{1}{2(p_i \cdot p_j ) } \rightarrow \frac{1}{(p_i+p_j)^2 } \end{aligned}$$\end{document}12(pipj)1(pi+pj)2 in any possible singular denominator of the real matrix element squared , integrate over the aforementioned massive phase space , and take the limit \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu \rightarrow 0$$\end{document}0.fig .
the one - gluon cut in a contributes to the virtual part , the two - gluon cut in b to the real radiation .
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				 \begin{document}$$h \rightarrow gg(g)$$\end{document}hgg(g ) 
  gluon splitting ir / cl singularities regulated by massive ( thick ) gluons .
the one - gluon cut in a contributes to the virtual part , the two - gluon cut in b to the real radiation .
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				 \begin{document}$$h \rightarrow gg(g)$$\end{document}hgg(g ) 
  as for the ir divergences , the reasoning follows the same lines . for example , the only ir / cl divergent scalar one - loop three - point function generated by the cut in fig . 
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				 \begin{document}$$\begin{aligned } c(s)= \int [ d^4q ]
\frac{1}{\bar{d}_0 \bar{d}_1 \bar{d}_2 } = \lim _ { \mu \rightarrow 0 } \int d^4q \frac{1}{\bar{d}_0 \bar{d}_1 \bar{d}_2 } , \end{aligned}$$\end{document}c(s)=[d4q]1d0d1d2=lim0d4q1d0d1d2,with18\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } m_0 ^ 2 = m_1 ^ 2 = m_2 ^ 2 = p_1 ^ 2= p_2 ^ 2= 0,~~~s= -2(p_1 \cdot p_2).\nonumber \\
\end{aligned}$$\end{document}m02=m12=m22=p12=p22=0,s=-2(p1p2).by denoting19\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \mu _ 0=\frac{\mu ^2}{s } , \end{aligned}$$\end{document}0=2s , one computes20\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } c(s)&= \lim _ { \mu \rightarrow 0 } \frac{i \pi ^2}{2 s } \ln ^2\bigg ( \frac{\sqrt{1 - 4\mu _ 0}+1}{\sqrt{1 - 4\mu _ 0}-1}\bigg ) \nonumber \\&= \frac{i \pi ^2}{s } \left [ \frac{\ln ^2(\mu _ 0)-\pi ^2}{2 } + i\,\pi \ln ( \mu _ 0 ) \right ] , \end{aligned}$$\end{document}c(s)=lim0i22sln2(1 - 40 + 11 - 40 - 1)=i2sln2(0)-22+iln(0),which is indeed fully matched by the inclusive real contribution in fig . 
i use the described approach to uv / cl / ir infinities to compute \documentclass[12pt]{minimal } 
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				 \begin{document}$$\vargamma _ { v}(h \rightarrow gg)$$\end{document}v(hgg ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\vargamma _ { r}(h \rightarrow ggg)$$\end{document}r(hggg ) .
the calculation is greatly simplified by eqs .  ( 13 ) and  ( 14 ) .
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				 \begin{document}$$v_1$$\end{document}v1 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$v_2$$\end{document}v2 in fig .  1
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				 \begin{document}$$\begin{aligned } \vargamma _ { v}(h \rightarrow gg)=-3 \frac{\alpha _ s}{\pi } \,\vargamma ^{(0)}(\alpha _ s)\,m^2_h\ , \mathcal{r}e \left [ \frac{c(m^2_h)}{i \pi ^2}\right ] .\nonumber \\
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				 \begin{document}$$\mu ^2$$\end{document}2  [ 15 , 21 ] : for consistency with eq . 
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				 \begin{document}$$q^2$$\end{document}q2 appearing in the numerator of a diagram should also be deformed,22\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } q^2 \rightarrow \bar{q}^2 , \end{aligned}$$\end{document}q2q2,and integrals involving \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu ^2$$\end{document}2 , such as23\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \tilde{b}}(p_1 ^ 2,m_0 ^ 2,m_1 ^ 2 ) = \int [ d^4q ]
\frac{\mu ^2}{\bar{d}_0 \bar{d}_1 } , \end{aligned}$$\end{document}b~(p12,m02,m12)=[d4q]2d0d1,require the same integrand expansion as if\documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu ^2= q^2$$\end{document}2=q2 .
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				 \begin{document}$$\begin{aligned}&{\tilde{b}}(p_1 ^ 2,m_0 ^ 2,m_1 ^ 2)\nonumber \\&\quad = \lim _ { \mu \rightarrow 0 } \int d^4q \ , \mu ^2 \left ( \frac{d_1 ^ 3}{\bar{q}^8 \bar{d}_1 } + \frac{d_0 d_1}{\bar{q}^6 \bar{d}_1 } + \frac{d_0 ^ 2}{\bar{q}^4 \bar{d}_0\bar{d}_1 } \right ) \nonumber \\&\quad = \frac{i \pi ^2}{2}\left ( m_0 ^ 2+m_1 ^ 2-\frac{p_1 ^ 2}{3}\right ) .
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				 \begin{document}$$\begin{aligned } \vargamma _ { v}(h \rightarrow gg)=\frac{3}{2 } \frac{\alpha _ s}{\pi } \,\vargamma ^{(0)}(\alpha _ s)\ , \left ( \pi ^2-\ln ^2\frac{m^2_h}{\mu ^2 } \right ) .
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				 \begin{document}$$\begin{aligned}&|m|^2 = 192\,\pi \alpha _
sa^2 \bigg [ \frac{s^3_{23}}{s_{12}s_{13 } } + \frac{s^3_{13}}{s_{12}s_{23 } } + \frac{s^3_{12}}{s_{13}s_{23 } } \nonumber \\&\quad + \,\ , \frac{2(s^2_{13}+s^2_{23})+3 s_{13}s_{23}}{s_{12 } } + \frac{2(s^2_{12}+s^2_{23})+3 s_{12}s_{23}}{s_{13 } } \nonumber \\&\quad + \,\ , \frac{2(s^2_{12}+s^2_{13})+3 s_{12}s_{13}}{s_{23}}+ 6 ( s_{12 } + s_{13 } + s_{23})\bigg ] , \end{aligned}$$\end{document}|m|2=192sa2[s233s12s13+s133s12s23+s123s13s23 + 2(s132+s232)+3s13s23s12 + 2(s122+s232)+3s12s23s13 + 2(s122+s132)+3s12s13s23 + 6(s12+s13+s23)],where \documentclass[12pt]{minimal } 
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				 \begin{document}$$s_{ij}= ( p_i+ p_j)^2$$\end{document}sij=(pi+pj)2 .
this expression is obtained from the massless result with the replacement \documentclass[12pt]{minimal } 
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				 \begin{document}$$2(p_i \cdot p_j ) \rightarrow s_{ij}$$\end{document}2(pipj)sij , in accordance with eq . 
( 16 ) . as described in sect .  3 , in order to match the virtual ir / cl singularities , \documentclass[12pt]{minimal } 
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				 \begin{document}$$|m|^2$$\end{document}|m|2 should be integrated over a massive three - gluon phase space with \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_i^2= \mu ^2$$\end{document}pi2=2 , which can be parametrized as27\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \int d \phi _ 3 = \frac{\pi ^2}{4s } \int ds_{12 } ds_{13 } ds_{23}\ , \delta ( s - s_{12}-s_{13}-s_{23}+3\mu ^2 ) , \end{aligned}$$\end{document}d3=24sds12ds13ds23(s - s12-s13-s23 + 32),where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\sqrt{s}$$\end{document}s is the higgs mass .
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				 \begin{document}$$\begin{aligned } x= \frac{s_{12}}{s}-\mu _ 0,~~y= \frac{s_{13}}{s}-\mu _ 0,~~z= \frac{s_{23}}{s}-\mu _ 0 , \end{aligned}$$\end{document}x = s12s-0,y = s13s-0,z = s23s-0,with \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu _ 0$$\end{document}0 given in eq . 
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				 \begin{document}$$\begin{aligned } x+y+z = 1 , \end{aligned}$$\end{document}x+y+z=1,all ir / cl divergent bremsstrahlung integrals can be reduced to the following ones:30\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&i(s ) = \int _ { r } \mathrm{d}x \mathrm{d}y\ , \frac{1}{(x+\mu _ 0 ) ( y+\mu _ 0 ) } , \nonumber \\&j_{p}(s ) = \int _ { r } \mathrm{d}x \mathrm{d}y\ , \frac{x^p}{(y+\mu _ 0)}\,\,~~(p \ge 0 ) , \end{aligned}$$\end{document}i(s)=rdxdy1(x+0)(y+0),jp(s)=rdxdyxp(y+0)(p0),where the integration region reads , in the fully inclusive case,31\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \int _ { r } \mathrm{d}x \mathrm{d}y \equiv \int _ { 3 \mu _ 0}^{1 - 2\sqrt{\mu _ 0 } } \mathrm{d}x \int
_ { y_-}^{y_+ } \mathrm{d}y , \end{aligned}$$\end{document}rdxdy301 - 20dxy - y+dy , with32\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } y_{\pm } & = \frac{1}{4(x + \mu _ 0 ) } \left [ ( 1-\mu _ 0)^2-(r_0 \mp r_1)^2\right ] -\mu
\mu _ 0}.\nonumber \\ \end{aligned}$$\end{document}y=14(x+0)(1-0)2-(r0r1)2-0,r0=(x-0)2 - 402,r1=(1-x)2 - 40.thus33\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&\vargamma _ { r}(h \rightarrow ggg)= 3 \frac{\alpha
_ s}{\pi } \,\vargamma ^{(0)}(\alpha _ s)\ , \nonumber \\&\quad \times \left ( \frac{1}{4}+i(m^2_h)-\frac{3}{2}j_0(m^2_h)-j_2(m^2_h ) \right ) .
\end{aligned}$$\end{document}r(hggg)=3s(0)(s)14+i(mh2)-32j0(mh2)-j2(mh2).finally , one computes , up to terms which vanish in the limit \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu _ 0 \rightarrow 0$$\end{document}00,34\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } i(s ) = \frac{\ln ^2(\mu _ 0)-\pi ^2}{2 } \end{aligned}$$\end{document}i(s)=ln2(0)-22and35\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&j_{p}(s ) = -\frac{1}{p+1 } \ln ( \mu _ 0 ) + \int _ 0 ^ 1 \mathrm{d}x \,x^p\left [ \ln ( x)+ 2 \ln ( 1-x ) \right ] \nonumber \\&\quad = -\frac{1}{p+1 } \ln ( \mu _ 0 ) \!-\!\frac{1}{p+1}\left [ \frac{1}{p+1}\!+\!2 \sum _ { n=1}^{p+1}\frac{1}{n } \right ] ( p \ge 0),\nonumber \\ \end{aligned}$$\end{document}jp(s)=-1p+1ln(0)+01dxxpln(x)+2ln(1-x)=-1p+1ln(0)-1p+11p+1 + 2n=1p+11n(p0),so that36\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&\vargamma _ { r}(h \rightarrow ggg)= \frac{3}{2 } \frac{\alpha _ s}{\pi } \,\vargamma ^{(0)}(\alpha _ s)\ , \nonumber \\&\quad \times \left ( \ln ^2\frac{m^2_h}{\mu ^2}-\pi ^2+\frac{73}{6 } -\frac{11}{3}\ln \frac{m^2_h}{\mu ^2 } \right ) .
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				 \begin{document}$$\begin{aligned } \vargamma ( h \rightarrow gg ) = \vargamma ^{(0)}(\alpha _ s ) \left [ 1+\frac{\alpha _ s}{\pi } \left ( \frac{95}{4}-\frac{11}{2 } \ln \frac{m^2_h}{\mu ^2 } \right ) \right ] .\nonumber \\ \end{aligned}$$\end{document}(hgg)=(0)(s)1+s954 - 112lnmh22.all cl / ir \documentclass[12pt]{minimal } 
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				 \begin{document}$$\ln ( \mu ^2)$$\end{document}ln(2 ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\ln ^2(\mu ^2)$$\end{document}ln2(2 ) cancel in eq .  ( 37 ) , so that the remaining \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu $ $ \end{document} is directly interpreted as the renormalization scale .
this is a typical procedure in fdr : since the uv infinities are subtracted from the very beginning , the unphysical left - over \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu $ $ \end{document} dependence is eliminated , on the perturbative level one is working at , by a finite renormalization , which fixes the bare parameters in terms of the observables  .
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				 \begin{document}$$\vargamma ^{(0)}(\alpha _ s ) \rightarrow \vargamma ^{(0)}(\alpha _
then the logarithm is reabsorbed in the gluonic running of the strong coupling constant,38\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \alpha _
s(\mu ^2)}{1+\frac{\alpha _ s}{2 \pi } \frac{11}{2 } \ln \frac{m_h^2}{\mu ^2 } } , \end{aligned}$$\end{document}s(mh2)=s(2)1+s2112lnmh22,and eq . 
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				 \begin{document}$$\alpha _ s$$\end{document}s and parton densities , conventionally derived in dr .
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				 \begin{document}$$1/\epsilon $ $ \end{document}1/ dr regulator and the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\ln ( \mu ^2)$$\end{document}ln(2 ) appearing in fdr . as for the uv infinities ,
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				 \begin{document}$$\begin{aligned } \int d^nq \frac{1}{q^2(q+p)^2}=
i \pi ^2 \int _ { 0}^{1 } \mathrm{d}x\ , [ \delta -\ln ( -p^2 x ( 1-x))],\nonumber \\ \end{aligned}$$\end{document}dnq1q2(q+p)2=i201dx[-ln(-p2x(1-x))],where40\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } n = 4 + \epsilon \,~~~\mathrm{and}~~~\delta = -\frac{2}{\epsilon } - \gamma _ e -\ln \pi .
\end{aligned}$$\end{document}n=4+and=-2-e - ln.thus , dr and fdr uv regulators are linked through the simple \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{ms}}$$\end{document}ms replacement41\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \delta \rightarrow \ln ( \mu ^2 ) .
virtual singularities follow the same pattern , as can be inferred from the exact uv / cl cancelation in eqs . 
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				 \begin{document}$$j_{p}$$\end{document}jp reads42\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned}&j^\mathrm{dr}_{p}(s ) \!=\ ! \frac{(\pi s)^{\frac{\epsilon } { 2}}}{\vargamma \left ( 1+\frac{\epsilon } { 2}\right ) } \int \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\frac{x^p}{y}\delta ( 1\!-\!x\!-\!y\!-\!z ) ( xyz)^{\frac{\epsilon } { 2 } } \nonumber \\&\quad = -\frac{1}{p+1 } ( \delta -\ln ( s ) ) -\frac{1}{p+1}\left [ \frac{1}{p+1}+2 \sum _ { n=1}^{p+1}\frac{1}{n } \right ] , \nonumber \\ \end{aligned}$$\end{document}jpdr(s)=(s)21+2dxdydzxpy(1-x - y - z)(xyz)2=-1p+1(-ln(s))-1p+11p+1 + 2n=1p+11n , which indeed coincides with eq .  ( 35 ) if \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta = \ln ( \mu ^2)$$\end{document}=ln(2 ) . finally , the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\ln ^2(\mu ^2)$$\end{document}ln2(2 ) terms  generated by overlapping ir / cl singularities  drop , together with the full constant part , when adding virtual and fully inclusive real contributions , which can be traced back to the following relation:43\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \mathcal{r}e \left [ \frac{c(s)}{i \pi ^2}\right ] = \frac{1}{s } i(s)\ , \end{aligned}$$\end{document}rec(s)i2=1si(s)between eqs . 
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				 \begin{document}$$\begin{aligned}&\mathcal{r}e \left [ \frac{1}{i \pi ^2 } \int d^nq \frac{1}{q^2(q+p_1)^2(q+p_2)^2 } \right ] \nonumber \\&\quad = \frac{1}{s } ( \pi s)^{\frac{\epsilon } { 2}}\ , \vargamma \left ( 1-\frac{\epsilon } { 2}\right ) \left [ \frac{4}{\epsilon ^2}-\frac{2}{3 } \pi ^2 \right ] , \end{aligned}$$\end{document}re1i2dnq1q2(q+p1)2(q+p2)2=1s(s)21-242 - 232,where \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_1 ^ 2= p_2 ^ 2= 0$$\end{document}p12=p22=0 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$s= -2(p_1\cdot p_2)$$\end{document}s=-2(p1p2 ) , and45\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } i^\mathrm{dr}(s)&= \frac{(\pi s)^{\frac{\epsilon } { 2}}}{\vargamma \left ( 1+\frac{\epsilon } { 2}\right ) } \int \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\frac{1}{xy}\delta ( 1\!-\!x\!-\!y\!-\!z ) ( xyz)^{\frac{\epsilon } { 2 } } \nonumber \\&= ( \pi s)^{\frac{\epsilon } { 2}}\,\vargamma \left ( 1-\frac{\epsilon } { 2}\right ) \left [ \frac{4}{\epsilon ^2}-\frac{2}{3 } \pi ^2 \right ] .
however , an important difference between dr and fdr follows from self - contractions of metric tensors coming from the feynman rules . in dr \documentclass[12pt]{minimal } 
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				 \begin{document}$$g_{\alpha \beta } g^{\alpha \beta } = n$$\end{document}gg=n , while \documentclass[12pt]{minimal } 
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				 \begin{document}$$g_{\alpha \beta } g^{\alpha \beta } = 4$$\end{document}gg=4 in fdr .
( 41 ) , and the fdr treatment of \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mu ^2$$\end{document}2 discussed in sect . 
4 , makes explicit the equivalence between fdr and dimensional reduction in the \documentclass[12pt]{minimal } 
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				 \begin{document}$$\overline{\mathrm{\mathrm ms}}$$\end{document}ms scheme . having established this , all the well - known transition rules between dimensional reduction and dr  [ 26 , 27 ] can be directly applied to fdr . in the case of eq . 
( 37 ) , it turns out that the expression is the same in both dimensional reduction ( or fdr ) and dr .
therefore , the correct strong coupling constant to be used is the customary \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha _
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				 \begin{document}$$\overline{\mathrm{ms}}$$\end{document}ms scheme , proving that the fdr result coincides with eq . 
i have presented an fdr calculation of the gluonic qcd corrections to \documentclass[12pt]{minimal } 
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				 \begin{document}$$h \rightarrow gg$$\end{document}hgg in the large top effective theory , demonstrating that ultraviolet , collinear , and infrared divergences can be simultaneously and successfully regulated in four dimensions .
i have proved the equivalence , at the one - loop level , of dimensional reduction and fdr , making the latter approach attractive also in supersymmetric calculations , where the fermionic and bosonic sectors must share the same number of degrees of freedom .
the advantage of directly working in the four - dimensional minkowsky space is expected to lead to considerable simplifications in higher - order qft computations , especially in connection with numerical techniques .
this issue , together with the extension of fdr to more loops , is currently under study .
 i apply fdr  a recently introduced four - dimensional approach to quantum field theories ( qfts)to the computation of the nlo qcd corrections to \documentclass[12pt]{minimal } 
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				 \begin{document}$$h \rightarrow gg$$\end{document}hgg in the large top mass limit . 
 the calculation involves all key ingredients of qcd  namely ultraviolet , infrared , and collinear divergences , besides \documentclass[12pt]{minimal } 
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				 \begin{document}$$\alpha _ s$$\end{document}s renormalization  and paves the way for successful use of fdr in massless one - loop qft computations . 
 i show in detail how the correct result emerges in fdr , and discuss the translation rules to dimensional regularization . 
Introduction
The model for 
FDR versus infinities
The virtual part 
The real radiation 
FDR versus DR
Conclusions