In this paper we consider the fragmentation of a parton into a jet with small jet radius R. Perturbatively, logarithms of R can appear, which for narrow jets can lead to large corrections. Using soft-collinear effective theory, we introduce the fragmentation function to a jet (FFJ), which describes the fragmentation of a parton into a jet. We discuss how these objects are related to the standard jet functions. Calculating the FFJ to next-to-leading order, we show that these objects satisfy the standard Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations, with a natural scale that depends upon R. By using the standard renormalization group evolution, we can therefore resum logarithms of R. We further use the soft-collinear effective theory to prove a factorization theorem where the FFJs naturally appear, for the fragmentation of a hadron within a jet with small R. Finally, we also show how this formalism can be used to resum the ratio of jet radii for a subjet to be emitted from within a fat jet. * where Ψ n = W † n ξ n , W n is a collinear Wilson line in SCET [6,7], and R is the jet radius to be determined by specific jet algorithm. X J−1 are the final states included in the observed jet except the primary jet parton k and X / ∈J are final states not included in the jet. Throughout, we will work in D = 4 − 2ε dimensions, and use the convention, p + ≡ n · p = p 0 +n J · p, p − ≡ n · p = p 0 −n J · p, wheren J is an unit vector in the jet direction. The lightcone vectors n and n satisfy n 2 = n 2 = 0 and n · n = 2. Therefore p + ∼ 2E for a collinear particle in n J direction. The expression of FFJ in Eq. (1) is displayed in the parton frame, where the transverse momentum of the mother parton, p ⊥ , is zero.If we consider FFJ in the jet frame, where the transverse momentum of the observed jet, p ⊥ J = 0, we can do the integral on p ⊥ J using the relation p ⊥ = −p ⊥ J /z. As a result we can express FFJ asThe normalization is chosen so that at lowest order (LO) in α s , the FFJ is given by

Conventional, hadronic matter consists of baryons and mesons made of three quarks and a quark–antiquark pair, respectively1,2. Here, we report the observation of a hadronic state containing four quarks in the Large Hadron Collider beauty experiment. This so-called tetraquark contains two charm quarks, a $$\overline{{{{{u}}}}}$$ u ¯ and a $$\overline{{{{{d}}}}}$$ d ¯ quark. This exotic state has a mass of approximately 3,875 MeV and manifests as a narrow peak in the mass spectrum of D0D0π+ mesons just below the D*+D0 mass threshold. The near-threshold mass together with the narrow width reveals the resonance nature of the state.

We analyze the recent LHCb measurement of the distribution of the fraction of the transverse momentum, z(J/ψ), carried by the J/ψ within a jet. LHCb data is compared to analytic calculations using the fragmenting jet function (FJF) formalism for studying J/ψ in jets. Logarithms in the FJFs are resummed using DGLAP evolution. We also convolve hard QCD partonic cross sections, showered with PYTHIA, with leading order Non-Relativistic Quantum Chromodynamics (NRQCD) fragmentation functions and obtain consistent results. Both approaches use Madgraph to calculate the hard process that creates the jet initiating parton. These calculations give reasonable agreement with the z(J/ψ) distribution that was shown to be poorly described by default PYTHIA simulations in the LHCb paper. We compare our predictions for the J/ψ distribution using various extractions of nonperturbative NRQCD long-distance matrix elements (LDMEs) in the literature. NRQCD calculations agree with LHCb data better than default PYTHIA regardless of which fit to the LDMEs is used. LDMEs from fits that focus exclusively on high transverse momentum data from colliders are in good agreement with the LHCb measurement.The production of quarkonium is a challenging test of Quantum Chromodynamics due to the mutiple length scales involved. The LHCb collaboration [1] published the first study of J/ψ produced within jets. The distribution of the fraction of the jet's transverse momentum, p T , carried by the J/ψ, z(J/ψ), was found to disagree significantly with predictions from the PYTHIA monte carlo [2, 3] using leading order calculations of J/ψ production in the Non-Relativistic Quantum Chromodynamics (NRQCD) factorization formalism [4]. This letter is provides improved theoretical calculations of the z(J/ψ) distribution and to discuss the implications of the LHCb results for the NRQCD factorization formalism.Production of quarkonium in hadron colliders has been the subject of experimental and theoretical studies for decades. The problem is challenging because it involves several disparate scales. These include p T , which can be much larger than the mass of the bound state, ≈ 2m Q , where m Q is the mass of the heavy quark, as well as scales that are much smaller: the relative momenta, m Q v (v is the typical velocity of the heavy quarks in the bound state), the kinetic energy, m Q v 2 , and the nonperturbative scale Λ QCD .The most common approach to calculating quarkonium production is the NRQCD factorization formalism [4]. In this formalism, the cross section for J/ψ in a pp collision is written aswhere dσ[pp → cc(n)X] is the short distance cross section for producing the cc pair in a state n with definite color and angular momentum quantum numbers and O J/ψ (n) is a long distance matrix element (LDME) that describes the nonperturbative transition of the cc pair in the state n into a final state containing J/ψ. X denotes other possible particles in the final state. The quantum numbers n will be denoted 2S+1 L [i]J where the notation for angular momentum is st...

The W boson mass is measured using proton-proton collision data at $$ \sqrt{s} $$ s = 13 TeV corresponding to an integrated luminosity of 1.7 fb−1 recorded during 2016 by the LHCb experiment. With a simultaneous fit of the muon q/pT distribution of a sample of W → μν decays and the ϕ* distribution of a sample of Z → μμ decays the W boson mass is determined to be$$ {m}_w=80354\pm {23}_{\mathrm{stat}}\pm {10}_{\mathrm{exp}}\pm {17}_{\mathrm{theory}}\pm {9}_{\mathrm{PDF}}\mathrm{MeV}, $$ m w = 80354 ± 23 stat ± 10 exp ± 17 theory ± 9 PDF MeV , where uncertainties correspond to contributions from statistical, experimental systematic, theoretical and parton distribution function sources. This is an average of results based on three recent global parton distribution function sets. The measurement agrees well with the prediction of the global electroweak fit and with previous measurements.

We consider the fragmentation of a parton into a jet with small radius R in the large z limit, where z is the ratio of the jet energy to the mother parton energy. In this region of phase space, large logarithms of both R and 1 − z can appear, requiring resummation in order to have a well defined perturbative expansion. Using soft-collinear effective theory, we study the fragmentation function to a jet (FFJ) in this endpoint region. We derive a factorization theorem for this object, separating collinear and collinear-soft modes. This allows for the resummation using renormalization group evolution of the logarithms ln R and ln(1 − z) simultaneously. We show results valid to next-toleading logarithmic order for the global Sudakov logarithms. We also discuss the possibility of non-global logarithms that should appear at two-loops and give an estimate of their size. *In order to satisfy gauge invariances at each order in λ ∼ O(R) and η, following the procedure considered in Ref. [38], we redefine the collinear gluon field,