The nucleon to Delta electromagnetic transition form factors in lattice QCD
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。
axions
AXIONSGEORG RAFFELTMax-Planck-Institut für Physik(Werner-Heisenberg-Institut),Föhringer Ring6,80805München,Germany(e-mail:*****************.de)(Received7August2001;accepted29August2001)Abstract.Axions are one of the few particle-physics candidates for dark matter which are well motivated independently of their possible cosmological role.A brief review is given of the theoreticalmotivation for axions,their possible role in cosmology,the existing astrophysical limits,and thestatus of experimental searches.1.IntroductionDespite its uncanny success,the particle-physics standard model has many looseends,among them the CP problem of quantum chromodynamics(QCD).The non-trivialfield structure of the QCD ground state(‘ -vacuum’)and a phase of thequark mass matrix each induce a non-perturbative CP-violating term in the QCDLagrangian which is proportional to the coefficient = QCD+arg det M quark, where could lie anywhere between0and2π.The experimental upper limit to aputative neutron electric dipole moment,a CP-violating quantity,informs us that 10−9,a severefine-tuning problem given that is a sum of two unrelated terms which would be expected to be of order unity each.One particularly elegant solution was proposed by Peccei and Quinn,where theparameter is re-interpreted as a dynamical variable, →a(x)/f a,where a(x)isthe axionfield and f a an energy scale called the Peccei-Quinn scale or axion decayconstant(Peccei and Quinn,1977a,b;Weinberg,1977;Wilczek,1977).The previ-ous CP-violating term automatically includes a potential for the axionfield whichdrives it to its CP-conserving minimum(dynamical symmetry restoration).Whilethis may sound complicated,Sikivie(1996)has constructed a beautiful mechanicalanalogy which nicely explains the main features of axion physics.While axions would be very weakly interacting,they are still a QCD phenom-enon.They share their quantum numbers with neutral pions;all generic axionproperties are roughly determined by those ofπ0,scaled with fπ/f a where fπ= 93MeV is the pion decay constant.For example,the axion mass is roughly given by m a f a=mπfπ,and the coupling to photons or nucleons is roughly suppressed by fπ/f a relative to the pion couplings.Axions have not been found during the quarter century since they werefirstproposed,but the interest in this hypothesis is well alive because other proposed Space Science Reviews100:153–158,2002.©2002Kluwer Academic Publishers.Printed in the Netherlands.154G.RAFFELTsolutions of the strong CP problem are not clearly superior,and mainly because axions are one of the few well-motivated particle candidates for the cold dark matter which apparently dominates the dynamics of the universe.The current status of axions physics and astrophysics was reviewed at a recent conference(Sikivie,1999).Particle-physics aspects,the status of astrophysical limits,and that of current search experiments are summarized in three separate mini-reviews in the Review of Particle Physics(Groom et al.,2000).Chapters on axions are also found in some textbooks(Kolb and Turner,1990;Raffelt,1996).For theoretical reviews see Kim(1987)and Cheng(1988),for a review of experimental searches see Rosenberg and van Bibber(2000).2.Stellar-Evolution LimitsThe main argument which proves that the Peccei-Quinn scale f a must be very large, corresponding to a very small axion mass m a,is related to stellar evolution.Axions would be produced by various processes in the hot and dense interior of stars and would thus carry away energy directly,much in analogy to the standard thermal neutrino losses.The strength of the axion interaction with photons,electrons,and nucleons can be constrained from the requirement that stellar-evolution time scales are not modified beyond observational limits(Raffelt,1996).For example,the helium-burning lifetime of horizontal-branch stars inferred from number counts in globular clusters reveals that the Primakoff processγ+Ze→Ze+a must not be too efficient in these stars,leading to a limit of m a 0.4eV(Figure1).Very restrictive limits arise from the observed neutrino signal of the supernova (SN)1987A.After collapse,the SN core is so hot and dense that neutrinos are trapped and escape only by diffusion so that it takes several seconds to cool a roughly solar-mass object the size of a few ten kilometers.The emission of axions would remove energy from the deep inner core which should show up in late-time neutrinos.Therefore,the observed duration of the SN1987A neutrino signal provides the most restrictive limits on the axion-nucleon coupling(Figure1).In the early papers on this topic,the difficulty of calculating the axion emission from a dense and hot nuclear medium had been underestimated;the most recent discussions attempt an inclusion of dense-medium effects(Janka et al.,1996).If axions are too‘strongly’interacting,they are trapped in a SN core,inval-idating the energy-loss argument and implying a mass above which axions are not excluded by the SN1987A signal(Turner,1988;Burrows et al.,1990).They would still carry away some of the energy and would cause excess counts in the water Cherenkov detectors which registered the neutrinos,allowing one to exclude another interval of axion masses(Engel et al.,1990).Probably there is a small crack of allowed axion masses between these two SN1987A arguments(Figure1), sometimes called the‘hadronic axion window’.Therefore,infine-tuned axion models where the tree-level coupling to photons nearly vanishes,eV-mass axionsAXIONS155Figure1.Astrophysical and cosmological exclusion regions(hatched)for the axion mass m a or the Peccei–Quinn scale f a.An‘open end’of an exclusion bar means that it represents a rough estimate. The globular cluster limit depends on the axion-photon coupling;it was assumed that E/N=83as in GUT models or the DFSZ model.The SN1987A limits depend on the axion-nucleon couplings; the shown case corresponds to the KSVZ model and approximately to the DFSZ model.The dot-ted‘inclusion regions’indicate where axions could plausibly be the cosmic dark matter.Most of the allowed range in the inflation scenario requiresfine-tuned initial conditions.Also shown is the projected sensitivity range of the search experiments for galactic dark-matter axions.may be allowed and could thus play the role of a cosmological hot dark matter component(Moroi and Murayama,1998).The axion coupling to electrons can be constrained from the properties of glob-ular-cluster stars and the white-dwarf luminosity function.However,the tree-level existence of such a coupling is not generic,and the resulting limits on m a and f a do not extend the range covered by the previous arguments.3.CosmologyIn the early universe,axions come into thermal equilibrium only if f a 108GeV, a region excluded by the stellar-evolution limits.For f a 108GeV cosmic axions are produced nonthermally.If inflation occurred after the Peccei-Quinn symmetry breaking or if T reheat<f a,the‘misalignment mechanism’(Preskill et al.,1983; Abbott and Sikivie,1983;Dine and Fischler,1983;Turner,1986)leads to a contri-bution to the cosmic critical density of a h2≈1.9×3±1(1µeV/m a)1.175 2i F( i)156G.RAFFELTwhere h is the Hubble constant in units of100km s−1Mpc−1.The stated range re-flects recognized uncertainties of the cosmic conditions at the QCD phase transition and of the temperature-dependent axion mass.The function F( )with F(0)=1 and F(π)=∞accounts for anharmonic corrections to the axion potential.Be-cause the initial misalignment angle i can be very small or very close toπ,there is no real prediction for the mass of dark-matter axions even though one wouldexpect 2i F( i)∼1to avoidfine-tuning the initial conditions.A possiblefine-tuning of i is limited by inflation-induced quantumfluctu-ations which in turn lead to temperaturefluctuations of the cosmic microwave background(Lyth,1990;Turner and Wilczek,1991;Linde,1991).In a broad class of inflationary models one thusfinds an upper limit to m a where axions could be the dark matter.According to the most recent discussion(Shellard and Battye,1998) it is about10−3eV(Figure1).If inflation did not occur at all or if it occurred before the Peccei-Quinn symme-try breaking with T reheat>f a,cosmic axion strings form by the Kibble mechanism (Davis,1986).Their motion is damped primarily by axion emission rather than gravitational waves.After axions acquire a mass at the QCD phase transition they quickly become nonrelativistic and thus form a cold dark matter component.The axion density is similar to that from the misalignment mechanism,but in detail the calculations are difficult and somewhat controversial between one group of authors(Davis,1986;Davis and Shellard,1989;Battye and Shellard,1994a,b)and another(Harari and Sikivie,1987;Hagmann and Sikivie,1991;Hagmann et al., 2001).Taking into account the uncertainty in various cosmological parameters one arrives at a plausible range for dark-matter axions as indicated in Figure1.4.Experimental SearchIf axions are indeed the dark matter of our galaxy one can search for them by the ‘haloscope’method(Sikivie,1983).The generic two-photon vertex which axions posess in analogy to neutral pions allows for the Primakoff conversion a↔γin the presence of external electromagneticfields.Therefore,the galactic axions should excite a microwave resonator which is placed in a strong magneticfield, i.e.,one expects a narrow line above the thermal noise of the cavity.While this line would not be difficult to identify once it has been found,searching for it requires to step a tunable cavity through many resonance intervals in order to cover a given m a range.In the late1980s,this method was pioneered in two pilot experiments (Wuensch et al.,1989;Hagmann et al.,1990).At the present time two full-scale ‘second generation’axion haloscopes are in operation,one in Livermore,Califor-nia(Hagmann et al.,1998,2000)and one in Kyoto,Japan(Ogawa et al.,1996; Yamamoto et al.,2001),the latter one using a beam of Rydberg atoms as a low-noise microwave detector.The projected sensitivity shown in Figure1covers the lower end of the plausible mass range for dark-matter axions.If axions are indeedAXIONS157 the galactic dark matter,these experiments for thefirst time are in a position to actually detect them.Axions or axion-like particles are currently also searched by the‘helioscope’method(Sikivie,1983;van Bibber et al.,1989).Axions would be produced in the Sun by the Primakoff effect,and could be back-converted into X-rays in a long dipole magnet oriented toward the Sun.A dedicated experiment of this sort in Tokyo has recently reported new limits(Inoue et al.,2000)while a much larger ef-fort using a decommissioned LHC test magnet,the CAST experiment,is currently under construction at CERN(Zioutas et al.,1999).It should be noted,however, that these searches are unrelated to axion dark matter,i.e.,if axions were to show up at CAST they almost certainly could not provide the cosmic dark matter.The evidence for the reality of dark matter has mounted for several decades,and most recently culminated with the determination of the cosmological parameters by cosmic-microwave precision experiments and other arguments.On the other hand, the physical nature of dark matter remains as mysterious as it was two decades ago. Therefore,the direct search experiments for particle candidates such as axions are among the most important efforts in the area of experimental cosmology.AcknowledgementsThis research was supported,in part,by the Deutsche Forschungsgemeinschaft under grant No.SFB-375and by the ESF network Neutrino Astrophysics.ReferencesAbbott,L.and Sikivie,P.:1983,‘A Cosmological Bound on the Invisible Axion’.Phys.Lett.B120, 133–136.Battye,R.A.and Shellard,E.P.S.:1994a,‘Global String Radiation’.Nucl.Phys.B423,260–304. Battye,R.A.and Shellard,E.P.S.:1994b,‘Axion String Constraints’.Phys.Rev.Lett.73,2954–2957;(E)ibid.76,2203–2204(1996).Burrows,A.,Ressel,T.and Turner,M.:1990,‘Axions and SN1987A:Axion trapping’.Phys.Rev.D42,3297–3309.Cheng,H.-Y.:1988,‘The Strong CP Problem Revisited’.Phys.Rept.158,1–89.Davis,R.L.:1986,‘Cosmic Axions from Cosmic Strings’.Phys.Lett.B180,225–230.Davis,R.L.and Shellard,E.P.S.:1989,‘Do Axions Need Inflation?’.Nucl.Phys.B324,167–186. Dine,M.and Fischler,W.:1983,‘The Not So Harmless Axion’.Phys.Lett.B120,137–141. Engel,J.,Seckel,D.and Hayes,A.C.:1990,‘Emission and Detectability of Hadronic Axions from SN1987A’.Phys.Rev.Lett.65,960–963.Groom,D.E.et al.:2000,‘The Review of Particle Physics’.Eur.Phys.J.C15,1–878.See also /Hagmann,C.and Sikivie,P.:1991,‘Computer Simulation of the Motion and Decay of Global Strings’.Nucl.Phys.B363,247–280.Hagmann,C.,Chang,S.and Sikivie,P.:2001,‘Axion Radiation from Strings’.Phys.Rev.D63, 125018(12pp).158G.RAFFELTHagmann,C.et al.:1990,‘Results from a Search for Cosmic Axions’.Phys.Rev.D42,1297–1300. Hagmann,C.et al.:1998,‘Results from a High-Sensitivity Search for Cosmic Axions’.Phys.Rev.Lett.80,2043–2046.Hagmann,C.et al.:2000,‘Cryogenic Cavity Detector for a Large-Scale Cold Dark-Matter Axion Search’.Nucl.Instrum.Meth.A444,569–583.Harari,D.and Sikivie,P.:1987,‘On the Evolution of Global Strings in the Early Universe’.Phys.Lett.B195,361–365.Inoue,Y.et al.:2000,‘Recent Results from the Tokyo Axion Helioscope Experiment’.astro-ph/0012338.Janka,H.-T.,Keil,W.,Raffelt,G.and Seckel,D.:1996,‘Nucleon Spin Fluctuations and the Supernova Emission of Neutrinos and Axions’.Phys.Rev.Lett.76,2621–2624.Kim,J.E.:1987.‘Light Pseudoscalars,Particle Physics and Cosmology’.Phys.Rept.150,1–177. Kolb,E.W.and Turner,M.S.:1990,‘The Early Universe’.Addison-Wesley,Reading,Mass. Linde,A.:1991,‘Axions in Inflationary Cosmology’.Phys.Lett.B259,38–47.Lyth,D.H.:1990,‘A Limit on the Inflationary Energy Density from Axion Isocurvature Fluctua-tions’.Phys.Lett.B236,408–410.Moroi,M.and Murayama,H.:1998,‘Axionic Hot Dark Matter in the Hadronic Axion Window’.Phys.Lett.B440,69–76.Ogawa,I.,Matsuki,S.and Yamamoto,K.:1996,‘Interactions of Cosmic Axions with Rydberg Atoms in Resonant Cavities Via the Primakoff Process’.Phys.Rev.D53,R1740–R1744. Peccei,R.D.and Quinn,H.R.:1977a,‘CP Conservation in the Presence of Instantons’.Phys.Rev.Lett.38,1440–1443.Peccei,R.D.and Quinn,H.R.:1977b,‘Constraints Imposed by CP Conservation in the Presence of Instantons’.Phys.Rev.D16,1791–1797.Preskill,J.,Wise,M.and Wilczek,F.:1983,‘Cosmology of the Invisible Axion’.Phys.Lett.B120, 127–132.Raffelt,G.G.:1996,‘Stars as Laboratories for Fundamental Physics’.University of Chicago Press. Rosenberg,L.and van Bibber,K.:2000,‘Searches for Invisible Axions’.Phys.Rept.325,1–39. Sikivie,P.:1983,‘Experimental Tests of the“Invisible”Axion’.Phys.Rev.Lett.51,1415–1417;(E) ibid.52,695(1984).Sikivie,P.:1996,‘The Pool Table Analogy to Axion Physics’.Physics Today49,22–27. Sikivie,P.(ed.):1999,‘Proceedings Axion Workshop’.Nucl.Phys.B(Proc.Suppl.)72,1–238. Shellard,E.P.S.and Battye,R.A.:1998,‘Cosmic Axions’.astro-ph/9802216.Turner,M.S.:1986,‘Cosmic and Local Mass Density of“Invisible”Axions’.Phys.Rev.D33,889–896.Turner,M.S.:1988,‘Axions from SN1987A’.Phys.Rev.Lett.60,1797–1800.Turner,M.S.and Wilczek,F.:1991,‘Inflationary Axion Cosmology’.Phys.Rev.Lett.66,5–8.van Bibber,K.et al.:1989,‘Design for a Practical Laboratory Detector for Solar Axions’.Phys.Rev.D39,2089–2099.Weinberg,S.:1978,‘A New Light Boson?’.Phys.Rev.Lett.40,223–226.Wilczek,F.:1978,‘Problem of Strong P and T Invariance in the Presence of Instantons’.Phys.Rev.Lett.40,279–282.Wuensch,W.U.et al.:1989,‘Results of a Laboratory Search for Cosmic Axions and Other Weakly Coupled Light Particles’.Phys.Rev.D40,3153–3167.Yamamoto,K.et al.:2001,‘The Rydberg Atom Cavity Axion Search’.hep-ph/0101200. Zioutas,K.et al.:1999,‘A Decommissioned LHC Model Magnet as an Axion Telescope’.Nucl.Instrum.Meth.A425,482–489.See also http://axnd02.cern.ch/CAST/。
Search for a new state of matter – the Quark-Gluon Plasma一个物质–夸克胶子等离子体的新状态搜索 43
-- multi-parton dynamics (recombination or coalescence or …)
-- Hydrodynamics (constituent quarks ? parton dynamics from gluons to constituent quarks? )
Nucleus-Nucleus Collisions and Volcanic Eruption
Volcanic high pT -- Strombolian eruption
Volcanic mediate pT – Spatter (clumps)
6
Volcanic low pT – Bulk matter flows
Evidence for Multi-parton Dynamics in Hadronization of Bulk Partonic Matter at RHIC
Huan Zhong Huang (黄焕中) Department of Physics and Astronomy
University of California Los Angeles
Rafelski+Danos, Molnar+Voloshin …..) Quark Recombination – (R.J. Fries et al, R. Hwa et al)
13
Constituent Quark Scaling
Constituent (n) Quark Scaling -- Meson n=2 and Baryon n=3 grouping
p td d td p N y d 2 1 π p td dtd p N 1 y i 12ic vo s ψ R () i)1( 0
梁伟红(广西师范大学)
16
f0(500) production is clearly dominant.
f0(980) shows up as a small peak.
17
Exp.
Conclusion: Our results agree with the experimental results. This gives a strong support to the idea of the low lying scalar mesons as being formed from the interaction of pairs of pseudoscalar mEexspo.ns.
(2)
1 12 f2
Tr[(
)2
M4 ],
( f 93 MeV) ——the pion decay constant
There are 5 coupled channels:
(1), 0 0(2), K K (3), K 0K 0(4), (5)
The V matrix elements are taken from
VP ApJ / cos , ( A 1, for an arbitrary normalizat ion)
invariant mass (Minv ) distributi on :
~t t BJ /
B J /
( pJ / cos ),
(in the B rest frame )
• Formalism
• Results and discussions • Summary
2n
• The nature of the light scalar mesons is a topic of longstanding debate.
激光原理答案
激光原理答案测验1.11、梅曼（TheodoreH.Maiman）于1960年发明了世界上第一台激光器——红宝石激光器,其波长为694.3nm。
其频率为:A：4.74某10∧14(14是上标)HzB：4.32某10∧14(14是上标)HzC：3.0某10∧14(14是上标)Hz 您的回答：B参考答案：Bnull满分：10分得分：10分2、下列说法错误的是：A：光子的某一运动状态只能定域在一个相格中，但不能确定它在相格内部的对应位置B：微观粒子的坐标和动量不能同时准确测定C：微观粒子在相空间对应着一个点您的回答：C参考答案：Cnull满分：10分得分：10分3、为了增大光源的空间相干性，下列说法错误的是：A：采用光学滤波来减小频带宽度B：靠近光源C：缩小光源线度您的回答：B参考答案：Bnull满分：10分得分：10分4、相干光强取决于：A：所有光子的数目B：同一模式内光子的数目C：以上说法都不对您的回答：B参考答案：Bnull满分：10分得分：10分5、中国第一台激光器——红宝石激光器于1961年被发明制造出来。
其波长为A：632.8nmB：694.3nmC：650nm您的回答：B参考答案：Bnull满分：10分得分：10分6、光子的某一运动状态只能定域在一个相格中，这说明了A：光子运动的连续性B：光子运动的不连续性C：以上说法都不对您的回答：B参考答案：Bnull满分：10分得分：10分7、3-4在2cm的空腔内存在着带宽（Δλ）为1某10m、波长为0.5m的自发辐射光。
求此光的频带范围Δν。
A：120GHzB：3某10^18（18为上标）Hz您的回答：B参考答案：Anull满分：10分得分：0分8、接第7题，在此频带宽度范围内，腔内存在的模式数？A：2某10^18（18为上标）B：8某10^10（10为上标）您的回答：A参考答案：Bnull满分：10分得分：0分9、由两个全反射镜组成的稳定光学谐振腔腔长为L，腔内振荡光的中心波长为求该光的波长带宽的近似值。
反应堆物理题库
西安交通大学——核反应堆物理分析(共470题)从反应堆物理的角度看，良好的慢化剂材料应具有什么样的性能？答案：慢化剂是快中子与它的核发生碰撞后能减速成热中子的材料，这与它的三种中子物理性能有关：δ－平均对数能量缩减；Σs－宏观散射截面；Σa－宏观吸收截面。
综合评价应是δ和Σs都比较大而Σa又较小的材料才是较好的慢化材料，定量地用慢化能力δΣs和慢化比δ和Σs/Σa来比较。
试列出常用慢化剂的慢化能力和慢化比。
核力所具有的特点是什么？答案：基本特点是：核力是短程力，作用范围大约是1～2×10-13cm；核力是吸引力，中子与中子，质子与中子，质子与质子之间均是强吸引力。
核力与电荷无关。
核力具有饱和性，每一核子只与其邻近的数目有限的几个核子发生相互作用。
4. 定性地说明：为什么燃料温度Tf越高逃脱共振吸收几率P越小？答案：逃脱共振吸收几率P是快中子慢化成热中子过程中逃脱238U共振吸收峰的几率，在燃料温度低的时候，ζa共振峰又高又窄，如图所示，当燃料温度升高后，238U的ζa的共振峰高度下降了，然而却变宽了，因而不仅原来共振峰处能量的中子被吸收，而且该能量左右的中子也会被吸收。
温度越高共振峰变得越宽，能被该共振峰吸收的中子越多，逃脱共振吸收几率P就越小，这种效应也称为多谱勒展宽。
试定性地解释燃料芯块的自屏效应。
答案：中子在燃料中穿行一定距离时的吸收几率，可表示为：P(a)=1-e-X/λ其中λ为吸收平均自由程，X为中子穿行距离。
一般认为X＝5λ时，中子几乎都被吸收了[P(a)→1]。
对于压水堆，燃料用富集度为3.0%的UO2，中子能量为6.7ev，穿行距离在5λa=0.0315cm内被吸收的几率为99.3%，所以很难有6.7ev的中子能进入到燃料芯块中心，这种现象称为自屏效应。
6. 什么是过渡周期？什么是渐近周期？答案：在零功率时，当阶跃输入－正反应性ρ0（ρ0<β）后，反应堆功率的上升速率（或周期）是随ρ0输入后的时间t而改变的（如图所示）。
Random Matrix Theory and Chiral Symmetry in QCD
KEYWORDS: effective low energy theories, finite volume partition function, lattice QCD
ABSTRACT: Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenvalues of the QCD Dirac operator. We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix theory with the global symmetries of the QCD partition function. The total number of such eigenvalues increases with the square root of the Euclidean four-volume. The spectral density for larger eigenvalues (but still well below a typical hadronic mass scale) also follows from the same low-energy effective partition function. The validity of the random matrix approach has been confirmed by many lattice QCD simulations in a wide parameter range. Stimulated by the success of the chiral random matrix theory in the description of universal properties of the Dirac eigenvalues, the random matrix model is extended to nonzero temperature and chemical potential. In this way we obtain qualitative results for the QCD phase diagram and the spectrum of the QCD Dirac operator. We discuss the nature of the quenched approximation and analyze quenched Dirac spectra at nonzero baryon density in terms of an effective partition function. Relations with other fields are also discussed.
Vub from Exclusive Semileptonic B to pi Decays
a School
arXiv:hep-ph/0703284v2 17 May 2007
Abstract
We use Omn` es representations of the form factors f+ and f0 for exclusive semileptonic B → π decays, paying special attention to the treatment of the B∗ pole and its effect on f+ . We apply them to combine experimental partial branching fraction information with theoretical calculations of both form factors to extract |Vub |. The precision we achieve is competitive with the inclusive determination and we do not find a significant discrepancy between our result, |Vub | = (3.90 ± 0.32 ± 0.18) × 10−3 , and the inclusive world average value, (4.45 ± 0.20 ± 0.26) × 10−3 [1].
SHEP–0710
|Vub| from Exclusive Semileptonic B → π Decays
Jonathan M Flynna and Juan Nievesb
of Physics and Astronomy, University of Southampton Highfield, Southampton SO17 1BJ, UK b Departay Nuclear, Universidad de Granada, E–18071 Granada, Spain
由质子单粒子效应截面预测重离子单粒子效应截面方法研究
由重离子 !00 截面预测 质子 !00 截面 的 研究开展得较早!已经发展了多个计算 模 型 ! &(' 如 "2428:59&U'.D\O&S'.0=?59=:&%'.D>8>_模型&b' 等"另外!在重离子 !00 实验 数据的基 础上建 立器件的平行 六 面 体 $O""%模 型!然 后 进 行 蒙 特卡 罗 仿 真 得 到 质 子 !00 截 面 也 是 一 种 重 要 的方法&R'"然而!使 用 重 离 子 对 器 件 进 行 辐 照 前 需 要 将 器 件 开 盖!尤 其 是 目 前 很 多 先 进 器 件 为 倒 装 器 件!它 们 在 开 盖 后 还 需 减 薄 衬 底 才 能 进行重离子 !00 实验!这样做 既会增加 实验成 本 !也 可 能 损 坏 器 件 +并 且 因 为 重 离 子 在 空 气 中 的射程短!重离子 !00 实验需 要在真空 环境中 进 行!所 以 使 用 重 离 子 辐 照 前 需 要 对 辐 照 靶 室 抽真空!较为费时"使用质子 进行 !00 实验 的 优 点 为 )由 于 中 高 能 质 子 的 穿 透 能 力 较 强 !所 以 使 用 中 高 能 质 子 进 行 实 验 无 需 开 盖 减 薄!避 免 损 坏 器 件 +质 子 辐 照 实 验 能 在 大 气 中 进 行 !无 需 抽真空!节省时间"使用质子 进行 !00 实验 不
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原 子 能 科 学 技 术 ! ! 第U%卷
存在上 述 重 离 子 !00 实 验 中 出 现 的 问 题!因 此!探 索 由 质 子 !00 截 面 预 测 重 离 子 !00 截 面的方法.拓展重离子 !00 截面 获取途 径对 评 估器件抗辐射性能具有重要价值"
超短强激光场中双原子分子的振动布居转移
ｅ ｃｔ ｔｏ ｎ ｈ ｏｒ ｅ ｐ ｎ ｉ ｏ ｅ ｉ ｌ ｘ ｉａ ｉ ｎ ａ ｄ ｔ ｅ ｃ ｒ ｓ ｏ ｄ ｎｇ ｐ ｔ ｎｔ ｓ ａ
较 低 的振动 能级 耦 合 ， 即大 部分 布 居 被 转 移 到 激
发 态 Ａ 上 较 小 的振 动 能级 ， 如图 ２ ａ 所示 。减 （） 小 激光 强度 则减 弱 了上下 ２个绝 热光 诱导 势能 面 之 间的相互 作用 ， 因此 左 右 ２个 势 阱之 间 的 势 垒
ｑ ｎｔ ｍ ｖ ｃ ｔ ｍｅ ｈｏ ｕａ ｕ ｗａ ｅ ｐａ ｋｅ ｔ ｄ．Ｂｙ ｖ ｒ ｉ ｈｅ ｉ ｔ ｎ ｉｙ ｏ ｈｅ ｄ ｉ ｉ ｅ ｏ ｅ ｏ ｄ ｌ ｓ ｒ ａ ｙ ｎｇ ｔ ｎ ｅ ｓ ｔ ｆｔ ｒｖ ｎｇ ｆ ｍｔ ｓ ｃ ｎ ａ ｅ ｐｕ ｓ ｌｅ，ｔ ｉ ｒ ｔｏ ｌｐｏ ｌ ｔｏ ａ ｅ ｓ ｌ ｃ ｉ ｌ ｒ ｎｓ ｅ ｒ ｄ． Ｔｈｅ ｅ ｄ ｎａ ｃ ｒ ｎ ｈｅ ｖ ｂ ａ ｉ ｎａ ｐｕ ａ ｉ ｎ ｃ ｎ ｂ ｅ ｅ ｔｖｅｙ ｔ ａ ｆ ｒ ｅ ｓ ｙ ｍｉｓ ａ ｅ ｉ — ｔ ｒ ｅ ｅ ｓ ｄ ｏ ｈｅ ｌｇ —ｎｄ ｅ ｔ ｎｔａ ｎ ａ ｉ ｂ ｔｃ ｒ ｐ ｅ ｅ ａ ｉ ｎ． ｅ ｐｒ ｔ ｄ ｂａ ｅ ｎ ｔ ｉ ｈｔｉ ｕｃ ｄ ｐｏ ｅ ｉｌｉ ｄ ａ ａ ｉ ｅ ｒ ｓ ｎｔ ｔｏ
光 脉 冲与原 子 和分 子 之 间的 作 用 ， 们 在 实 验上 人
和理 论 上广 泛研 究 了许 多 奇特 而有 趣 的现 象 ， 比 如键 的 软化 和硬化 、 阈值 电离 和超 阈值 解离 、 超 光
快 、 强激 光 场 中的振 动 布 居 转 移 。发 现分 子 末 超 振动 能级 随着 激光 脉 冲强 度 的减小 而增 大 。