Heavy-light quark pseudoscalar and vector mesons at finite temperature

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The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules

a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we ﬁnd that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark eﬀective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coeﬃcients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suﬃces to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the eﬀective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures theﬁnite mass diﬀerence between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)deﬁned by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus eﬀectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the eﬀective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have reﬁned the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such eﬀects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only brieﬂy sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”oﬀ-shell momenta of the heavy quarks.Due to the phase redeﬁnition of the eﬀective heavy quark ﬁelds in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coeﬃcient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which deﬁne the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the eﬀective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coeﬃcient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one ﬁrst writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coeﬃcient function.possible subtraction terms.Because of theﬂavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The eﬀective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suﬃces to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonﬁeld,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usﬁrst present the result for the nonperturbative power corrections.WeﬁndK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its eﬀect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we ﬁnd that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our ﬁnal result,there is one more eﬀect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the eﬀective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the eﬀective theory[16,18].Oneﬁnds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,weﬁnd stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourﬁnal result forχ3(y)is not very diﬀerent from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisﬁgure one can also deduce how changes in the values of the vacuum condensates would aﬀect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Weﬁnd that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonﬁeld and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesﬁnancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientiﬁc,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisﬁgure and are both represented by the narrow dots.11。

research statement的范例_jiangyun

Research StatementYun JiangPh.D.candidate in Physics at U.C.Davis2013LHC-TI Graduate FellowMy current research concentrates on Large Hadron Collider(LHC)phenomenology,especially that related to Higgs physics and dark matter.The phenomenology of the125.5GeV Higgs boson,which was discovered at the LHC,in the next-to-minimal supersymmetric standard model(NMSSM)and two-Higgs-doublet model(2HDM)has constituted the main part of my work towards my Ph.D. degree.I am in the process of expanding my research work to include the topics of extra dimensions and inﬂation of the early universe.Accomplished workWeﬁrst assessed the extent to which various semi-constrained NMSSM(scNMSSM)scenarios with a∼125GeV lightest CP-even Higgs h1are able to describe the LHC signal.We found that enhancedγγrates are most natural when the h1has mass similar to the second lightest CP-even Higgs,h2,with enhancement particularly likely if the h1and h2are degenerate.To experimentally probe this possibility,we developed diagnostic tools that could discriminate whether or not there are two(or more)Higgs bosons versus just one contributing to the LHC signals at125.5GeV.In addition,we considered the case where the lightest Higgs h1provides a consistent description of the small LEP excess at98GeV whereas the heavier Higgs h2possesses the primary features of the LHC Higgs-like signals at125GeV.Besides the NMSSM studies,the2HDM,one of the simplest extensions of the Higgs sector,is another focus of my research work.We recently performed an exhaustive analysis for Type I and Type II models to address an important question:to what extent are the latest measurements of the125.5GeV Higgs-like signal at the LHC compatible with the2HDM,assuming that the observed 125.5GeV state is one of the two CP-even Higgs bosons?We also discussed the implications for future colliders,including expectations regarding other lighter or heavier Higgs bosons.In an earlier study,we examined the maximum Higgs signal enhancements that can be achieved in the2HDM in which either a single Higgs boson or multiple Higgs bosons have mass(es)near125.5GeV.We found that the constraints requiring vacuum stability,unitarity and perturbativity substantially restrict possibilities for signal enhancement.Furthermore,we extend the2HDM by adding a real gauge-singlet scalar(2HDMS),which couldbe stable under the extra Z2symmetry and thereby a possible dark matter(DM)candidate. Comparing with the simplest singlet extension this model has richer phenomenology.For heavy DM (mass above55GeV)which generates the desired relic abundance,the predicted cross section for DM-nucleon scattering is below the current LUX limit and even the future XENON1T projection. In contrast,this model can accommodate light DM,even if the constraint on Higgs invisible decay is taken into account,and describe the CDMS II and CoGeNT positive signal regions. More impressively,the tension with the LUX/SuperCDMS exclusion can be alleviated in the Type II2HDMS in which the DM-nucleon interaction could be isospin-violating.In the process of completing this project,we independently worked out the modelﬁles for the FeynRules program and will make the model database publicly available soon.Ongoing projectsBased on the comprehensive studies we have accomplished,we focus on the light(pseudo)scalar Higgs boson region in the2HDM.We are also pursuing whether the current LHC8TeV-run data pushes the2HDM to the alignment limit and/or the decoupling limit.In the meanwhile we are developing a routine to simplify the calculation for gluon-fusion and bottom-quark associated production cross sections.Besides,we consider the decoupling2HDM to determine if the vacuum could be stable above the inﬂation or GUT scale,assuming the2HDM is a low-energy eﬀective theory.If it is stable,then the inﬂation driven by the2HDM Higgs would be possible and a topic for future study.One of the most important extensions of the standard model(SM)is the inclusion of additional particle(s)that comprise the DM of the universe.So far a number of collaborations have been devoted to working on the direct detection of DM.They typically translate the limit on the event rate against recoil energy they directly detect into a limit on the DM-proton cross section as a function of DM mass.However,there are several standard assumptions hidden in this translation that might not be correct.In particular,it is normally assumed that DM has equal coupling to neutrons and protons.In fact,the tension between the null LUX/SuperCDMS exclusions and the positive signal regions favored by CDMS II and CoGeNT could be alleviated if the DM interactions with nucleons are allowed to violate isospin symmetry.Thus,we are now interested in exploring the possibilities of a light isospin-violating DM(IVDM)in the2HDMS and NMSSM even though such an isospin-violation eﬀect in supersymmetry(SUSY)models has been claimed to be negligible.If present,such light annihilating IVDM may explain the origin of the excess of gamma rayﬂux from the galactic center,as indicated in the previous studies.Another project I am now involved in is warped DM.In view of the success of extra dimensions in resolving the hierarchy andﬂavor problems of the SM,we are studying DM in warped extra dimensions in particular with Randall-Sundrum like geometries.We consider the case that all SM ﬁelds live in the bulk.In our model theﬁrst Higgs excited state is a possible stable DM candidate due to the presence of a geometric KK-parity.Our focus is on phenomenological implications of the DM after imposing constraints from current experimental data.Future planIn the near future I will continue investigating LHC implications of various Higgs models beyond the SM both within and outside the framework of SUSY.Potential extensions to my previous studies include the future prospects of2HDM at the100TeV collider and the related analyses in the framework of phenomenological NMSSM,a version of NMSSM without GUT-scale uniﬁcation assumptions.Additionally,dark matter physics and inﬂation of the early universe driven by the Higgs boson,Higgs portal DM,axion,etc.will be important topics of exploration in my post-doctoral research.It is well-known that Higgs inﬂation is unlikely to occur within the pure SM given the latest LHC measurement on the top quark mass.To remedy this issue,I am considering the additional loop contribution from Higgs portal interactions to raise the tensor-to-scalar ratio at the inﬂation scale.Another probable direction of my future work is in Higgs triplet and neutrino physics.I wish to construct a model that contains a LHC observed Higgs and a DM candidate and is also able to explain the neutrino mass by means of Type-II seesaw mechanism.Rather than being the end of the story,the discovery of the125.5GeV Higgs boson has marked a new era in particle physics.I anticipate that this discovery will provide a key window into theories beyond the SM,and that additional Higgs bosons and SUSY particles may well be found.A variety of ongoing experiments aimed at detecting dark matter will either provide further limits or succeed in detecting dark matter.Either way,DM models will be constrained and/or eliminated,thereby providing guidance to ongoing theoretical work.As a young researcher,I am fortunate to be in the midst of an exciting time and will certainly work extremely hard to contribute to our high energy physics community.。

ResearchObjectives

Research ObjectivesThe MILC Collaboration is engaged in a broad research program in Quantum Chromodynamics (QCD).This research addresses fundamental questions in high energy and nuclear physics,and is directly related to major experimental programs in theseﬁelds.It includes studies of the mass spectrum of strongly interacting particles,the weak interactions of these particles,and the behavior of strongly interacting matter under extreme conditions.The Standard Model of High Energy Physics encompasses our current knowledge of the funda-mental interactions of subatomic physics.It consists of two quantumﬁeld theories:the Weinberg-Salaam theory of electromagnetic and weak interactions,and QCD,the theory of the strong interac-tions.The Standard Model has been enormously successful in explaining a wealth of data produced in accelerator and cosmic ray experiments over the past thirty years;however,our knowledge of it is incomplete because it has been difﬁcult to extract many of the most interesting predictions of QCD,those that depend on the strong coupling regime of the theory,and therefore require non-perturbative calculations.At present,the only means of carrying out non-perturbative QCD calculations fromﬁrst principles and with controlled errors,is through large scale numerical sim-ulations within the framework of lattice gauge theory.These simulations are needed to obtain a quantitative understanding of the physical phenomena controlled by the strong interactions,to de-termine a number of the fundamental parameters of the Standard Model,and to make precise tests of the Standard Model’s range of validity.Despite the many successes of the Standard Model,it is believed by high energy physicists that to understand physics at the shortest distances,a more general theory,which uniﬁes all four of the fundamental forces of nature,will be required.The Standard Model is expected to be a limiting case of this more general theory,just as classical mechanics is a limiting case of the more general quantum mechanics.A central objective of the experimental program in high energy physics,and of lattice QCD simulations,is to determine the range of validity of the Standard Model,and to search for new physics beyond it.Thus,QCD simulations play an important role in efforts to obtain a deeper understanding of the fundamental laws of physics.QCD is formulated in the four-dimensional space-time continuum;however,in order to carry out numerical calculations one must reformulate it on a lattice or grid.It should be emphasized that the lattice formulation of QCD is not merely a numerical approximation to the continuum formu-lation.The lattice regularization of QCD is every bit as valid as continuum regularizations.The lattice spacing a establishes a momentum cutoffπ/a that removes ultraviolet divergences.Stan-dard renormalization methods apply,and in the perturbative regime they allow a straightforward conversion of lattice results to any of the standard continuum regularization schemes.Lattice QCD calculations proceed in two steps.In theﬁrst,one uses importance sampling tech-niques to generate gauge conﬁgurations,which are representative samples from the Feynman path integrals that deﬁne QCD.These conﬁgurations are saved,and in the second step they are used to calculate a wide variety of physical quantities.It is necessary to generate conﬁgurations with a range of lattice spacings,and then perform extrapolations to the zero lattice spacing limit.Fur-thermore,the computational cost of calculations rises as the masses of the quarks,the fundamental constituents of strongly interacting matter,decrease.Until recently,it has been too expensive to carry out calculations with the masses of the two lightest quarks,the up and the down,set to their physical values.Instead,one has performed calculations for a range of up and down quark masses, and extrapolated to their physical values guided by chiral perturbation theory,an effectiveﬁeld theory that determines how physical quantities depend on the masses of the lightest quarks.The extrapolations in lattice spacing(continuum extrapolation)and quark mass(chiral extrapolation) are the major sources of systematic errors in QCD calculations,and both must be under control in order to obtain trustworthy results.In our current simulations,we are,for theﬁrst time,working at or near the physical masses of the up and down quarks.The gauge conﬁgurations produced in these simulations greatly reduce,and will eventually eliminate,the systematic errors associatedwith the chiral extrapolation.A number of different formulations of QCD on the lattice are currently in use by lattice gauge theorists,all of which are expected to give the same results in the continuum limit.In recent years, major progress has been made in theﬁeld through the development of improved formulations(im-proved actions)which reduceﬁnite lattice spacing artifacts.Approximately twelve years ago,we developed one such improved action called asqtad[1],which signiﬁcantly increased the accuracy of our simulations for a given amount of computing resources.We have used the asqtad action to generate an extensive library of gauge conﬁgurations with small enough lattice spacings and light enough quark masses to perform controlled calculations of a number of physical quantities. Computational resources provided by the DOE and NSF have enabled us to complete our program of generating asqtad gauge conﬁgurations.These conﬁgurations are publicly available,and have been used by us and by other groups to study a wide range of physical phenomena of importance in high energy and nuclear physics.Ours was theﬁrst set of full QCD ensembles that enabled control over both the continuum and chiral extrapolations.We have published a review paper describing the asqtad ensembles and the many calculations that were performed with them up to2009[2]. Over the last decade,a major component of our work has been to use our asqtad gauge conﬁg-urations to calculate quantities of importance to experimental programs in high energy physics. Particular emphasis was placed on the study of the weak decays and mixings of strongly interact-ing particles in order to determine some of the least well known parameters of the standard model and to provide precise tests of the standard model.The asqtad ensembles have enabled the calcu-lation of a number of physical quantities to a precision of1%–5%,and will enable many more quantities to be determined to this precision in the coming years.These results are already having an impact on experiments in high energy physics;however,in some important calculations,partic-ularly those related to tests of the standard model,higher precision is needed than can be provided by the existing asqtad ensembles.In order to obtain the required precision,we are now working with the Highly Improved Staggered Quark(HISQ)action developed by the HPQCD Collabora-tion[3].We have performed tests of scaling in the lattice spacing using HISQ valence quarks with gauge conﬁgurations generated with HISQ sea quarks[4].We found that lattice artifacts for the HISQ action are reduced by approximately a factor of2.5from those of the asqtad action for the same lattice spacing,and taste splittings in the pion masses are reduced by approximately a factor of three,which is sufﬁcient to enable us to undertake simulations with the mass of the Goldstone pion at or near the physical pion mass.(“Taste”refers to the different ways one can construct the same physical particle in the staggered quark formalism.Although particles with different tastes become identical in the continuum limit,their masses can differ atﬁnite lattice spacing).More-over,the improvement in the quark dispersion relation enables us to include charm sea quarks in the simulations.The properties of the HISQ ensembles are described in detail in Ref.[5],and the ﬁrst physics calculations using the physical quark mass ensembles in Refs.[6,7,8].The current status of the HISQ ensemble generation project is described at the link HISQ Lattice Generation and some initial calculations with them at Recent Results.The HISQ action also has major advan-tages for the study of QCD at high temperatures,so we have started to use it in our studies of this subject.Projects using the HISQ action will be a major component of our research for the next several years.Our research is currently focused on three major areas:1)the properties of light pseudoscalar mesons,2)the decays and mixings of heavy-light mesons,3)the properties of strongly interacting matter at high temperatures.We brieﬂy discuss our research in each of these areas at the link Recent Results.References[1]The MILC Collaboration:C.Bernard et al.,Nucl.Phys.(Proc.Suppl.),60A,297(1998);Phys.Rev.D58,014503(1998);G.P.Lepage,Nucl.Phys.(Proc.Suppl.),60A,267(1998);Phys.Rev.D59,074501(1999);Kostas Orginos and Doug Toussaint(MILC),Nucl.Phys.(Proc.Suppl.),73,909(1999);Phys.Rev.D59,014501(1999);Kostas Orginos,Doug Tou-ssaint and R.L.Sugar(MILC),Phys.Rev.D60,054503(1999);The MILC Collaboration:C.Bernard et al.,Phys.Rev.D61,111502(2000).[2]The MILC Collaboration: A.Bazavov et al.,Rev.Mod.Phys.82,1349-1417(2010)[arXiv:0903.3598[hep-lat]].[3]The HPQCD/UKQCD Collaboration: E.Follana et al.,Phys.Rev.D73,054502(2007)[arXiv:hep-lat/0610092].[4]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D82,074501(2010)[arXiv:1004.0342].[5]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D87,054505(2013)[arXiv:1212.4768].[6]The MILC Collaboration: A.Bazavov et al.,Phys.Rev.Lett.110,172003(2013)[arXiv:1301.5855].[7]The Fermilab Lattice and MILC Collaborations:A.Bazavov,et al.,Phys.Rev.Lett.112,112001(2014)[arXiv:1312.1228].[8]The MILC Collaboration:A.Bazavov et al.,Proceedings of Science(Lattice2013)405(2013)[arXiv:1312.0149].。

重型燃气轮机高雷诺数CDA_叶型转捩特性数值计算

收稿日期：2021-08-26作者简介：王润禾（1997），女，硕士。

引用格式：王润禾，童歆，羌晓青，等.重型燃气轮机高雷诺数CDA 叶型转捩特性数值计算[J].航空发动机，2023，49（5）：136-142.WANG Runhe ，TONG Xin ，QIANG Xiaoqing ，et al.Numerical calculation of controlled diffusion airfoils of transition characteristics for heavy-duty gas turbine at high Reynolds number[J].Aeroengine ，2023，49（5）：136-142.航空发动机Aeroengine重型燃气轮机高雷诺数CDA 叶型转捩特性数值计算王润禾1，童歆1，羌晓青2，3，杜朝辉1，3，欧阳华1，3（上海交通大学机械与动力工程学院1，航空航天学院2：上海200240；3.燃气轮机与民用航空发动机教育部工程研究中心，上海201306）摘要：为研究重型燃气轮机的压气机叶片在高雷诺数工况下的气动性能，基于Gamma-Theta 转捩模型的雷诺时均方程对某可控扩散叶型进行了数值计算。

通过对比不控制马赫数与控制马赫数，分析高雷诺数对可控扩散叶型气动性能及转捩特性的影响。

结果表明：在不控制马赫数条件下，在零攻角时，雷诺数从7×105增大为9×105，总压损失增加了约391.95%；在高雷诺数工况下随着雷诺数的增大，叶片流动损失不断增大，叶片可用攻角范围减小，同时在叶片吸力面出现激波，干扰转捩的产生。

在控制马赫数条件下，当Ma =0.6时，在零攻角工况下，雷诺数从8.2×105增大为1×107，总压损失减小了约38.98%，吸力面转捩起始点从4.78%弦长处前移至1.11%弦长处；在高雷诺数工况下，叶片流动损失随着雷诺数的增大不断减小，吸力面转捩位置前移。

用QCD因子化方法研究B→PV两体弱衰变过程

第26卷增刊高能物理与核物理V o1.26，Supp. 2002年12月HIGH ENERGY PHYSICS AND NUCLEAR PHYSICS Dec.，2002 Two-Body B Decays to Pseudoscalar and Vector Mesonsin QCD Factorization ApproachYANG Mao-Zhi1 YANG Ya-Dong21 (Institute of High Energy Physics, CAS, Beijing 100039, China)2 (Department of Physics, Technion, Haifa 32000, Israel)Abstract Motivated by recent CELO measurements and the progress of the theory of B decays,B→PV(P=π, K; V= K*, ρ, ω) decay modes are studied in the framework of QCD factorization.All the measured branching ratios are well accommodated in the reasonable parameter space andpredictions for other decay modes are well below the experimental upper limits.Key words factorization, weak decay, mesonB physics is one of the most important fields nowadays because it is of great help for testing the quark flavor mixing theory of the standard model and exploring the source of CP violation. Most of the theoretical studies of B decays to pseudocalar and vector final states are based on the popular Naive Factorization approach[1]. As it was ponited out years ago in Ref. [2], the dominant contribution in B decays comes from the so-called Feynman mechanism, where the energetic quark created in the weak decay picks up the soft spectator softly and carries nearly all of the final-state meson's momentum. It is also shown that Pion form factor in QCD at intermediate engery scale is dominated by Feynman mechanism[3—5]. From this point, we can understand why the naive factorization approach have worked well for B and D decays, and the many existing predictions for B decays based on naive factorization and spectator ansatz do have taken in the dominant physics effects although there are shortcommings. However, with the many new data available from CLEO and an abundance of data to arrive within few years from the B factories BaBar and Belle, it is demanded highly to go beyond the naive factorization approach.Recently, Beneke et al., have formed an interesting QCD factorization formula for B exclusive nonleptonic decays[6,7]. The factorization formula incorporates elements of the naive factorization approach (as leading contribution) and the hard-scattering approach (as subleading corrections), which allows us to calculate systematically radiative(subleading nonfactorizable) corrections to naive factorization for B exclusive nonleptonic decays. An important product of the formula is that the strong final-state interaction phases are calculable, which arise from the2 高能物理与核物理(HEP &NP) 第26卷hard-scattering kernel and hence process dependent. The strong phases are very important for studying CP violation in B decays.The amplitude of B decays to two light mesons, say M 1 and M 2, is obtained through the hadronic matrix element <M 1(p 1) M 2(p 2)⏐O i ⏐B (p )>, here M 1 denotes the final meson that picks up the light spectator quark in the B meson, and M 2 is the another meson which is composed of the quarks produced from the weak decay point of b quark. Since the quark pair, forming M 2, is ejected from the decay point of b quark carrying the large energy of order of m b , soft gluons with the momentum of order of ΛQCD decouple from it at leading order of ΛQCD /m b in the heavy quark limit. As a consequence any interaction between the quarks of M 2 and the quarks out of M 2 is hard at leading power in the heavy quark expansion. On the other hand, the light spectator quark carries the momentum of the order of ΛQCD , and is softly transferred into M 1 unless it undergoes a hard interaction.Any soft interaction between the spectator quark and other constituents in B and M 1 can be absorbed into the transition form factor of B →M 1. The non-factorizable contribution to B →M 1 M 2 can be calculated through the diagrams in Fig.1.Fig. 1. Order αs non-factorizable contributions in B →M 1M 2 decays.The O i 's incorporated in Fig.1 are the operators in the effective Hamiltonian for B decays [8], ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+++⎟⎟⎠⎞⎜⎜⎝⎛++=∑∑∑∑====21103g g ccqcb 21103g g uuq ub F eff2i i i i i i *i i i i i i *O C O C O C V V O C O C O C V V G H , (1)Where()()A V A V O －－ββααu 1b u u q ⋅=, ()()A V A V O －－αββαu 2b u u q ⋅=, ()()A V A V O －－ββααc 1b c c q ⋅=, ()()A V A V O －－αββαc 2b c c q ⋅=,()()AV A V O －－ββq αα3q q b q ′′⋅=∑′,()()AV A V O －－αβq βα4q q b q ′′⋅=∑′,()()AV A V O +′′′⋅=∑ββq αα5q q b q －, ()()A V A V O +′′′⋅=∑αβq βα6q q b q －, ()()A V A V e O +′′′′⋅=∑ββq q αα7q q b q 23－, ()()A V A V e O +′′′′⋅=∑αq q b q 23βq q βα8－,增刊杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 3()()A V A V e O －－ββq q αα9q q b q 23′′⋅=∑′′, ()()A V A V e O －－αβq q βα10q q b q 23′′⋅=∑′′, ()()AA a g G b R m g O µνβαβµνλσ2/d π8/b 2s =. (2)Here q=d, s and (q'ε {u, d, s, c, b}), α and β are the SU (3) color indices and , A =1,...,8 are the Gell-Mann matrices, and denotes the gluonic field strength tensor. The Wilson coefficients evaluated at µ=m AαβλAG µνb scale are[8]C 1= 1.082, C 2=−0.185, C 3= 0.014, C 4=−0.035, C 5= 0.009, C 6=−0.041,C 7=−0.002/137, C 8=0.054/137, C 9=−1.292/137, C 10=0.262/137, C g =−0.143. (3) The non-factorizable contributions to B →M 1M 2 can be calculated through the diagrams in Fig.1. The details of the calculations can be found in Ref. [9]. In the numerical calculations we use[10]τ (B +) = 1.65×10－12s, τ (B 0) = 1.56×10－12s,M B = 5.2792GeV , m b = 4.8GeV , m c = 1.4GeV , f B= 0.180GeV , f π = 0.133GeV , f K = 0.158GeV , f K * = 0.214GeV , f ρ = 0.21GeV , f ω = 0.195GeV .For the chiral enhancement factors for the pseudoscalar mesons, we takeR π ±π= R K ±, 0 = －1.2 ,which are consistent with the values used in [6, 11, 12]. We should take care for R π0. As pointedout in Ref. [7], R π0 for π0 should be －2M /(m 2b (m u + m d )) and equal to R π± due to inclusion ofisospin breaking effects correctly.For the form factors, we take the results of light-cone sum rule[13,14]F B →π(0)=0.3, F B →K (0)=1.13F B →π(0), A =0.372, A =0.470,ρB 0→*K B 0→and assume (0)=1.2(0) since we find larger (0) is preferred by experimental data.ωB 0→A ρB 0→A ωB 0→A We take the leading-twist distribution amplitude (DA) φ(x ) and the twist-3 DA φ0(x ) of light pseudoscalar and vector mesons as the asymptotic form[15]φP,V (x ) =6x (1－x ), (x ) =1. (4) 0P φFor the B meson, the wave function is chosen as[16,17]()(),xM x x N x ⎥⎥⎦⎤⎢⎢⎣⎡=2B 22B 22B B 2exp 1ωφ－－ (5)with ωB =0.4GeV , and N B is the normalization constant to make(x ) =1. φ∫1Bd φx B (x ) is stronglypeaked around x =0.1, which is consistent with the observation of Heavy Quark Effective Theory that the wave function should be peaked around ΛQCD /M B .We have used the unitarity of the CKM matrix V *uq V ub +V *V cq cb +V *tq V tb =0 to decompose the4 高能物理与核物理(HEP &NP)第26卷amplitudes into terms containing , V *uq V ub and V *V cq cb , and⏐V us ⏐=λ=0.2196, ⏐V ub /V cb ⏐=0.085±0.02, ⏐V cb ⏐=0.0395±0.0017, ⏐V ud ⏐=1－λ2/2 . (6) We leave the CKM angle γ as a free parameter.The numerical results of the branching ratios B →PV are shown in Fig.2 as the function of CKM angle γ. We can see from Fig. 2(a), (b) and (c) that for the three detected channels the predicted branching ratios agree well with the CLEO experiment data [18]. Our predictions for other decay modes are well below their 90% C.L. upper limits.There are several works available with detailed analysis of the CLEO new data of the decays of B to charmless PV states[11,12,19]. It is worth to note that the shortcomings in the “generalizedfactorization” are resolved in the framework of QCD Factorization. Nonfactorizable effects are calculated in a rigorous way here instead of being parameterized by effective color number. Since the hard scattering kernals are convoluted with the light cone DAs of the mesons, gluon virtualityk 2=2b m x in the penguin diagram Fig. 1(e) has well defined meaning and leaves no ambiguity as tothe value of k 2, which has usually been treated as a free phenomenological parameter in the estimations of the strong phase generated though the BSS mechanism [20]. So that CP asymmetries are predicted soundly in this approach. We present the numerical result of the branching ratios of B →PV decays in Table 1 with the relevant strong phases shown explicitly. It shows that the strong phases are generally mode dependent.Table 1. Strong phases in the branching ratios (in units of 10－6) for thecharmless decays modes studied by CLEO. (γ =Arg V *u b )B (B －→π－ρ0)=6.65⏐0.11e －i86.5°+e －i γ⏐2B (0B →π+ρ－)=19.79⏐0.11e i9.02°+e －i γ⏐2B (0B →π－ρ+)=13.43⏐0.03e i172°+e －i γ⏐2B (B －→π－ω)=10.59⏐0.065e i26.01°+e －i γ⏐2B (0B →π0ρ0)=0.11⏐0.21e 2.90°+e －i γ⏐2B (B －→π0ρ－)=10.81⏐0.176e i7.20°+e －i γ⏐2B (0B →π－ω)=1.49×10－3⏐1.64e i148°+e －i γ⏐2B (B －→K －ρ0)=0.55⏐0.24e －i162°+e －i γ⏐2B (B －→π－⎯K *0)=0.0012⏐56.4e －i15.7°+e －i γ⏐2B (B －→K －K *0)=0.030⏐2.86e i164°+e －i γ⏐2B (B －→π0K *－)=0.59⏐2.80e －i169°+e －i γ⏐2B (B －→K －ω)=0.80⏐0.48e －i9.23°+e －i γ⏐2B (0B →K 0ω)=0.72⏐0.81e －i 11.8°+e －i γ⏐2B (⎯B 0→K －ρ+)=0.96⏐0.63e －i7.20°+e －i γ⏐2B (0B →π0⎯K *0)=0.004⏐12.89e i67.61°+e －i γ⏐2Hou, Smith and W ürthwein have performed a model dependent fit using the recent CLEOdata and found γ =114degree. Using SU (3) flavor symmetry, Gronau and Rosner have analyzedthe decays of B to charmless PV final states extensively and found several processes are consistent with cos γ < 0. In this paper we find cos γ < 0 is favored by the B 2521+－－→π－ρ0 and ⎯B 0→π－ρ++π+ρ－ if their experimental center values are taken seriously. To meet its center value with cos γ < 0 , B －→π增刊杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 5－ω would indicate larger form factor i.e. A (0) > A (0). In our numerical calculation, wehave taken A (0) = 0.446 which is still consistent with the LCSR results 0.372 ± 0.074ω→B 0ρ→B 0ω→B 0[13]. It isalso interesting to note that ⎯B 0→π+ρ－ is suppressed by cos γ < 0 while ⎯B 0→π－ρ+ is enchanced. The defference between Br (⎯B 0→π+ρ－) and Br (⎯B 0→π－ρ+) is much more sensitive to γ than their sum.6 高能物理与核物理(HEP &NP) 第26卷Summarywe have calculated the branching ratios and CP asymmetries of the charmless decays B →PV(P = (π, K), V= (ρ,ω, K *)) in QCD factorization approach. We have used LCSR formfactors F B →π,K (0) and A (0) as inputs. The results of Br (B *K ,0ρ－→π－ρ0) and Br (⎯B 0→π±ρ) agree with CLEO m [18]very well and favor cos γ < 0 if their experimental center values are taken seriously. To meet its experimental center value and cos γ < 0, the decay B －→π－ω will prefer larger form factor (0). For the other decay modes, the branching ratios are predicted well below their 90% C.L. upper limits given in Ref. [18].ωB 0→A References1 Bauer M, Stech B, Wirbel M. Z. Phys., 1985, C29: 637; Z. Phys., 1987, C34: 1032 Chernyak V L, Zhitnitsky L R. Nucl. Phys., 1990, B345: 1373 Isgur N, Llewelyn-Smith C H. Phys. Rev. Lett., 1984, 52: 1080; Nucl. Phys., 1989, B317: 5264 Radyushkin A V . Acta Phys., 1984, Pol. 15: 4035 Stefanis N G . hep-ph/99113756 Beneke M, Buchalla G , Neubert M. Phys. Rev. Lett., 1999, 83: 19147 Beneke M, Buchalla G , Neubert M et al. hep-ph/00061248 Buchalla G , Buras A J, Lautenbacher M E. Rev. Mod. Phys., 1996, 68: 1125 9 YANG M Z, YANG Y Y . Phys. Rev., 2000, D62: 114019 10 Particle Data Group. Eur. Phys. J., 1998, C3: 1 11 CHENG H Y , YANG K C. hep-ph/991029112 HOU W S, Smith J G , W ürthwein F W. hep-ex/9910014 13 Ball P, Braun V M. Phys. Rev., 1998, D58: 094016 14 Ball P. JHEP09, 005(1998)15Lepage G P, Brodsky S J. Phys. Lett., 1979, B87: 359; Chernyak V L, Zhitinissky A R. Phys. Rep., 1983, 112: 173; Braun V M, Filyanov I E. Z. Phys., 1990, C48: 239 16 Keum Y Y , LI H -N, Sanda A I. Phys. Lett., 2001, B504: 2; Phys. Rev., 2001, D63: 054008 17 LÜ C D, Ukai D, YANG M Z. Phys. Rev., 2001, D63: 07400918 CLEO Collaboration. CLEO CONF 99-13; CLEO Collaboration. CLNS 99/1652 and CLEO 99-19 19 Gronau M, Rosner J L. Phys. Rev., 2000, D61: 073008 20Bander M, Silverman D, Soni A. Phys. Rev. Lett., 1979, 43: 242增刊杨茂志等：用QCD因子化方法研究B→PV两体弱衰变过程7 用QCD因子化方法研究B→PV两体弱衰变过程杨茂志1 杨亚东21 (中国科学院高能物理研究所北京 100039)2 (Department of Physics, Technion, Haifa 32000, Israel)摘要基于最近CLEO实验和B介子物理中理论研究的进展, 在QCD因子化方案下研究了B介子到一个赝标π, K和一个矢量介子ρ, ω的两体弱衰变过程.在合理的参数范围内, 理论计算与实验相符得很好.关键词因子化弱衰变介子。

赛米控丹佛斯 SEMITRANS 全碳化硅功率模块 SKM260MB170SCH17 数据表

© by SEMIKRONRev. 1.0–04.11.20201SEMITRANS ®3SiC MOSFET ModuleSKM260MB170SCH17Features*•Full Silicon Carbide (SiC) power module•High reliability 2nd Generation SiC MOSFETs•Optimized for fast switching and lowest power losses•External SiC Schottky Barrier Diode embedded•Insulated copper baseplate using DBC technology (Direct Bonded Copper)•Improved thermal performances with Aluminum Nitride (AlN) substrate •UL recognized, file no. E63532Typical Applications•High frequency power supplies •AC inverters •Traction APU •EV Chargers•Industrial Test SystemsRemarks•Case temperature limited to T C = 125°C max.•Recommended T jop = -40 ... +150°C •Gate-Source SURGE VOLTAGE(t surge <300ns), V GS_surge = -10V ... +26VAbsolute Maximum Ratings SymbolConditions Values UnitMOSFET V DSS T j =25°C 1700VI D T j =175°CT c =25°C 378 A T c =80°C301 A I DMPW ≤ 10µs, Duty cycle ≤ 1%980A I DM,repetitive790A V GS -6...22V T j-40 (175)°CAbsolute Maximum Ratings SymbolConditionsValuesUnitInverse diodeV RRM T j =25°C 1700V I F T j =175°CT c =25°C 552A T c =80°C428A I Fnom 300A I FRM 900A I FSM t p =10ms, sin 180°, T j =150°C, including body diode2030A T j-40 (175)°CAbsolute Maximum Ratings SymbolConditions Values UnitModule I t(RMS)500A T stg module without TIM -40...125°C V isolAC sinus 50 Hz, t =1min4000V2Rev. 1.0–04.11.2020© by SEMIKRONSEMITRANS ®3SiC MOSFET ModuleSKM260MB170SCH17Features*•Full Silicon Carbide (SiC) power module•High reliability 2nd Generation SiC MOSFETs•Optimized for fast switching and lowest power losses•External SiC Schottky Barrier Diode embedded•Insulated copper baseplate using DBC technology (Direct Bonded Copper)•Improved thermal performances with Aluminum Nitride (AlN) substrate •UL recognized, file no. E63532Typical Applications•High frequency power supplies •AC inverters •Traction APU •EV Chargers•Industrial Test SystemsRemarks•Case temperature limited to T C = 125°C max.•Recommended T jop = -40 ... +150°C •Gate-Source SURGE VOLTAGE(t surge <300ns), V GS_surge = -10V ... +26VMOSFET V (BR)DSS V GS =0V,I D =1mA, T j =25°C 1700V V GS(th)V DS =V GS , I D =57.75mA1.62.84V I DSS V GS =0V,V DS =1700V, T j =25°C 1.8mA I GSS V GS =22V,V DS =0V 700nA R DS(on)V GS =18V I D =161AchiplevelT j =25°C 8.110m ΩT j =150°C 14m ΩC iss V GS =0V V DS =800Vf =1MHzT j =25°C 27nF C oss T j =25°C 0.88nF C rss T j =25°C0.105nF R Gint T j =25°C2.1ΩQ G V DD =1000V, V GS =-5 ... 20V, I D =300A 1470nC t d(on)V DD =900V I D =300A V GS =-5 / +20VR Gon =0.7ΩR Goff =0.7Ωdi/dt on =12kA/µs di/dt off =9.5kA/µsdv/dt off =22kV/µs T j =150°C 64ns t r T j =150°C 60ns t d(off)T j =150°C162ns t f T j =150°C 32ns E on T j =150°C 7.59mJ E off T j =150°C6.21mJ R th(j-c)per MOSFET0.065K/W R th(c-s)per MOSFET (λgrease =0.81 W/(m*K))0.03K/WCharacteristics SymbolConditionsmin.typ.max.UnitInverse diodeV F = V SD I F =300A chiplevel T j =25°C 1.65 1.95V T j =150°C 2.51 2.86V V F0chiplevel T j =25°C 1.00 1.10V T j =150°C 0.860.96V r F chiplevelT j =25°C2.2 2.8m ΩT j =150°C5.56.3m ΩC j parallel to C oss , f =1MHz, V R =1700V, T j =25°C1.026nF Q c V R =800V, di/dt off =500A/µs, T j =25°C 0.95µCR th(j-c)per diode0.056K/W R th(c-s)per diode (λgrease =0.81 W/(m*K))0.027K/W© by SEMIKRONRev. 1.0–04.11.20203SEMITRANS ®3SiC MOSFET ModuleSKM260MB170SCH17Features*•Full Silicon Carbide (SiC) power module•High reliability 2nd Generation SiC MOSFETs•Optimized for fast switching and lowest power losses•External SiC Schottky Barrier Diode embedded•Insulated copper baseplate using DBC technology (Direct Bonded Copper)•Improved thermal performances with Aluminum Nitride (AlN) substrate •UL recognized, file no. E63532Typical Applications•High frequency power supplies •AC inverters •Traction APU •EV Chargers•Industrial Test SystemsRemarks•Case temperature limited to T C = 125°C max.•Recommended T jop = -40 ... +150°C •Gate-Source SURGE VOLTAGE(t surge <300ns), V GS_surge = -10V ... +26VModule L DS 15nH R DD'+SS'measured per switchT C =25°C0.55m ΩT C =125°C0.85m ΩR th(c-s)1calculated without thermal coupling (λgrease =0.81 W/(m*K))0.008K/W R th(c-s)2including thermal coupling, T s underneath module (λgrease =0.81 W/(m*K))0.013K/W M s to heat sink M635Nm M tto terminals M62.55Nm Nmw325g4Rev. 1.0–04.11.2020© by SEMIKRON© by SEMIKRON Rev. 1.0–04.11.202056Rev. 1.0–04.11.2020© by SEMIKRON© by SEMIKRON Rev. 1.0–04.11.20207This is an electrostatic discharge sensitive device (ESDS) due to international standard IEC 61340.*IMPORTANT INFORMATION AND WARNINGSThe specifications of SEMIKRON products may not be considered as guarantee or assurance of product characteristics ("Beschaffenheitsgarantie"). The specifications of SEMIKRON products describe only the usual characteristics of products to be expected in typical applications, which may still vary depending on the specific application. Therefore, products must be tested for the respective application in advance. Application adjustments may be necessary. The user of SEMIKRON products is responsible for the safety of their applications embedding SEMIKRON products and must take adequate safety measures to prevent the applications from causing a physical injury, fire or other problem if any of SEMIKRON products become faulty. The user is responsible to make sure that the application design is compliant with all applicable laws, regulations, norms and standards. Except as otherwise explicitly approved by SEMIKRON in a written document signed by authorized representatives of SEMIKRON, SEMIKRON products may not be used in any applications where a failure of the product or any consequences of the use thereof can reasonably be expected to result in personal injury. No representation or warranty is given and no liability is assumed with respect to the accuracy, completeness and/or use of any information herein, including without limitation, warranties of non-infringement of intellectual property rights of any third party. SEMIKRON does not assume any liability arising out of the applications or use of any product; neither does it convey any license under its patent rights, copyrights, trade secrets or other intellectual property rights, nor the rights of others. SEMIKRON makes no representation or warranty of non-infringement or alleged non-infringement of intellectual property rights of any third party which may arise from applications. Due to technical requirements our products may contain dangerous substances. For information on the types in question please contact the nearest SEMIKRON sales office. This document supersedes and replaces all information previously supplied and may be superseded by updates. SEMIKRON reserves the right to make changes.8。

超低功耗微型超声时间差测距传感器CH101说明书

CH101 Ultra-low Power Integrated Ultrasonic Time-of-Flight Range SensorChirp Microsystems reserves the right to change specifications and information herein without notice.Chirp Microsystems2560 Ninth Street, Ste 200, Berkeley, CA 94710 U.S.A+1(510) 640–8155Document Number: DS-000331Revision: 1.2Release Date: 07/17/2020CH101 HIGHLIGHTSThe CH101 is a miniature, ultra-low-power ultrasonic Time-of-Flight (ToF) range sensor. Based on Chirp’s patented MEMS technology, the CH101 is a system-in-package that integrates a PMUT (Piezoelectric Micromachined Ultrasonic Transducer) together with an ultra-low-power SoC (system on chip) in a miniature, reflowable package. The SoC runs Chirp’s advanced ultrasonic DSP algorithms and includes an integrated microcontroller that provides digital range readings via I2C.Complementing Chirp’s long-range CH201 ultrasonic ToF sensor product, the CH101 provides accurate range measurements to targets at distances up to 1.2m. Using ultrasonic measurements, the sensor works in any lighting condition, including full sunlight to complete darkness, and provides millimeter-accurate range measurements independent of the target’s color and optical transparency. The sensor’s Field-of-View (FoV) can be customized and enables simultaneous range measurements to multiple objects in the FoV. Many algorithms can further process the range information for a variety of usage cases in a wide range of applications.The CH101-00ABR is a Pulse-Echo product intended for range finding and presence applications using a single sensor for transmit and receive of ultrasonic pulses. The CH101-02ABR is a frequency matched Pitch-Catch product intended for applications using one sensor for transmit and a second sensor for receiving the frequency matched ultrasonic pulse.DEVICE INFORMATIONPART NUMBER OPERATION PACKAGECH101-00ABR Pulse-Echo 3.5 x 3.5 x 1.26mm LGA CH101-02ABR Pitch-Catch 3.5 x 3.5 x 1.26mm LGA RoHS and Green-Compliant Package APPLICATIONS•Augmented and Virtual Reality•Robotics•Obstacle avoidance•Mobile and Computing Devices•Proximity/Presence sensing•Ultra-low power remote presence-sensing nodes •Home/Building automation FEATURES•Fast, accurate range-finding•Operating range from 4 cm to 1.2m•Sample rate up to 100 samples/sec• 1.0 mm RMS range noise at 30 cm range•Programmable modes optimized for medium and short-range sensing applications•Customizable field of view (FoV) up to 180°•Multi-object detection•Works in any lighting condition, including full sunlight to complete darkness•Insensitive to object color, detects opticallytransparent surfaces (glass, clear plastics, etc.) •Easy to integrate•Single sensor for receive and transmit•Single 1.8V supply•I2C Fast-Mode compatible interface, data rates up to 400 kbps•Dedicated programmable range interrupt pin•Platform-independent software driver enables turnkey range-finding•Miniature integrated module• 3.5 mmx 3.5 mm x 1.26 mm, 8-pin LGA package•Compatible with standard SMD reflow•Low-power SoC running advanced ultrasound firmware•Operating temperature range: -40°C to 85°C •Ultra-low supply current• 1 sample/s:o13 µA (10 cm max range)o15 µA (1.0 m max range)•30 samples/s:o20 µA (10 cm max range)o50 µA (1.0 m max range)Table of ContentsCH101 Highlights (1)Device Information (1)Applications (1)Features (1)Simplified Block Diagram (3)Absolute Maximum Ratings (4)Package Information (5)8-Pin LGA (5)Pin Configuration (5)Pin Descriptions (6)Package Dimensions (6)Electrical Characteristics (7)Electrical Characteristics (Cont’d) (8)Typical Operating Characteristics (9)Detailed Description (10)Theory of Operation (10)Device Configuration (10)Applications (11)Chirp CH101 Driver (11)Object Detection (11)Interfacing to the CH101 Ultrasonic Sensor (11)Device Modes of Operation: (12)Layout Recommendations: (13)PCB Reflow Recommendations: (14)Use of Level Shifters (14)Typical Operating Circuits (15)Ordering Information (16)Part Number Designation (16)Package Marking (17)Tape & Reel Specification (17)Shipping Label (17)Revision History (19)SIMPLIFIED BLOCK DIAGRAMFigure 1. Simplified Block DiagramABSOLUTE MAXIMUM RATINGSPARAMETER MIN. TYP. MAX. UNIT AVDD to VSS -0.3 2.2 V VDD to VSS -0.3 2.2 V SDA, SCL, PROG, RST_N to VSS -0.3 2.2 V Electrostatic Discharge (ESD)Human Body Model (HBM)(1)Charge Device Model (CDM)(2)-2-5002500kVV Latchup -100 100 mA Temperature, Operating -40 85 °C Relative Humidity, Storage 90 %RH Continuous Input Current (Any Pin) -20 20 mA Soldering Temperature (reflow) 260 °CTable 1. Absolute Maximum RatingsNotes:1.HBM Tests conducted in compliance with ANSI/ESDA/JEDEC JS-001-2014 Or JESD22-A114E2.CDM Tests conducted in compliance with JESD22-C101PACKAGE INFORMATION8-PIN LGADESCRIPTION DOCUMENT NUMBER CH101 Mechanical Integration Guide AN-000158CH101 and CH201 Ultrasonic Transceiver Handling andAssembly Guidelines AN-000159Table 2. 8-Pin LGAPIN CONFIGURATIONTop ViewFigure 2. Pin Configuration (Top View)PIN DESCRIPTIONSPIN NAME DESCRIPTION1 INT Interrupt output. Can be switched to input for triggering and calibration functions2 SCL SCL Input. I2C clock input. This pin must be pulled up externally.3 SDA SDA Input/Output. I2C data I/O. This pin must be pulled up externally.4 PROG Program Enable. Cannot be floating.5 VSS Power return.6 VDD Digital Logic Supply. Connect to externally regulated 1.8V supply. Suggest commonconnection to AVDD. If not connected locally to AVDD, b ypass with a 0.1μF capacitor asclose as possible to VDD I/O pad.7 AVDD Analog Power Supply. Connect to externally re gulated supply. Bypass with a 0.1μFcapacitor as close as possible to AVDD I/O pad.8 RESET_N Active-low reset. Cannot be floating.Table 3. Pin DescriptionsPACKAGE DIMENSIONSFigure 3. Package DimensionsELECTRICAL CHARACTERISTICSAVDD = VDD = 1.8VDC, VSS = 0V, T A = +25°C, min/max are from T A = -40°C to +85°C, unless otherwise specified.PARAMETER SYMBOL CONDITIONS MIN TYP MAX UNITSPOWER SUPPLYAnalog Power Supply AVDD 1.62 1.8 1.98 V Digital Power Supply VDD 1.62 1.8 1.98 VULTRASONIC TRANSMIT CHANNELOperating Frequency 175 kHzTXRX OPERATION (GPR FIRMWARE USED UNLESS OTHERWISE SPECIFIED)Maximum Range Max Range Wall Target58 mm Diameter Post1.2(1)0.7mm Minimum Range Min Range Short-Range F/W used 4(2)cm Measuring Rate (Sample/sec) SR 100 S/s Field of View FoV Configurable up to 180º deg Current Consumption (AVDD +VDD) I SSR=1S/s, Range=10 cmSR=1S/s, Range=1.0mSR=30S/s, Range=10 cmSR=30S/s, Range=1.0m13152050μAμAμAμA Range Noise N R Target range = 30 cm 1.0 mm, rms Measurement Time 1m max range 18 ms Programming Time 60 msTable 4. Electrical CharacteristicsNotes:1.Tested with a stationary target.2.For non-stationary objects. While objects closer than 4cm can be detected, the range measurement is not ensured.ELECTRICAL CHARACTERISTICS (CONT’D)AVDD = VDD = 1.8VDC, VSS = 0V, T A = +25°C, unless otherwise specified.PARAMETERSYMBOL CONDITIONS MINTYP MAX UNITS DIGITAL I/O CHARACTERISTICS Output Low Voltage V OL SDA, INT,0.4 V Output High Voltage V OH INT 0.9*V VDD V I 2C Input Voltage Low V IL_I2C SDA, SCL 0.3*V VDDV I 2C Input Voltage High V IH_I2C SDA, SCL 0.7*V VDD V Pin Leakage Current I L SDA,SCL, INT(Inactive), T A =25°C±1μA DIGITAL/I 2C TIMING CHARACTERISTICSSCL Clock Frequencyf SCLI 2C Fast Mode400kHzTable 5. Electrical Characteristics (Cont’d)TYPICAL OPERATING CHARACTERISTICSAVDD = VDD = 1.8VDC, VSS = 0V, T A = +25°C, unless otherwise specified.Typical Beam Pattern – MOD_CH101-03-01 Omnidirectional FoV module(Measured with a 1m2 flat plate target at a 30 cm range)Figure 4. Beam pattern measurements of CH101 moduleDETAILED DESCRIPTIONTHEORY OF OPERATIONThe CH101 is an autonomous, digital output ultrasonic rangefinder. The Simplified Block Diagram, previously shown, details the main components at the package-level. Inside the package are a piezoelectric micro-machined ultrasonic transducer (PMUT) and system-on-chip (SoC). The SoC controls the PMUT to produce pulses of ultrasound that reflect off targets in the sensor’s Field of View (FoV). The reflections are received by the same PMUT after a short time delay, amplified by sensitive electronics, digitized, and further processed to produce the range to the primary target. Many algorithms can further process the range information for a variety of usage cases in a wide range of applications.The time it takes the ultrasound pulse to propagate from the PMUT to the target and back is called the time-of-flight (ToF). The distance to the target is found by multiplying the time-of-flight by the speed of sound and dividing by two (to account for the round-trip). The speed of sound in air is approximately 343 m/s. The speed of sound is not a constant but is generally stable enough to give measurement accuracies within a few percent error.DEVICE CONFIGURATIONA CH101 program file must be loaded into the on-chip memory at initial power-on. The program, or firmware, is loaded through a special I2C interface. Chirp provides a default general-purpose rangefinder (GPR) firmware that is suitable for a wide range of applications. This firmware enables autonomous range finding operation of the CH101. It also supports hardware-triggering of the CH101 for applications requiring multiple transceivers. Program files can also be tailored to the customer’s application. Contact Chirp for more information.CH101 has several features that allow for low power operation. An ultra-low-power, on-chip real-time clock (RTC) sets the sample rate and provides the reference for the time-of-flight measurement. The host processor does not need to provide any stimulus to the CH101 during normal operation, allowing the host processor to be shut down into its lowest power mode until the CH101 generates a wake-up interrupt. There is also a general-purpose input/output (INT) pin that is optimized to be used as a system wake-up source. The interrupt pin can be configured to trigger on motion or proximity.APPLICATIONSCHIRP CH101 DRIVERChirp provides a compiler and microcontroller-independent C driver for the CH101 which greatly simplifies integration. The CH101 driver implements high-level control of one or more CH101s attached to one or more I2C ports on the host processor. The CH101 driver allows the user to program, configure, trigger, and readout data from the CH101 through use of C function calls without direct interaction with the CH101 I2C registers. The CH101 driver only requires the customer to implement an I/O layer which communicates with the host processor’s I2C hardware and GPIO hardware. Chirp highly recommends that all designs use the CH101 driver.OBJECT DETECTIONDetecting the presence of objects or people can be optimized via software, by setting the sensor’s full-scale range (FSR), and via hardware, using an acoustic housing to narrow or widen the sensor’s field-of-view. The former means that the user may set the maximum distance at which the sensor will detect an object. FSR values refer to the one-way distance to a detected object.In practice, the FSR setting controls the amount of time that the sensor spends in the listening (receiving) period during a measurement cycle. Therefore, the FSR setting affects the time required to complete a measurement. Longer full-scale range values will require more time for a measurement to complete.Ultrasonic signal processing using the CH101’s General Purpose Rangefinder (GPR) Firmware will detect echoes that bounce off the first target in the Field-of-View. The size, position, and material composition of the target will affect the maximum range at which the sensor can detect the target. Large targets, such as walls, are much easier to detect than smaller targets. Thus, the associated operating range for smaller targets will be shorter. The range to detect people will be affected by a variety of factors such as a person’s size, clothing, orientation to the sensor and the sensor’s field-of-view. In general, given these factors, people can be detected at a maximum distance of 0.7m from the CH101 sensor.For additional guidance on the detection of people/objects using the NEMA standard, AN-000214 Presence Detection Application Note discusses the analysis of presence detection using the Long-Range CH201 Ultrasonic sensor.INTERFACING TO THE CH101 ULTRASONIC SENSORThe CH101 communicates with a host processor over the 2-wire I2C protocol. The CH101 operates as an I2C slave and responds to commands issued by the I2C master.The CH101 contains two separate I2C interfaces, running on two separate slave addresses. The first is for loading firmware into the on-chip program memory, and the second is for in-application communication with the CH101. The 7-bit programming address is0x45, and the 7-bit application address default is 0x29. The application address can be reprogrammed to any valid 7-bit I2C address. The CH101 uses clock stretching to allow for enough time to respond to the I2C master. The CH101 clock stretches before the acknowledge (ACK) bit on both transmit and receive. For example, when the CH101 transmits, it will hold SCL low after it transmits the 8th bit from the current byte while it loads the next byte into its internal transmit buffer. When the next byte is ready, it releases the SCL line, reads the master’s ACK bit, and proceeds accordingly. When the CH101 is receiving, it holds the SCL line low after it receives the 8th bit in a byte. The CH101 then chooses whether to ACK or NACK depending on the received data and releases the SCL line.The figure below shows an overview of the I2C slave interface. In the diagram, ‘S’ indicates I2C start, ‘R/W’ is the read/write bit, ‘Sr’ is a repeated start, ‘A’ is acknowledge, and ‘P’ is the stop condition. Grey boxes indicate the I2C master actions; white boxes indicate the I2C slave actions.Figure 5. CH101 I2C Slave Interface DiagramDEVICE MODES OF OPERATION:FREE-RUNNING MODEIn the free-running measurement mode, the CH101 runs autonomously at a user specified sample rate. In this mode, the INT pin is configured as an output. The CH101 pulses the INT pin high when a new range sample is available. At this point, the host processor may read the sample data from the CH101 over the I2C interface.HARDWARE-TRIGGERED MODEIn the hardware triggered mode, the INT pin is used bi-directionally. The CH101 remains in an idle condition until triggered by pulsing the INT pin. The measurement will start with deterministic latency relative to the rising edge on INT. This mode is most useful for synchronizing several CH101 transceivers. The host controller can use the individual INT pins of several transceivers to coordinate the exact timing.CH101 BEAM PATTERNSThe acoustic Field of View is easily customizable for the CH101 and is achieved by adding an acoustic housing to the transceiver that is profiled to realize the desired beam pattern. Symmetric, asymmetric, and omnidirectional (180° FoV) beam patterns are realizable. An example beam pattern is shown in the Typical Operating Characteristics section of this document and several acoustic housing designs for various FoV’s are available from Chirp.LAYOUT RECOMMENDATIONS:RECOMMENDED PCB FOOTPRINTDimensions in mmFigure 6. Recommended PCB FootprintPCB REFLOW RECOMMENDATIONS:See App Note AN-000159, CH101 and CH201 Ultrasonic Transceiver Handling and Assembly Guidelines.USE OF LEVEL SHIFTERSWhile the use of autosense level shifters for all the digital I/O signal signals is acceptable, special handling of the INT line while using a level shifter is required to ensure proper resetting of this line. As the circuit stage is neither a push-pull nor open-drain configuration (see representative circuit below), it is recommended that level shifter with a manual direction control line be used. The TI SN74LVC2T45 Bus Transceiver is a recommended device for level shifting of the INT signal line.Figure 7. INT Line I/O Circuit StageTYPICAL OPERATING CIRCUITSFigure 8. Single Transceiver OperationFigure 9. Multi- Transceiver OperationORDERING INFORMATIONPART NUMBER DESIGNATIONFigure 10. Part Number DesignationThis datasheet specifies the following part numbersPART NUMBER OPERATION PACKAGE BODY QUANTITY PACKAGING CH101-00ABR Pulse-Echo 3.5 mm x 3.5 mm x 1.26 mmLGA-8L 1,000 7” Tape and ReelCH101-02ABR Pitch-Catch 3.5 mm x 3.5 mm x 1.26 mmLGA-8L 1,000 7” Tape and ReelTable 6. Part Number DesignationCH101-xxABxProduct FamilyProduct Variant Shipping CarrierR = Tape & Reel 00AB = Pulse-Echo Product Variant02AB = Pitch-Catch Product VariantCH101 = Ultrasonic ToF SensorPACKAGE MARKINGFigure 11. Package MarkingTAPE & REEL SPECIFICATIONFigure 12. Tape & Reel SpecificationSHIPPING LABELA Shipping Label will be attached to the reel, bag and box. The information provided on the label is as follows:•Device: This is the full part number•Lot Number: Chirp manufacturing lot number•Date Code: Date the lot was sealed in the moisture proof bag•Quantity: Number of components on the reel•2D Barcode: Contains Lot No., quantity and reel/bag/box numberDimensions in mmDEVICE: CH101-XXXXX-XLOT NO: XXXXXXXXDATE CODE: XXXXQTY: XXXXFigure 13. Shipping LabelREVISION HISTORY09/30/19 1.0 Initial Release10/22/19 1.1 Changed CH-101 to CH101. Updated figure 7 to current markings.07/17/20 1.2 Format Update. Incorporated “Maximum Ratings Table” and “Use of LevelShifters” section.This information furnished by Chirp Microsystems, Inc. (“Chirp Microsystems”) is believed to be accurate and reliable. However, no responsibility is assumed by Chirp Microsystems for its use, or for any infringements of patents or other rights of third parties that may result from its use. Specifications are subject to change without notice. Chirp Microsystems reserves the right to make changes to this product, including its circuits and software, in order to improve its design and/or performance, without prior notice. Chirp Microsystems makes no warranties, neither expressed nor implied, regarding the information and specifications contained in this document. Chirp Microsystems assumes no responsibility for any claims or damages arising from information contained in this document, or from the use of products and services detailed therein. This includes, but is not limited to, claims or damages based on the infringement of patents, copyrights, mask work and/or other intellectual property rights.Certain intellectual property owned by Chirp Microsystems and described in this document is patent protected. No license is granted by implication or otherwise under any patent or patent rights of Chirp Microsystems. This publication supersedes and replaces all information previously supplied. Trademarks that are registered trademarks are the property of their respective companies. Chirp Microsystems sensors should not be used or sold in the development, storage, production or utilization of any conventional or mass-destructive weapons or for any other weapons or life threatening applications, as well as in any other life critical applications such as medical equipment, transportation, aerospace and nuclear instruments, undersea equipment, power plant equipment, disaster prevention and crime prevention equipment.©2020 Chirp Microsystems. All rights reserved. Chirp Microsystems and the Chirp Microsystems logo are trademarks of Chirp Microsystems, Inc. The TDK logo is a trademark of TDK Corporation. Other company and product names may be trademarks of the respective companies with which they are associated.©2020 Chirp Microsystems. All rights reserved.。

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a rXiv:077.2844v1[he p-ph]19J ul27UCT-TP-268/07July 2007Heavy-light quark pseudoscalar and vector mesons at ﬁnite temperature 1Cesareo A.Dominguez (a ),Marcelo Loewe ,(b ),J.Cristobal Rojas ,(c )(a )Centre for Theoretical Physics and Astrophysics University of Cape Town,Rondebosch 7700,South Africa (b )Facultad de F´ısica,Pontiﬁcia Universidad Cat´o lica de Chile,Casilla 306,Santiago 22,Chile (c )Departamento de F´ısica,Universidad Cat´o lica del Norte,Casilla 1280,Antofagasta,Chile Abstract The temperature dependence of the mass,leptonic decay constant,and width of heavy-light quark peseudoscalar and vector mesons is obtained in the framework of thermal Hilbert moment QCD sum rules.The leptonic decay constant of both pseudoscalar and vector mesons decreases with increasing T ,and vanishes at a critical temperature T c ,while the mesons develop a width which increases dra-matically and diverges at T c ,where T c is the temperature for chiral-symmetry restoration/quark-gluon deconﬁnement.These results indicate the disappear-ance of hadrons from the spectral function,which then becomes a smooth func-tion of the energy.This is interpreted as a signal for deconﬁnement at T =T c .In contrast,the masses show little dependence on the temperature,except very close to T c ,where the pseudoscalar meson masses increase slightly by 10-20%,and the vector meson masses decrease by some 20-30%.KEYWORDS:Thermal QCD Sum Rules,Finite Temperature QCD.1IntroductionThe thermal behaviour of hadronic Green’s functions,obtainable in a variety of theoretical frameworks,plays a fundamental role in understanding the dynamics of the quark-gluon plasma.One such framework is that of QCD sum rules[1], based on the Operator Product Expansion(OPE)of current correlators beyond perturbation theory,and on the notion of quark-hadron duality.The extension of this program toﬁnite temperature wasﬁrst discussed long ago in[2].It is based on two basic assumptions,(a)that the OPE remains valid at T=0, with perturbative QCD(PQCD)and the vacuum condensates developing a temperature dependence,and(b)that the notion of quark-hadron duality also remains valid.Additional evidence,from solvable quantumﬁeld theory models, supporting these assumptions was provided later in[3].Numerous applications of this technique have been made over the years[5],leading to the following consistent picture.(i)With increasing temperature,particles that are stable at T=0develop a non-zero width,and resonances become broader,diverging at a critical temperature interpreted as the deconﬁnement temperature(T c).This width is a result of particle absorption in the thermal bath.(ii)The onset of the continuum in hadronic spectral functions,traditionally accounted for by PQCD, decreases and approaches threshold near T c.In other words,as T→T c hadrons melt and disappear from the hadronic spectral functions,which become perfectly smooth.(iii)This scenario is further supported by results for hadronic and electromagnetic mean-squared radii,which also diverge at T c[4].In addition, QCD sum rules in the axial-vector channel have provided(analytical)evidence for the near equality of the critical temperatures for deconﬁnement and chiral-symmetry restoration[6].Regarding the temperature dependence of a hadronic mass,this parameter does not appear to be a relevant signal for deconﬁnement. Conceptually,given either the emergence or the broadening of an existing width, together with its divergence at T c,the concept of mass looses its meaning.In practical applications,in some cases the mass increases slightly with increasing T,and in others it decreases.In this paper we use Hilbert moment QCD sum rules for heavy-light quark pseudoscalar and vector meson correlators to determine the thermal behaviour of the hadronic masses,couplings,and widths.At T=0this problem was discussed long ago in[7]-[9].While there are only four ground-state pseudoscalar heavy-light quark mesons in the spectrum(D,D s,B,and B s),and similarly for vector mesons,it is possible to determine the decay constants for arbitrary meson masses in a self-consistent way[8].The result is that the leptonic decay constants obey a scaling law as a function of the meson mass.Since we are only interested in the temperature behaviour of the hadronic parameters,we shall normalize all our results to those at T=0,thus obtaining a universal functional relation.Weﬁnd that the meson masses are basically independent of T,except very close to T c where they increase slightly(pseudoscalars)by10-20%,or decrease(vector mesons)by20-30%.Here T c is the critical temperature for chiral-symmetry restoration/quark-gluon deconﬁnement.The leptonic decay1constants decrease with increasing T,and vanish at the critical temperature. Pseudoscalar mesons develop a non-zero hadronic width that increases with increasing T and diverges at T c,while vector meson widths exhibit a similar beahaviour.These results may be interpreted as providing(analytical)evidence for quark deconﬁnement at T=T c.2Pseudoscalar mesonsWe begin by deﬁning the correlator of axial-vector divergences atﬁnite temper-ature,i.e.the retarded Green’s functionψ5(q2,T)=i d4x e iqxθ(x0)<<|[∂µAµ(x),∂νA†ν(0)]|>>,(1) where∂µAµ(x)=m Q:¯q(x)iγ5Q(x):,q(Q)refers to the light(heavy)quark ﬁeld,and m Q>>m q is assumed.The matrix element above is the Gibbs average<<A·B>>= n exp(−E n/T)<n|A·B|n>/T r(exp(−H/T)),(2)where|n>is any complete set of eigenstates of the(QCD)Hamiltonian.We use here the quark-gluon basis,which allows for a smooth extension of the QCD sum rule program to non-zero temperature[3].At T=0and to leading order in PQCD[9]18π2m4Q(1−x)2πImψ5(x,T)=12T(1+x) −n F ωwhere Imψ5(x,0)is given by Eq.(3),n F(z)=(1+e z)−1is the Fermi thermal function,and in the rest frame(q=0)of the thermal bath x=m2Q/ω2. At temperatures below T≃200MeV one can safely assume the heavy quark mass to be temperature independent[10].Theﬁrst thermal function above is exponentially suppressed and can be safely neglected for temperatures of order O(100−200MeV),but the second one does contribute near threshold.Up to dimension d=6the non-perturbative expansion of the correlator at T=0is given by[7],[9]ψ5(q2)|NP=m2Q4q26× 2(m2Q−q2)3−m4Q12π<αs G2>−m Q<¯q q>,(6) C5<O5>=<g s¯q iσµνG aµνλa q>≡2M20<¯q q>,(7) C6<O6>=παs<(¯qγµλa q) q¯qγµλa q>V S=⇒−16as that for deconﬁnement.The ratio R (T )=<<¯q q >>/<¯q q >from[11]as a function of T/T c is shown in Fig.1.The low temperature expansion of the gluon condensate is proportional to the trace of the energy-momentum tensor,and it starts only at order T 8[13],viz.<<αs12πG 2>−αs405N 2F (N 2F −1)T −1)T 812πG 2>>=<αsT c 8 .(10)Figure 1:The light-quark condensate ratio R (T )=<<¯q q >>/<¯q q >as a function of T/T c from [11].Because of this T -dependence,the gluon condensate remains essentially con-stant up to temperatures very close to T c .Hence,the thermal non-perturbative4QCD correlator is basically driven by the quark condensate.Concerning the dimension d=6condensate,it has been argued that the vacuum saturation approximation breaks down atﬁnite temperature[14].This is based on the comparison between the slopes of the low temperature expansion(chiral pertur-bation theory)with and without assuming vacuum saturation.They are in fact numerically diﬀerent.However,this result is only valid at very low temperatures (T<<fπ);hence it cannot be extrapolated to T≃T c.In fact,both the quark condensate and the four-quark condensate should vanish at the same tempera-ture T=T c.In any case,numerically,at temperatures of order T≃100MeV the quark condensate dominates over the gluon condensate,the dimension d=5 condensate is comparable to<<¯q q>>,and the dimension d=6condensate is almost two orders of magnitude smaller.Hence,potential violations of vacuum saturation can be safely ignored.Finally,atﬁnite temperature it is possible, in principle,to have non-zero values of non-diagonal(Lorentz non-invariant) vacuum condensates.There is one example discussed in the literature[15]with enough detail to make a numerical estimate of their importance,and it refers to operators of spin-two(quark and gluon energy momentum tensors).The low temperature expansion of these terms starts at order O(T4),in contrast to a T2dependence for the diagonal condensates.Weﬁnd that at temperatures of order T≃100MeV both non-diagonal condensates are three orders of magni-tude smaller than the corresponding diagonal equivalents.We shall then ignore non-diagonal condensates in the sequel.Turning now to the hadronic sector,the spectral function at T=0can be written as1Imψ5(s)|P QCD,(11)πwhere M P and f P are the mass and leptonic decay constant of the pseudoscalar meson,and the continuum,starting at some threshold s0,is modeled by per-turbative QCD.With this normalization,fπ≃93MeV.Anticipating the pseu-doscalar mesons to develop a sizable widthΓP(T)atﬁnite temperature(particle absorption in the thermal bath),and using a Breit-Wigner parametrization,the following replacement will be understood1δ(s−M2P)=⇒constFigure2:The ratio s0(T)/s0(0),Eq.(14),as a function of T/T c for m Q=m c. ature;to a very good approximation it scales universally as the quark condensate [16],i.e.s0(T),(13)<¯q q>where s0(0)is clearly channel dependent.At the critical temperature we expect s0(T c)=m2Q,in which case Eq.(13)can be rewritten ass0(T)<¯q q> 1−m2Q s0(0),(14) This is shown in Fig.2for the case m Q=m c and s0(0)=5GeV2;a qualita-tively similar behaviour is obtained for m Q=m b and s0(0)≃(1.1−1.3)M2B.6Figure3:The ratio M P(T)/M P(0)as a function of T/T c.The correlation functionψ5(q2,T),Eq.(1),satisﬁes a twice subtracted dispersion relation.To eliminate the subtractions one can use Hilbert moments at Q2≡−q2=0,i.e.ϕ(N)(T)≡(−)N+1dQ2 N+1ψ5(Q2,T)|Q2=0=1s N+2Imψ5(s,T),(15)where N=1,2,....Invoking quark-hadron dualityϕ(N)(T)|HAD=ϕ(N)(T)|QCD,(16) and combining the continuum contribution in the hadronic spectral function with the PQCD piece of the QCD counterpart leads to theﬁnite energy Hilbert moments7Figure4:The ratio f P(T)/f P(0)as a function of T/T c.1s N+2Imψ5(s,T)|P OLE=1s N+2Imψ5(s,T)|P QCD+ϕ(N)(T)|NP,(17)where Imψ5(s,T)|P OLE is given by theﬁrst term in Eq.(11)modiﬁed according to Eq.(12),the PQCD spectral function corresponds to Eq.(4),andϕ(N)(T)|NP=−m Q<<¯q q>>12π<<αs G2>>4(N+2)(N+1)×M2081(N+2)(N2+10N+9)παsρ<<¯q q>>Figure5:The widthΓP(T))as a function of T/T c,withΓP(0)=0.Using theﬁrst three moments one obtains the temperature dependence of the mass,the leptonic decay constant,and the width.Results from this procedure are shown in Figs.3-5for the charm case;in the case of beauty mesons,results are qualitatively similar.3Vector mesonsWe consider the correlator of the heavy-light quark vector currentΠµν(q2,T)=i d4x e iqxθ(x0)<<|[Vµ(x),V†ν(0)]|>>=−(gµνq2−qµqν)Π(1)(q2,T)+qµqνΠ(0)(q2,T),(19) where Vµ(x)=:¯q(x)γµQ(x):.In the sum rule analysis we shall use the function9−Q2Π(1)(Q2,T),which is free of kinematical singularities.A straightforward calculation gives1(1−x)2(2+x) 1−n F z+ −n F z− ,(20)8π2where z±≡ω2M V f Vǫµ,(21)so that the pole contribution to the hadronic spectral function is2f2Vδ(s−M2V).At T=0the vector meson D∗(2010)has a very small width in the keV range(96±22keV),which we expect to increase with increasing T,so that the replacement in Eq.(12)will be made.Figure6:The ratio M V(T)/M V(0)as a function of T/T c.10Figure7:The widthΓV(T)as a function of T/T c.The Hilbert moments at Q2=0of the function−Q2Π(1)(Q2,T)are given byϕ(N)(T)≡(−)N+1dQ2 N+1[−Q2Π(1)(Q2,T)]|Q2=0=1s N+1ImΠ(1)(s,T).(22)Following the same procedure as for the pseudoscalar mesons(see Eq.(17)), the sum rules become1s N+1ImΠ(1)(s,T)|P OLE=1s N+1ImΠ(1)(s,T)|P QCD+ϕ(N)(T)|NP,(23)11whereϕ(N)(T)|NP is given byϕ(N)(T)|NP=−m Q<<¯q q>>12πm Q<<¯q q>>−(N+2)(N+3)m2Q +4m3Q .(24)Using theﬁrst three Hilbert moments toﬁnd the temperature dependence of the hadronic parameters,we obtain for the mass and the width of D∗(2010) the results shown in Figs.6-7.The behaviour of the vector-meson leptonic de-cay constant will not be shown,as it is essentially the same as that of the pseudoscalar-meson in Fig.4.Similar results are found for the case of the beauty vector meson B∗.4ConclusionsThe thermal behaviour of pseudoscalar and vector meson decay constants,masses, and widths was obtained in the framework of Hilbert momentﬁnite energy QCD sum rules.This behaviour is basically determined by the thermal light quark condensate on the QCD sector,and by the T-dependent continuum threshold on the hadronic sector.Normalizing to values at T=0,and using the method of[8]for arbitrary masses,there follows a universal relation for the hadronic pa-rameters as a function of T/T c.Results show that the decay constants decrease with increasing 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