diff --git a/draft.tex b/draft.tex index 58eb033..e7ef9c9 100644 --- a/draft.tex +++ b/draft.tex @@ -122,7 +122,7 @@ \allowdisplaybreaks \preprint{ZU-TH-} -\title{Towards establishing Lepton Flavour Universality breaking in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays} +\title{Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays} %\title{Novel approach for probing Lepton Flavour Universality in $B\to K^*\ell^+\ell^-$ decays} %\title{Probing Lepton Flavour Universality in $B\to K^*\ell^+\ell^-$ decays} \author{Andrea Mauri} @@ -134,32 +134,41 @@ \affiliation{Physik-Institut, Universit\"at Z\"urich, Winterthurer Strasse 190, 8057 Z\"urich, Switzerland} \begin{abstract} - Rare semileptonic $b \to s \ell^+ \ell^-$ transitions provide some of the most promising framework to search for New Physics effects. + Rare semileptonic $b \to s \ell^+ \ell^-$ transitions provide some of the most promising framework to search for new physics effects. Recent analyses have indicated an anomalous pattern in measurements of lepton-flavour-universality observables. We propose a novel approach to independently and complementary clarify the nature of these effects by performing a simultaneous amplitude analysis of $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays. - This method enables the direct determination of the difference of the Wilson Coefficients ${\cal{C}}_{9}$ and ${\cal{C}}_{10}$ between electrons and muons, - and are found to be insensitive to both local and non-local hadronic contributions. + This method enables the direct determination of + observables that encode potential non-equal couplings of muons and electrons, + %observables that encode potential difference in the coupling for muons and electrons, + %This method enables the direct determination of the difference + %of observables that encode the effects of high energy scales + %of the Wilson Coefficients ${\cal{C}}_{9}$ and ${\cal{C}}_{10}$ + %between electrons and muons, + and are found to be insensitive to non-perturbative QCD effects. We show that considering the current preferred New Physics scenario a first observation of LFU breaking in a single measurement is possible with LHCb Run-II dataset. \end{abstract} \maketitle -Flavour changing neutral current processes of {\textit{B}} meson decays, dominantly mediated by -$b \to s$ amplitudes, are crucial probes for the Standard Model (SM), -since as-yet undiscovered particles may contribute and cause observables to deviate +Flavour changing neutral current processes of {\textit{B}} meson decays +%, dominantly mediated by $b \to s$ amplitudes, +are crucial probes for the Standard Model (SM), +since as-yet undiscovered particles may contribute to these transitions and cause observables to deviate from their SM predictions~\cite{Grossman:1996ke,Fleischer:1996bv,London:1997zk,Ciuchini:1997zp}. The decay mode $\bar{B}\to \bar{K}^*\ell^+\ell^-$ is a prime example (\textit{i.e.} $\ell = \mu, e$), which offers a rich framework to study from differential decay widths to angular observables. %{\color{red} phenomenology to study, formed by differential decay widths and angular observables.} An anomalous behaviour in angular and branching fraction analyses of the decay channel $\bar{B}^{0} \to \bar{K}^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, -notably in one of the observables with reduced form-factor uncertainties, +%notably in one of the observables with reduced form-factor uncertainties, +notably in one of the observables with reduced theoretical uncertainties, $P^{\prime}_{5}$~\cite{Descotes-Genon:2015uva}. -Several models have been suggested in order to interpret these results as New Physics (NP) +Several models have been suggested in order to interpret these results as new physics (NP) signatures~\cite{Gauld:2013qja,Buras:2013qja,Altmannshofer:2013foa,Crivellin:2015era,Hiller:2014yaa,Biswas:2014gga,Gripaios:2014tna}. Nonetheless, the vector-like nature of this pattern could be also explained by -large hadronic contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) +%large hadronic contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) +non-perturbative QCD contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) that are able to either mimic or camouflage NP effects~\cite{Jager:2012uw,Jager:2014rwa}. Non-standard measurement in ratios of $b \to s \ell^+ \ell^-$ processes - such as of $R_{K}$~\cite{Aaij:2014ora} and $R_{K^{*}}$~\cite{Aaij:2017vbb} - @@ -167,16 +176,27 @@ In this case an immediate interpretation of lepton flavour universality (LFU) breaking is suggested due to the small theoretical uncertainties in their predictions~\cite{Hiller:2003js,Bordone:2016gaq}. Whilst the individual level of significance of the present anomalies is still inconclusive, -there is an appealing non-trivial consistency shown in +there is an appealing non-trivial consistent pattern shown in global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. The formalism of {\textit{b}} decays is commonly described within -an effective field theory~\cite{Altmannshofer:2008dz} - -hereafter only a subset of the Wilson coefficients (WC) $C_i$ for the basis of -dimension-six field operators $O_i$ is used for the weak Lagrangian~\cite{Bobeth:2017vxj}. +an effective field theory~\cite{Altmannshofer:2008dz}, +which probes distinct energy scales; +with regimes classified into short-distance perturbative\footnote{ +At short-distance (high energies) quarks interact weakly, being considered in an asymptotic freedom, which +allows a pertubative calculation~\cite{PhysRevD.8.3633,Politzer:1974fr,Gross:1998jx}.} +and non-calculable long-distance effects. +These can be parametrised in the weak Lagrangian in terms of induced couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC) - +and effective vertex operators with different Lorentz structure, $\mathcal{O}_i$. +Only a subset of the operators that are most sensitive to NP is examined in this work~\cite{Bobeth:2017vxj}, +\textit{i.e.} $\mathcal{O}_7$ (virtual photon exchanges), $\mathcal{O}_{9,10}$ (vector and axial currents) +and corresponding right-handed couplings with flipped helicities. +% +%hereafter only a subset of the Wilson coefficients (WC) $C_i$ for the basis of +%dimension-six field operators $O_i$ is used for the weak Lagrangian~\cite{Bobeth:2017vxj}. In this framework NP effects are incorporated -by introducing deviations in the WCs~\cite{Ali:1994bf} -({\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$). +by introducing deviations in the WCs~\cite{Ali:1994bf}, +{\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$. %For instance, whilst the individual level of significance of the present anomalies is still inconclusive, %there is an appealing non-trivial consistency shown in %global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}; @@ -187,12 +207,13 @@ or $\mathcal{C}_9$ and $\mathcal{C}_{10}$ simultaneously~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. A direct experimental determination of the WCs is currently bounded by sizeable uncertainties that arise from -non-factorisable hadronic contributions. -Some promising approaches propose to extract these non-local hadronic elements +non-factorisable hadronic contributions that are difficult to assess reliably from first principles. +Some promising approaches suggest to extract these elements either from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} -or by using the analytical and dispersive properties of these correlators~\cite{Bobeth:2017vxj}. +%or by using the analytical and dispersive properties of these correlators~\cite{Bobeth:2017vxj}. +or by exploring properties (\textit{e.g.} analyticity) of the structure of these functions~\cite{Bobeth:2017vxj}. However, these models still have intrinsic limitations, in particular -in the assumption of the parametrisation of the di-lepton invariant mass. +in the assumptions that enter in parametrisation of the di-lepton invariant mass. In this Letter we propose a new \textit{model-independent} approach that from a simultaneous unbinned amplitude analysis of both @@ -204,7 +225,8 @@ This relies on the strong correlation between the muon and electron modes imposed by the lepton-flavour universality of the hadronic matrix elements. %when examining directly the difference in Wilson coefficients. -Furthermore, in this method the full set of observables available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ +Furthermore, in this method the full set of observables +(\textit{e.g} $R_{K^{*}}$, $P^{\prime}_{5}$ and branching fraction measurements) available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays is exploited, and therefore, most stringent constraints on LFU for a single measurement can be expected. %Let us consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ @@ -215,15 +237,15 @@ %being completely dominated by the on-shell $\bar{K}^{*0}$ ($p$-wave) contribution. %The differential decay rate is hence fully described by four kinematic variables; -the di-lepton invariant mass square, $q^2$, and the three angles +the di-lepton squared invariant mass, $q^2$, and the three angles $\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. The probability density function ($p.d.f.$) for this decay can be written as % \begin{equation} -p.d.f.^{(i)} = \frac{1}{\Gamma_i} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \ +p.d.f. = \frac{1}{\Gamma} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \ \quad \text{with}\quad - \Gamma_i = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}\,, + \Gamma = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}\,, \end{equation} % with different \qsq intervals depending on the lepton flavour under study. @@ -243,14 +265,16 @@ } where ${\cal{N}}_{\lambda}^{(\ell)}$ is a normalisation factor, and ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ and $\mathcal{H}_\lambda(q^{2})$ -are local and non-local hadronic matrix elements, respectively. -While the ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factor parameters set from~\cite{Straub:2015ica} -\footnote{Following a conservative approach, uncertainties on the form factors parameters are -doubled with respect to Ref.~\cite{Straub:2015ica}}, +are referred to ``local'' and ``non-local'' hadronic matrix elements, respectively. +While the ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factor parameters set from~\cite{Straub:2015ica}\footnote{In order +to guarantee a good agreement between Light-Cone Sum Rules~\cite{Ball:1998kk,Khodjamirian:2006st} +and Lattice results~\cite{Becirevic:2006nm,Horgan:2013hoa}, +%sure full agreement among the LCSRs and Lattice results. +uncertainties on the form factors parameters are doubled with respect to Ref.~\cite{Straub:2015ica}}, the $\mathcal{H}_\lambda(q^{2})$ are described using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw} - for brevity only a subset of results is shown for the latter approach. -In the following this correlator is expressed in terms of a conformal +In the following this function is expressed in terms of a ``conformal'' variable $z(q^{2})$~\cite{Bobeth:2017vxj,Boyd:1995cf,Bourrely:2008za}, with an analytical expansion truncated at a given order $z^n$ (herein referred to as $\mathcal{H}_\lambda[z^n]$). @@ -268,12 +292,12 @@ where the usual WCs $\mathcal{C}_i^{(\mu,e)}$ are renamed as $\widetilde{\mathcal{C}}_i^{(\mu,e)}$, since a precise disentanglement between the physical meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution is -impossible at the current stage of the theoretical knowledge. +impossible at the current stage of the theoretical knowledge~\cite{Mauri:2018}. The key feature of this strategy is to realise that all hadronic matrix elements are known to be lepton-flavour universal, and thus are shared among both semileptonic decays. This benefits from the large statistics available for $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ decays that is sufficient to enable the determination of these multi-space parameters.\footnote{Note -that an amplitude analysis of the electron mode only has been always previously disregarded, +that an amplitude analysis of the electron mode only has been previously disregarded, given the limited dataset foreseen in either LHCb or Belle-II experiments.} Therefore, in a common framework these hadronic contributions are treated as nuisance parameters, while only the Wilson coefficients $\widetilde{\WC}_9^{(\mu,e)}$ @@ -309,8 +333,8 @@ %$\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$, referred to as \texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. These points are favoured by several global fit -analyses with similar significance~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}, -but chosen with reduced SM tension in order to examine a more conservative hypothesis. +analyses with similar significance~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. +%but chosen with reduced SM tension in order to examine a more conservative hypothesis. An extended unbinned maximum likelihood fit is performed to these simulated samples, in which multivariate Gaussian terms are added to the likelihood to incorporate prior knowledge @@ -330,7 +354,9 @@ up to $z^3$ and $z^4$ - free floating; % \item[iv.] and re-parametrisation of its description as - proposed in~\cite{Hurth:2017sqw}. + proposed in Ref.~\cite{Hurth:2017sqw}, \textit{i.e.} + using instead a series of $1 + a_{\lambda} + b_{\lambda}(q^{2}/6\,$GeV$^{2})$. + % \end{enumerate} % @@ -388,7 +414,7 @@ %difference of Wilson coefficients $\Delta\WC_9$ and $\Delta\WC_{10}$. We quantify the maximal expected significance with respect to the SM to be $4.6$ and $5.3\,\sigma$ for \texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. -Notice that realistic experimental effects are necessary to determine the exact sensitivity achievable. +Realistic experimental effects are necessary to determine the exact sensitivity achievable. Nevertheless, these results suggest that a first observation (with a single measurement) of LFU breaking appears to be feasible with the expected recorded statistics by the end of LHCb Run II. Furthermore, it is interesting to examine the prospects for confirming this evidence in the upcoming LHCb/Belle upgrades. @@ -434,7 +460,6 @@ \textit{e.g.} $K\pi$ S-wave from a non-resonant decay and/or a scalar resonance (see detailed discussion in Ref.~\cite{Hurth:2017hxg}). These contributions are in general expected to be small~\cite{Aaij:2015oid,Aaij:2016flj}, and in the proposed formalism they benefit from the same description between the muon and electron mode. -%these are introduced in an identical manner for muons and electrons. Therefore, in this constrained framework these effects are even further suppressed and can then be neglected for the scope of this work. @@ -446,10 +471,10 @@ $\mathcal{H}_\lambda[z^3]$ and $\mathcal{H}_\lambda[z^4]$, and we perform the fit with $\mathcal{H}_\lambda[z^2]$. We vary the choice of the $\mathcal{H}_\lambda[z^{(3,4)}]$ generated parameters, -including a \textit{provocative} set of values that minimises the tension with the $P_5'$ +including a set of values that minimises the tension with the $P_5'$ anomaly~\cite{Aaij:2015oid}, while keeping $\WC_9^{(\mu)}$ and $\WC_{10}^{(\mu)}$ at their SM values. -Despite the mis-modelling of the non-local hadronic effects in the fit, we observe +Despite the mis-modelling of the non-local hadronic effects in the fit results, we observe that the determination of $\Delta\WC_9$ and $\Delta\WC_{10}$ is always unbiased, thanks to the relative cancellation of all the shared parameters between the two channels. It is worth emphasizing that a hypothetical determination of the individual @@ -462,15 +487,16 @@ all the available information from $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays for a precise determination of LFU-breaking differences of WCs, \textit{i.e.} $\Delta\WC_9$ and $\Delta\WC_{10}$. -This relies on a shared parametrisation of the local (form-factors) and non-local ($\mathcal{H}_\lambda(q^2)$) +This relies on a shared parametrisation of the local (form-factors) and non-local ($\mathcal{H}_\lambda[z^n]$) hadronic matrix elements between the muonic and electronic channels, that in turn enables the determination of the observables of interest free from any theoretical uncertainty. Figure~\ref{fig:allComponents} illustrates the usefulness of the newly-proposed observables by combining -the different information from angular analysis and branching ratio measurements. +the different information from angular analysis to branching ratio measurements. Due to the inclusiveness of the approach, the expected sensitivity surpasses any -of projection for the current $R_{K^{*}}$ and $P^{\prime}_{5}$ anomalies alone, given the benchmark points. -Therefore, this novel formalism can be the \textit{holy grail} to observe -NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays in near future. +of the projections for the foreseen measurements of \textit{e.g.} $R_{K^{*}}$ or $P^{\prime}_{5}$ alone - given the benchmark points. +%of projection for the current anomalous measurements alone given the benchmark points. +Therefore, this novel formalism can be the most immediate method to observe unambigously +NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays. A promising feature of this framework is the possibility to extend the analysis to include other decay channels involving flavour changing neutral currents. @@ -500,33 +526,31 @@ %We acknowledge useful contributions from Gino Isidori, Danny van Dyk and Patrick Owen. %This work is supported by the Swiss National Science Foundation (SNF) under contract PZ00P2-174182. -\bibliography{references} - -\FloatBarrier - -\appendix %\newpage +%\FloatBarrier + +\appendix \section{Supplemental material} An extension of the physics case of the proposed method is to investigate the sensitivity to the chirality-flipped counterparts of the usual Wilson coefficients, -\textit{i.e.} $\WC^{\prime \,(\mu)}_9$ and $\WC^{\prime \,(\mu)}_{10}$. +\textit{i.e.} $\WC^{\prime (\mu)}_9$ and $\WC^{\prime(\mu)}_{10}$. Following the formalism discussed in this letter, the primed WCs are examined by considering in addition to the -\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios: -$\WC_{9,10}^{\prime \,(\mu)} = \WC^{\prime \,\rm{SM}}_{9,10} = 0$; -$\WC^{\prime \,(\mu)}_9 = \WC^{\prime \,(\mu)}_{10} = 0.3$; -and $\WC^{\prime \,(\mu)}_9 = - \WC^{\prime \,(\mu)}_{10} = 0.3$. -Notice that for the electron mode the $\WC_{9,10}^{\prime \,(e)}$ is set and fixed to the SM value $\WC^{\prime \,\rm{SM}}_{9,10} = 0$. +\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios for the muon only: +$\WC_{9,10}^{\prime(\mu)} = \WC^{\prime \rm{SM}}_{9,10} = 0$; +$\WC^{\prime (\mu)}_9 = \WC^{\prime (\mu)}_{10} = 0.3$; +and $\WC^{\prime (\mu)}_9 = - \WC^{\prime (\mu)}_{10} = 0.3$. +Notice that for the electron mode the $\WC_{9,10}^{\prime(e)}$ is set and fixed to the SM value $\WC^{\prime \rm{SM}}_{9,10} = 0$. \begin{figure}[b] \includegraphics[width=.4\textwidth]{plots/ellipses_CpMu_Hz.pdf} \caption{% - Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{\,(\mu)}_9$ and - $\WC'^{\,(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario - with $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0.3$. + Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and + $\WC'^{(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario + with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$. The contours correspond to $3\,\sigma$ statistical-only uncertainty bands evaluated with the expected statistics after \lhcb Run II. \label{fig:Cp_Hz} @@ -534,30 +558,31 @@ \end{figure} Figure~\ref{fig:Cp_Hz} shows the fit results for different order of the analytic expansion for -the non-local hadronic contribution for the NP scenario with $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0.3$, -and yields corresponding to the expected \lhcb Run II statistics. -The dependency on the determination of $\WC'^{\,(\mu)}_9$ and $\WC'^{\,(\mu)}_{10}$ on the +the non-local hadronic contribution for a NP scenario with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$, +and yields corresponding to the \lhcb Run II expected statistics. +The dependency on the determination of $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ on the order of the expansion clearly saturates after $\mathcal{H}_\lambda[z^3]$ and allows a measurement of the primed Wilson coefficients for the muon decay channel $B^{0} \to K^{*0} \mumu$ independent on the theoretical hadronic uncertainty. % -Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{\,(\mu)}_9$ and -$\WC'^{\,(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at \lhcb -upgrades $50\,$fb$^{-1}$ and $\,300\,$fb$^{-1}$. +Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{(\mu)}_9$ and +$\WC'^{(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at Note that only with the full capability of the LHCb experiment it is possible to start disentangling the different NP hypotheses. % \begin{figure}[t] \includegraphics[width=.4\textwidth]{plots/ellipses_CpMu.pdf} \caption{% - Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{\,(\mu)}_9$ and $\WC'^{\,(\mu)}_{10}$ - for three NP scenarios: (blue) $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0$, (orange) $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0.3$ - and (magenta) $\WC'^{\,(\mu)}_9 = - \WC'^{\,(\mu)}_{10} = 0.3$. + Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ + for three NP scenarios: (blue) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0$, (orange) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ + and (magenta) $\WC'^{(\mu)}_9 = - \WC'^{(\mu)}_{10} = 0.3$. The contours correspond to $3\,\sigma$ statistical-only uncertainty bands expected for the LHCb Upgrade (dotted) $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics. \label{fig:Cp} } \end{figure} +\FloatBarrier +\bibliography{references} \end{document} diff --git a/references.bib b/references.bib index 258e6f8..893c318 100644 --- a/references.bib +++ b/references.bib @@ -605,3 +605,113 @@ primaryClass = "hep-ph", SLACcitation = "%%CITATION = ARXIV:1709.10308;%%" } +@article{PhysRevD.8.3633, + title = {Asymptotically free gauge theories. I}, + author = {Gross, David J. and Wilczek, Frank}, + journal = {Phys. Rev. D}, + volume = {8}, + issue = {10}, + pages = {3633--3652}, + numpages = {0}, + year = {1973}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevD.8.3633}, + url = {http://link.aps.org/doi/10.1103/PhysRevD.8.3633} +} +@article{Politzer:1974fr, + author = "Politzer, H. 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