We report two different types of backflow from jets by performing 2D special
relativistic hydrodynamical simulations. One is anti-parallel and
quasi-straight to the main jet (quasi-straight backflow), and the other is bent
path of the backflow (bent backflow). We find that the former appears when the
head advance speed is comparable to or higher than the local sound speed at the
hotspot while the latter appears when the head advance speed is slower than the
sound speed bat the hotspot. Bent backflow collides with the unshocked jet and
laterally squeezes the jet. At the same time, a pair of new oblique shocks are
formed at the tip of the jet and new bent fast backflows are generated via
these oblique shocks. The hysteresis of backflow collisions is thus imprinted
in the jet as a node and anti-node structure. This process also promotes
broadening of the jet cross sectional area and it also causes a decrease in the
head advance velocity. This hydrodynamic process may be tested by observations
of compact young jets.Comment: 9 pages, 5 figures, accepted for publication in ApJ

A Aeppli

A Denner

DY Bardin

E Maina

K Melnikov

M Böhm

RG Stuart

S Dittmaier

U Baur

VS Fadin

W Beenakker

Y Kurihara

English

arXiv

High-precision predictions for W-production processes are complicated by the instability of the W bosons, requirements of gauge invariance, and the necessity to include radiative corrections. Salient features and recent progress concerning these issues are discussed for the process ee->WW->4f.High-precision predictions for W-production processes are complicated by the instability of the W bosons, requirements of gauge invariance, and the necessity to include radiative corrections. Salient features and recent progress concerning these issues are discussed for the process ee->WW->4f

Dittmaier, Stefan

CERN Document Server

CERN-TH/97-302hep-ph/9710542Theoretical aspects of W physics?Stefan DittmaierTheory Division, CERNCH-1211 Geneva 23, SwitzerlandAbstract:High-precision predictions for W-production processes are compli-cated by the instability of the W bosons, requirements of gauge in-variance, and the necessity to include radiative corrections. Salientfeatures and recent progress concerning these issues are discussed forthe process ee!WW! 4f .CERN-TH/97-302October 1997? Contribution to the proceedings of The International Europhysics Conference onHigh-Energy Physics, 19-26 August 1997, Jerusalem, Israel.802: Theoretical aspects of W physicsStefan Dittmaier (Stefan.Dittmaier@cern.ch)CERN, Theory Division, SwitzerlandAbstract. High-precision predictions for W-production processes are complicatedby the instability of the W bosons, requirements of gauge invariance, and the neces-sity to include radiative corrections. Salient features and recent progress concerningthese issues are discussed for the process ee!WW! 4f .1 IntroductionThe investigation of the W boson and its properties at LEP2 [1] and possiblefuture linear e+e− colliders [2] is very promising. Together with the Fermiconstant and the LEP1 observables, an improvement of the empirical value ofthe W-boson mass MW will put better indirect constraints on the mass of theStandard-Model Higgs boson and on new-physics parameters. The W-bosonmass can be obtained by inspecting the total W-pair production cross-sectionnear threshold, where it is most sensitive to MW, or by reconstructing the in-variant masses of the W decay products. W-boson production in ee-, eγ-, andγγ-collisions also yields direct information on the vector-boson self-couplings,which are governed by the gauge symmetry. For low and intermediate centre-of-mass (CM) energies, useful information can be obtained by investigatingthe distributions over the W-production angles. For higher energies also thetotal cross-sections become very sensitive to anomalous couplings.The described experimental aims require the knowledge of the Standard-Model predictions for the mentioned observables to a high precision, e.g. forthe cross-section of W-pair production at LEP2 to 0.5% [3]. The instabilityof the W bosons, the issue of gauge invariance, and the relevance of radiativecorrections render this task highly non-trivial. In this short presentation thesesources of complications and their consequences for actual calculations arediscussed, and special emphasis is laid on recent developments. For denite-ness, we consider the process ee!WW! 4f , which is the most importantone for W physics at LEP2.2 Gauge invariance and nite-width eectsAt and beyond a per-cent accuracy, gauge-boson resonances cannot be treatedas on-shell states in lowest-order calculations, since the impact of a nite de-cay width ΓV for a gauge boson V of mass MV can be roughly estimated toΓV=MV, which is, for instance, 3% for the W boson. Therefore, the full setof tree-level diagrams for a given fermionic nal state has to be taken into ac-count. For ee!WW! 4f this includes graphs with two resonant W-boson2 S. Dittmaierlines (\signal diagrams") and graphs with one or no W resonance (\back-ground diagrams"), leading to the following structure of the amplitude [3{5]:M =R+−(k2+; k2−)(k2+ −M2W)(k2− −M2W)| {z }doubly-resonant+R+(k2+; k2−)k2+ −M2W+R−(k2+; k2−)k2− −M2W| {z }singly-resonant+ N(k2+; k2−)| {z }non-resonant:(1)Gauge invariance implies that M is independent of the gauge xing usedfor calculating Feynman graphs (gauge-parameter independence), and thatgauge cancellations between dierent contributions to M take place. Thesegauge cancellations are ruled by Ward identities. For a physical description ofthe W resonances, the nite W decay width has to be introduced in the reso-nance poles. However, since only the sum in (1), but not the single contribu-tions toM, possesses the gauge-invariance properties, the simple replacementk2 −M2W−1!k2 −M2W + iMWΓW(k2)−1(2)in general violates gauge invariance.Although such gauge-invariance-breaking terms are formally suppressedby a factor ΓV=MV [6], they can completely destroy the consistency of predic-tions if they disturb gauge cancellations [7{9]. Gauge cancellations can occurif a current u(p1)γu(p2) that is associated to a pair of external fermionsbecomes proportional to the momentum k of the attached gauge boson:p1p2kV1k2 −M2VkT V :T V represents the set of subgraphs hidden in the blob. The cancellations inkT V are governed by the Ward identitieskT γ = 0; kTZ = iMZT; kTW = MWT: (3)The rst one expresses electromagnetic current conservation and is relevant,e.g., for forward scattering of e (k ! 0). The others imply the Goldstone-boson equivalence theorem, which relates the amplitudes for high-energeticlongitudinal W and Z bosons (k0  MV) to the ones for their respectivewould-be Goldstone bosons  and .Among the proposed methods (see Refs. [3, 9] and references therein) tointroduce nite widths for W and Z bosons in tree-level amplitudes, the eld-theoretically most convincing one is provided by the \fermion-loop scheme".This scheme goes beyond a pure tree-level calculation by including and con-sistently Dyson-summing all closed fermion loops in O(). This procedureintroduces the running tree-level width in gauge-boson propagators via theimaginary parts of the fermion loops. The Ward identities (3) are not vio-lated, since the fermion-loop (as well as the tree-level) contributions to vertex802: Theoretical aspects of W physics 3functions obey the simple linear (also called \naive") Ward identities thatare related to the original gauge invariance rather than to the more involvedBRS invariance of the quantized theory. Owing to the linearity of the crucialWard identities for the vertex functions, the fermion-loop scheme works bothwith the full fermion loops and with the restriction to their imaginary parts.The full fermion-loop scheme has been worked out for ee ! WW ! 4f inRef. [9], where applications are discussed as well. Simplied versions of thescheme have been introduced in Ref. [8].The fermion-loop scheme is not applicable in the presence of resonant par-ticles that do not exclusively decay into fermions. For such particles, parts ofthe decay width are contained in bosonic corrections. The Dyson summationof fermionic and bosonicO() corrections leads to inconsistencies in the usualeld-theoretical approach, i.e. the Ward identities (3) are broken in general.This is due to the fact that the bosonicO() contributions to vertex functionsdo not obey the \naive Ward identities". The problem is circumvented by em-ploying the background-eld formalism [10], in which these naive identitiesare valid. This implies [11] that a consistent Dyson summation of fermionicand bosonic corrections to any order in  does not disturb the Ward identities(3). Therefore, the background-eld approach provides a natural generaliza-tion of the fermion-loop scheme. We recall that any resummation formalismgoes beyond a strict order-by-order calculation and necessarily involves am-biguities in relative order n if not all n-loop diagrams are included. Thiskind of scheme dependence, which in particular concerns gauge dependences,is only resolved by successively calculating the missing orders.Note that the consistent resummation of all O() loop corrections doesnot automatically lead to O() precision in the predictions if resonances areinvolved. The imaginary parts of one-loop self-energies generate only tree-level decay widths so that directly on resonance one order in  is lost. Toobtain also full O() precision in these cases, the imaginary parts of thetwo-loop self-energies are required. However, how and whether this two-loopcontribution can be included in a practical way without violating the Wardidentities (3) is still an open problem. Taking the imaginary parts of all two-loop contributions solves the problem in principle at least for the background-eld approach, but this is certainly impractical.Fortunately, the full o-shell calculation for the process ee!WW! 4fin O() is not needed for most applications. Suciently above the W-pairthreshold a good approximation should be obtained by taking into accountonly the doubly-resonant part of the amplitude (1), leading to an error of theorder of ΓW=(MW) < 0:1%. In such a \pole scheme" calculation [4,12] thenumerator R+−(k2+; k2−) has to be replaced by the gauge-independent residueR+−(M2W;M2W). The structure of this approach, which is in fact non-trivialand has not been completely carried out so far, is described below.4 S. Dittmaier3 Electroweak radiative correctionsPresent-day Monte Carlo generators for o-shell W-pair production (see e.g.Ref. [13]) include only universal electroweak O() corrections1 such as therunning of the electromagnetic coupling, (q2), leading corrections enteringvia the -parameter, the Coulomb singularity [15], which is important nearthreshold, and mass-singular logarithms  ln(m2e=Q2) from initial-state radi-ation. In leading order, the scale Q2 is not determined and has to be set to atypical scale for the process; for the following we take Q2 = s. Since the fullO() correction is not known for o-shell W pairs, the size of the neglectedO() contributions is estimated by inspecting on-shell W-pair production, forwhich the exact O() correction and the leading contributions were presentedin Refs. [16] and [17], respectively. These O() corrections have already beenimplemented in an event generator for on-shell W pairs [18]. The followingtable shows the dierence between an \improved Born approximation" IBA,which is based on the above-mentioned universal corrections, and the corre-sponding full O() correction  to the Born cross-section integrated over theW-production angle  for some CM energiesps. rangeps=GeV 161 175 200 500 1000 20000<<180 (IBA − )=% 1.5 1.3 1.5 3.7 6.0 9.310<<170 1.5 1.3 1.5 4.7 11 22Here the corrections IBA and  include only soft-photon emission. For moredetails and results we refer to Refs. [3, 19]. The quantity IBA− correspondsto the neglected non-leading corrections and amounts to 1{2% for LEP2energies, but to 10{20% in the TeV range. Thus, in view of the aimed 0:5%level of accuracy for LEP2 and all the more for energies of future linear collid-ers, the inclusion of non-leading corrections is indispensable. The large con-tributions in IBA−  at high energies are due to terms such as  ln2(s=M2W),which arise from vertex and box corrections and can be read o from thehigh-energy expansion [20] of the virtual and soft-photonic O() corrections.As explained above, a reasonable starting point for incorporating O()corrections beyond universal eects is provided by a double-pole approxima-tion. Doubly-resonant corrections to ee ! WW ! 4f can be classied intotwo types: factorizable and non-factorizable corrections [3{5]. The former arethose that correspond either to W-pair production [16] or to W decay [21].Since these corrections were extensively discussed in the literature, we focuson the non-factorizable corrections. They are furnished by diagrams in whichthe production subprocess and/or the decay subprocesses are not indepen-dent. Among such corrections, doubly-resonant contributions only arise if a\soft" photon of energy Eγ < ΓW is exchanged between the subprocesses.In Ref. [22] it was shown that the non-factorizable corrections vanish ifthe invariant masses of both W bosons are integrated over. Thus, these cor-1 The QCD corrections for hadronic nal states are discussed in Ref. [14].802: Theoretical aspects of W physics 5rections do not influence pure angular distributions, which are of particularimportance for the analysis of gauge-boson couplings. For exclusive quantitiesthe non-factorizable corrections are non-vanishing and have been calculatedin Refs. [23{25]2. It turns out [25] that the correction factor to the dieren-tial Born cross-section is non-universal in the sense that it depends on theparametrization of phase space. The calculations [23{25] have been carriedout using the invariant masses M of the W bosons, which are identiedwith the invariant masses of the respective nal-state fermion pairs, as in-dependent variables. Since all eects from the initial e+e− state cancel, theresulting correction factor does not depend on the W-production angle andis also applicable to processes such as γγ ! WW ! 4f . After integrat-ing over production and decay angles, i.e. when inspecting pure invariant-mass distributions, the non-factorizable corrections become independent ofthe fermionic nal state. Figure 1 shows that non-factorizable corrections to asingle invariant-mass distribution are of the order of 1% for LEP2 energies,shifting the maximum of the distribution by an amount of 1{2 MeV, which issmall with respect to the experimental uncertainty at LEP2 [26]. For higherenergies the non-factorizable correction is more and more suppressed.300 GeV200 GeV192 GeV184 GeV172 GeV90888684828078767472701:510:50 0:5 1 1:5nf=%M+=GeVFig. 1. Relative non-factorizable corrections to the invariant-mass distributiond=dM+ for some CM energies (plot taken from Ref. [25]).References[1] Report CERN 96-01, Physics at LEP2, eds. G. Altarelli, T. Sjo¨strandand F. Zwirner.[2] E. Accomando et al., hep-ph/9705442, to appear in Phys. Rep.2 The recent evaluations [24, 25] are in complete mutual agreement, but conrmthe analytical results of Ref. [23] only for the special nal state f ff 0 f 0.6 S. Dittmaier[3] W. Beenakker et al., in Ref. [1], hep-ph/9602351.[4] A. Aeppli, G.J. van Oldenborgh and D. Wyler, Nucl. Phys. B428 (1994)126.[5] W. Beenakker and A. Denner, Int. J. Mod. Phys. A9 (1994) 4837.[6] A. Aeppli, F. Cuypers and G.J. van Oldenborgh, Phys. Lett. B314(1993) 413.[7] Y. Kurihara, D. Perret-Gallix and Y. Shimizu, Phys. Lett. B349 (1995)367.[8] U. Baur and D. Zeppenfeld, Phys. Rev. Lett. 75 (1995) 1002;C. G. Papadopoulos, Phys. Lett. B352 (1995) 144;E.N. Argyres et al., Phys. Lett. B358 (1995) 339.[9] W. Beenakker et al., Nucl. Phys. B500 (1997) 255.[10] A. Denner, S. Dittmaier and G. Weiglein, Nucl. Phys. B440 (1995) 95;X. Li and Y. Liao, Phys. Lett. B356 (1995) 68.[11] A. Denner and S. Dittmaier, Phys. Rev. D54 (1996) 4499.[12] R.G. Stuart, Phys. Lett. B262 (1991) 113.[13] D. Bardin et al., in Ref. [1], hep-ph/9709270, and references therein.[14] E. Maina, R. Pittau and M. Pizzio, Phys. Lett. B393 (1997) 445 andhep-ph/9710375;R. Pittau, these proceedings.[15] V.S. Fadin, V.A. Khoze and A.D. Martin, Phys. Lett. B311 (1993) 311;D. Bardin, W. Beenakker and A. 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Hysteresis of Backflow Imprinted in Collimated Jets

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