The Baryon Fraction Distribution and the Tully-Fisher Relation

a rXiv:h ep-la t/9874v12J ul19981Can a flavour-conserving treatment improve things ?E.Mendel a ∗a FB Physik,Carl von Ossietzky Universit¨a t Oldenburg,26111Oldenburg,Germany In this work I would like to present some ideas on how to improve on the gauge sector in our lattice simulations at finite baryon density.The long standing problem,that we obtain an onset in thermodynamic quantities at a much smaller chemical potential than expected,could be related to an unphysical proliferation of flavours due to hard gluons close to the Brillouin edges.These hard gluons produce flavour non-conserving vertices to the fermion sector.They also produce excessive number of small instantons due to lattice dislocations.Both unphysical effects could increase the propagation in (di)-quarks to give the early onset in µ.Thus we will present here a modified action that avoids large fields close to the lattice cutoff.Some of these ideas have been tested for SU (2)and are being implemented for SU (3).1.Introduction There has been the long standing problem,since the introduction of a chemical potential µcoupled to the baryon density [1,2],that the onset of thermodynamic quantities happens in our lattice simulations [3–5]at a much smaller µthan at the expected µ=m N /3.Thebaryon density and the chiral condensate start changing from their vacuum values at µof the order of or earlier than m π/2,as if controlled by a Goldstone mode with net Baryon number,which is clearly unphysical in the low Temperature normal phase.The community had the hope that including properly the fermion determinant,which for non-zero µis complex and therefore very difficult to implement with lattice methods,will by itself solve this problem.The careful work by the Barbour group in ref.[6]and the follow up presented by I.Barbour and S.Morrison at this Workshop,seem to show a pinching at a higher µbut the onset still seems to be around m π/2.Several years ago I described another possible problem,on top of the fermion deter-minant one,that could be producing the early onset in our lattice simulations.As is well known,for Kogut Susskind fermions the flavour number is not conserved along their propagation due to hard gluons that can make jump the fermion propagator pole to other corners of the Brillouin zone,thus changing its flavour.This flavour changing would produce a proliferation of flavours even for the valence quarks,forming a much tighter bound nuclear matter than in nature[4,7].As we heard on various talks by F.Wilczeck,E.Shuryak and others at this workshop,diquark condensates could be produced easier the more flavour species,which could drive the observed early onset due to the flavour2proliferation.Furthermore,these hard gluons,with momenta p≈π/2a or higher,pro-duce easily lattice dislocations that give many unphysical“small instantons”[8].Again E. Shuryak in his model of quark propagating among instantons,gets increased propagation with a higher instanton density.We see then,that from several points of view it seems important tofind a way to suppress the gluons with momenta of the order of the lattice cutoff.We could regain flavour conservation and suppress unphysical mini instantons.Both unwanted effects tend to lower theµonset value from the physical one.In order to achieve such a hard gluon suppression I have been experimenting,first for SU(2),with an“improved”gauge action which has the same naive continuum limit as the usual one in thefield strengths,but diverges for plaquette phases of the order of the lattice cutoff.Even though this suppression method is gauge invariant,it is important to use a Metropolis updating scheme with small changes in the link phases,in order to guarantee that we stay locally without spurious dislocations.In this way we can hope to suppress small unphysical instantons.The physical gluon momentum transmitted to the fermions,given in the continuum by the Poynting vector,being essentially in the nonabelian case shaped as a“chair”formed with two contiguous plaquettes,could then also be kept low so as to conserveflavour.With L.Polley we had also tried out a new kind of lattice fermions[9,10],the t-asymmetric ones,where one can reduce to2flavours from the start(in contrast to the4staggered ones),but have several technical problems to surmount,related with regaining the O(4) symmetry,in order to calculate the hadron masses and compare them with the onset. This will be treated in the next section.In sec.3we will describe the new gauge action,mainly for SU(2),and discuss its implementation for SU(3).The ultimative test will be to generate configurations gener-ated with such an improved gauge action,to then compute the baryon density or chiral condensate for the full theory.2.t-asymmetric fermionsAn earlier attempt in order to reduce the number offlavours,was based on a new kind of fermions,also called t-asymmetric fermions,that don’t allow more than2flavours in contrast to the4flavours for staggered ones[9,10].This fermions are obtained by just taking a one-sided time derivative,which still produces an hermitian hamiltonian and eventually a positive transfer matrix(for aτ<a s/−iaτ eµU x,4δx,y−ˆ4−δx,y3 with the usual:Γν(x)=(−)x1+···+xν−1,after having thinned out to thefirst diagonal component.This fermionic action¯χKχhas then just2flavours.Susskind showed that there is a discrete version of chiral invariance left over,preventing mass counterterms in the interacting case,but not guaranteeing“Goldstone behaviour”for the pions.One could still get light pions,like in Aoki’s proposal for Wilson fermions[14].We had already donefirst simulations with the baryon density for these fermions, J4 = eµ¯χx U x,4χx+ˆ4 and the chiral condensate, ¯χχ ,as function ofµ.The operators have been calculated with the solved pseudofermion method[4].The curves for J are shown in ref.[10](Fig.1)and show a clear onset for two m q=.01,.04at the same µ≈0.2.This result could be deceiving though as it is not clear how mπscales with m q for these fermions,as already mentioned above.In order to really check if we are getting better results,we have to compute for these fermions theπand N masses.For this, considering that we need afiner lattice in theτdirection for positivity and afine aτfor charge conjugation symmetry in the propagators,we have to scale the couplings to regain the O(4)symmetry.Once this is done,one can attempt to calculate the masses and the baryon density for this O(4)symmetric action.This has proven to be a formidable task and presently I consider more promissory to improve on the gauge sector.3.Improvements on the gauge actionThe other way out,in order to preventflavour changing is to suppress the appearance of hard gluons with momenta p>π/2a.Without these hard gluons the quarks cannot change theirflavour by jumping to other corners of the Brillouin zone.As mentioned, these hard gluons can also cause unphysical mini-instantons that could enhance the quark propagation in Shuryak’s scheme,also affecting theµonset.From my investigations in SU(2)with instantons,in order to try to suppress mini-instantons produced by lattice dislocations,one can do so by modifying the action for plaquettes with a large phase.Instead of the usual gauge action:S=β 2(1−1/N Re tr(U2)),(3) which has a maximum plaquette action of2βfor a phaseθ2=π,one can take an improved action of the form:S=β 22/πtan(π/2(1−1/N Re tr(U2))),(4) which has the same naive continuum limit for small a,but diverges for plaquettes with phases close to the cuttof.One can easily device functions which start the same and grow even faster for larger plaquette phases.The point being that at relatively moderateβ’s one can suppress plaquette phases larger thanθ2≈π/2and therefore make sure that the effective tr(F2µν)stays away from the cutoff.(For the usual gaugefield action in SU(3), for example atβ=6.0,around1/3of the plaquettes still have a phaseθ2>π/2!).The actual gauge invariant definition for the gluon momentum,resembling the Poynting vector (tr E∧B)in the continuum,involves pairs of orthogonal plaquettes like a“chair”.If each of the plaquette phases is kept small also the combined one should be constrainable to get p<π/2a.4In order to avoid large phases in the gauge links,which produce dislocations carring a topological charge[4],it is convenient to use the Metropolis algorithm for the link up-date,with small changes in the SU(N)matrix on each iteration.In combination with an improved action as introduced above,one can obtain then fairly“smooth”gauge configu-rations,starting from the identity one.There is a problem though,related with a randomwalk like drifting of the links at a vertex corresponding to local gauge transformations, until we get large link phases close toπthat can produce dislocations.This can be re-duced by gauge transformations that try to keep all the link phases low.The trick I used in SU(2)is to check the phaseφof the new link U=exp(iφˆφ·σ)and if it is largerthan some phase,lets sayπ/6,make a local gauge transformation that shrinks that link’sphase toπ/8keeping its SU(2)direction,while applying the inverse element to the other links at that vertex.These local gauge transformations succeed in keeping almost allφsmaller thanπ/6,thus suppressing almost all dislocations.In this way I have been able to constraint the total topological charge to values Q V<|±1|in periodic lattices(thefield-theoretic definition does not give exactly in-tegers),for fairly large SU(2)lattices(123×24)withβ=2.5.For these lattices also the charge Qυwithυ=V/2half the volume is also smaller than|±1|showing no I−¯I pairs.One could also use more conventional action improvements in order to avoid hardgluons,by including terms in the action proportional to the Poynting operator(more generally the energy-momentum tensorΘµν)that suppress largefields.In other words,include“chair”diagram terms in the action.Another interesting alternative for an improved action in order to reduce theflavour breaking,was presented by Laga¨e and Sinclair[15].They smear the links by covariantdisplacements in all directions,in order to project the momenta to the central pole region in the Brillouin zone.In momentum space in the continuum they project thefieldsAµ(k)→1/16 i=1..4(1+cos k i)Aµ(k),which in coordinate space corresponds to the smearing Aµ(x)→1/256 i=1..4(2+D i+D−i)Aµ(x).They report a large reduction in theflavour symmetry violations for the(non)-Goldstoneπsplitting of masses.Up tonow the only introduce this projected gaugefields in the minimal coupling terms in their action and it would be interesting to see what happens with the topology if they consider such improvement in the gauge action.54.ConclusionsThe inclusion of the fermion determinant could be not the only ingredient needed in order to shift the onset chemical potential to physical values close to the Nucleon mass. Our lattice simulations atfiniteµhave been done with an unimproved gauge sector.This produces several unwanted effects,like the proliferation offlavours even for valence quarks due to the hard gluons that can change theflavour along the quark propagation.Also the topological sector is distorted due to unphysical mini-instantons appearing due to non smooth gaugefields.Both unwanted effects should be suppressed with an improvement in the gauge action.I introduced a simple to implement improvement that should help avoid hard gluons and dislocations.These ideas have been tested for SU(2)with encouraging results and could also be implemented for SU(3).In the Metropolis algorithm we cannot use anymore the trick to have the sum of staples precalculated as the action is not linear anymore,but the rest is very simple to implement.Once we have the gauge sector under control,with no hard gluons close to the cutoffand no unphysical mini-instantons,we could again calculate the baryon density operator or the chiral condensate tofind itsµdependence.This gauge updating could also be integrated in the unquenched code to redo Barbour’s group method to include the determinant.I would like to thank L.Polley for several interesting conversations on this gaugefield improvement.REFERENCES1.J.Kogut et al.,Nucl.Phys.B225(1983)93.2.P.Hasenfratz and F.Karsch,Physs.Lett.B125(1983)308.3.I.Barbour et al.,Nucl.Phys.B275(1986)296.4. E.Mendel,Nucl.Phys.B387(1992)485;Nucl.Phys.B20(Proc.Suppl.)(1991)343.5. C.Davies and E.Klepfish,Phys.Lett.B256(1991)68.6.I.Barbour,S.Morrison, E.Klepfish,J.Kogut and M.P.Lombardo,Nucl.Phys.60A(Proc.Suppl.)(1998)220.7. E.Mendel,Nucl.Phys.B34(Proc.Suppl.)(1994)304;Nucl.Phys.B30(Proc.Suppl.)(1993)944.8. E.Mendel and G.Nolte Nucl.Phys.B53(Proc.Suppl.)(1997)567;see also hep-lat:9511030.9. E.Mendel and L.Polley,Z.f¨u r Phys.C65(1995)127.10.E.Mendel and L.Polley,Nucl.Phys.B42(Proc.Suppl.)(1995)53511.L.Susskind,Phys.Rev.D16(1977)3031.12.T.Banks,S.Raby,L.Susskind,D.Jones,P.Scharbach and D.Sinclair,Phys.Rev.D15(1977)1111.13.I.Bender,H.Rothe,W.Wetzel and I.Stamatescu,Z.Phys.C58(1993)333.14.S.Aoki,Phys.Rev.D33(1984)2339.ga¨e and D.Sinclair,Nucl.Phys.B63(Proc.Suppl.)(1998)892.。

1Calculation of Areas Under the Normal CurveWe want to calculate the probability that a random event occurs, given that the probability distribution function is Normal with mean μ and standard deviation σ. We may numerically calculate the probabilityWe can do this as follows: The probability of exceeding a certain value x in a normal curve is denoted by Q(x). We, first, normalize the values of x by putting:σμ-←x xThen, the area under the normal curve to the right of x is:⎪⎩⎪⎨⎧<---=>=0)()(102/10)()()(x if x N x Z x if x if x N x Z x Qwhere)2/exp(21)(2x x Z -=π()pxt t t t t t b b b b b x N +=⋅⋅⋅⋅++++=11]})({[)(54321andp =+0.23164192b1=+0.319381530 b2=-0.356563782 b3=+1.781477937 b4=-1.821255978 b5=+1.330274429The probability that a given x is between to values a and b is given, simply,by )()()(b Q a Q b x a p -=≤≤where a < b.I. The Calculation of the Probabilities for a Normal Distribution0) The following are the constants for calculating the area under the Normal curve:p =+0.2316419 b1=+0.319381530 b2=-0.356563782 b3=+1.781477937 b4=-1.821255978 b5=+1.330274429 π2=k1) Normalize: a) offset = μ b) scale = σ2) Normalize the dataNb=(b-μ)/σ Na=(a-μ)/σNalpha=(alpha-μ)/ σ3) Calculate the probability of exceeding "b"fQ=Q(Nb)4) Calculate the probability of not reaching "b”P=1-fQ5) Calculate the probability of exceeding "a" and "b"Qa=Q(Na) Qb=Q(Nb)6) Calculate the probability of being between "a" and "b"a_b=Qa-Qb7) To calculate the prob. of a "point", define a very narrow band (0.1 sd's)alpha1=Nalpha-.05alpha2=Nalpha+.05Q1=Q(alpha1)Q2=Q(alpha2)P_alpha=Q1-Q2Function Q calculates the area under the normal curve by determining whether the sought for area is to the right or left of the mean value.a) If to the right, area is calculated from a direct numerical approximationb) If exactly on the middle p=1/2c) If to the left, calculate the probability from the symmetry of the curveIt uses two other functions:i) Z()ii) N()function Q(arg)if arg>0return Z(arg)*N(arg)endifif arg=0return 0.5endifif arg<03return 1-Z(-arg)*N(-arg)endiffunction Z(arg)return k*e(-arg*arg/2)function N(arg)t=1/(1+p*arg)return *(b1+t*(b2+t*(b3+t*(b4+t*b5))))45II. The Generation of Normally Distributed NumbersAn approximation to normally distributed random numbers y may be found from asequence of uniform random numbers [System/360 Scientific Subroutine Package] using the formula:12/2/1k k x y ki i ∑=-=wherea) x i is a uniformly distributed random numberb) 10≤<i xy approaches true normality asymptotically as k approaches infinity. To reduce execution time, make k=12.Then:66121-⎪⎪⎭⎫ ⎝⎛=∑=i i x yFor a given set μ and σ:μσ+=y y '7III. The 2χ Goodness_of_Fit TestThe theoretical distribution may be approximated using Chebyshev’s polynomials of degree 4. Then:443322102ννννχc c c c c ++++≈For 0.005 confidence level, the coefficients (c 0, ..., c 4) are:For 0.01 confidence level, the coefficients are:For 0.05 confidence level, the coefficients are:If the 2χ statistic, calculated from ∑=-=ri ii i E E O 122)(χ exceeds the theoretical 2χ, calculated from the polynomials, then the probability that the distribution is NOT normal equals the stipulated confidence level. In other words:8l c c c c c E E O Z P ri ii i =⎪⎪⎭⎫ ⎝⎛++++>-⌝∑=4433221012)(|νννν where P(Z) is the probability that the distribution is gaussian, r is the number of intervals and l is the confidence level.9Sampling Distribution of MeansStatistical MethodologyWe want to answer the following. Q : For any given algorithm A(i), what is the probability that we find a certain minimum value (denoted by κ) for any )(x γ given that A(i) is iterated G times?Since one of our premises is that the )(x γ be selected randomly from Ξ we do not know, a priori, anything about the probability distribution function of the κ’s. To answer Q we rely on the following known theorems from statistical theory.T1) Any sampling distribution of means (sdom) is distributed normally for a large enough sample size n .Remark: This is true, theoretically, as ∞→n . However, it is considered that any n >20 is satisfactory. We have chosen n =36.T2) In a normal distribution (with mean X μ and standard deviation X σ) approximately 1/10 of the observations lie in the intervals: X μ–5X σ to X μ-1.29X σ; X μ-1.29X σ to X μ–0.85X σ; X μ-0.85X σ to X μ–0.53X σ; X μ-0.53X σ to X μ–0.26X σ; X μ-0.26X σ to X μ and the symmetrical X μ to 0.26X σ, etc.Remark: These deciles divide, therefore, the area under the normal curve in 10 unequally spaced intervals. The expected number of observed events in each interval will, however, be equal.T3) The relation between the population distribution’s parameters [which we denote with μ (the mean) and σ(the standard deviation)] and the sdom’s parameters (which we denote with X μ and X σ) is given by X μμ= and X n σσ⋅=. Remark: In our case X σσ6=.T4) The proportion of any distribution found within k standard deviations of the mean is, at least, 1-1/k 2.Remark: Chebyshev’s bound generality makes it quite a loose one. Tighter bounds are achievable but they may depend on the characteristics of the distribution under study. We selected k = 4, which guarantees that our observations will occur with probability = 0.9375.T5) For a set of r intervals, a number of O i observed events in the i -th interval, a number of expected E i events in the i -th interval, p distribution parameters and 1--=p r ν degrees of freedom, the following equation holds.05.0)(4433221012=⎪⎪⎭⎫ ⎝⎛++++>-∑=ννννc c c c c E E O Z P ri ii i 98829512.10+≈c06290867.21+≈c 06021040.02-≈c00205163.03+≈c00002637.04-≈c(1)where ≡)(Z P probability that the distribution is normal.10Remarks: The summation on the left of (4) is the 2χ statistic; the polynomial to the right of the inequality sign (call it )(νT ) is a least squares Chebyshev polynomial approximation to the theoretical 2χ for a 95% confidence level. In our case, 7=ν for which ≈)(νT 14.0671. Furthermore, if we choose the deciles as above, we know that i E i ∀=10/η, where η is a sample of size n . A further condition normally imposed on this goodness-of-fit test is that a minimum number of observations θ (usually between 3 and 5) be required in each interval. Thus, (4) is replaced by()05.0)(4433221012=⎥⎥⎦⎤⎢⎢⎣⎡∀<∨⎪⎪⎭⎫ ⎝⎛++++>-∑=i O c c c c c E E O Z P i r i i i i θνννν(2) Making 5=θ and using the parameters’ values described above, equation (5) finally takes the following form.()95.05&0671.1410/)10/()5/(21011012=⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡≥⎪⎪⎪⎪⎪⎭⎫ ⎝⎛≤+-∑∑==ii i i i O O O Z P ηηη (3)We assume that we are exploring a set of algorithms. These are to be characterized by a set of parameters (from 1 to n ) and a method, which is, actually, a meta-heuristic (from 1 to m ).Algorithm for the Determination of the Distribution’s Parameters (of a meta -heuristic) 1. 1α← (determine the parameter set) 2. m ethod)the (determ ine ;1β←3. 1←i (count the number of samples)4. 1←j (count the elements of a sample)5. A function is selected randomly from the suite.6. Experiment βαE is performed with this function and a) the best value and b) the number of satisfied constraints are stored.7. 1+←j j8. If j ≤36, go to step 5 (a sample size of 36 guarantees normality).9. The average ∑=j)x (j f N 1i x of the best fitness’ values is calculated.10. 1+←i i11. If i ≤ 50, go to step 412. According to the central limit theorem, the i x distribute normally. We, therefore, define 10 intervals which are expected to hold 1/10 of the samples assuming a normal distribution: i.e., the intervals are standardized. If the samples are indeed normally distributed the following 2 conditions should hold.a) At least 5 observations should be found in each of the 10 intervals (which explains why we test for 50 in step 11).11b) The values of a 2χ goodness of fit test should be complied with (which we demand to be in the 95% confidence level).We, therefore, check for conditions (a) and (b) above. If they have not been reached, go to step 4.13. Once we are assured (with probability = 0.95) that the i x ’s are normally distributed, we calculate the mean X μ and standard deviation X σ of the sampling distribution of the measured mean values of the best fitnesses for this experiment. Moreover, we may calculate the mean μand the standard deviation σ of the distribution of the best values (rather than the means) from Xσ6σ;X μμ==. Notice that, therefore, we characterize the statistical behavior of experiment αβE quantitatively .14. 1ββ+←. If β< m , go to step 3.15. 1αα+←. If α< n , go to step 2.Program for the Sampling Distribution of MeansWe start by assuming the following data:Next, we define samples of size 3. For example: (3,3,3); (3,6,3); (4,6,10), etc. There are 1000 possible samples of this size.121. Directly from the table we calculate the mean and standard deviation, which turn out to be4.500 and 0.793 respectively. These are the parameters (μ) and (σ) of the population.2. Then we calculate the sampling distribution of means by generating all possible samples[(3,3,3);(3,3,4);(3,3,5);...;(10,9,10);(10,10,10)], the average (or mean) for every sample (3, 3.333,3.667,...,9.667, 10) and the resulting mean and standard deviation of the averages (means). This yields: x μ=4.500 and x σ = 0.458. From the CLT we know that x σ3σ⋅=; and that σ0.4581.732⋅≈0.793≈.3. Now we sample the population as per the algorithm above, to get 0.418x μ=, 0.442x σ= and 0.765σ=.In the figure above we show the intervals under the Normal curve, the number of observations in each interval and the percentage of the total number (81) for which 2χ= 3.814.4. Finally, we calculate the estimators ∑==N 1i i x N 1x and ∑=--=N 1i 2)x i (x 1N 1s , getting 3.666X = and 0.763s =.13。

a rXiv:h ep-ph/945231v15May1994Institute of Physics,Acad.of Sci.of the Czech Rep.PRA–HEP–94/3and hep-ph/9405231Nuclear Centre,Charles University May 5,1994Prague Feasibility of Beauty Baryon Polarization Measurement in Λ0J/ψdecay channel by ATLAS-LHC Julius Hˇr ivn´a ˇc ,Richard Lednick´y and M´a ria Smiˇz ansk´a Institute of Physics AS CR Prague,Czech Republic submitted to Zeitschrift f¨u r Physik CAbstractThe possibility of beauty baryon polarization measurement by cascade decay angu-lar distribution analysis in the channel Λ0J/ψ→pπ−l +l −is demonstrated.The error analysis shows that in the proposed LHC experiment ATLAS at the luminosity 104pb −1the polarization can be measured with the statistical precision better than δ=0.010for Λ0b and δ=0.17for Ξ0b .IntroductionThe study of polarization effects in multiparticle production provides an important infor-mation on spin-dependence of the quark confinement.Thus substantial polarization of the hyperons produced in nucleon fragmentation processes[1,2]as well as the data onthe hadron polarization asymmetry were qualitatively described by recombination quark models taking into account the leading effect due to the valence hadron constituents[3−6].Although these models correctly predict practically zero polarization ofΛandΩ−,they fail to explain the large polarization of antihyperonsΣ−recently discovered in Fermilab[7,8].The problem of quark polarization effects could be clarified in polarization measure-ments involving heavy quarks.In particular,an information about the quark mass de-pendence of these effects could be obtained[4,9].The polarization is expected to be proportional to the quark mass if it arises due to scattering on a colour charge[10−12]. The opposite dependence takes place if the quark becomes polarized due to the interac-tion with an”external”confiningfield,e.g.,due to the effect of spontaneous radiation polarization[13].The decrease of the polarization with increasing quark mass is expected also in the model of ref.[14].In QCD the polarization might be expected to vanish with the quark mass due tovector character of the quark-gluon coupling[10].It was shown however in Ref.[15] that the quark mass should be effectively replaced by the hadron mass M so that even the polarization of ordinary hadrons can be large.The polarization is predicted to be independent of energy and to vanish in the limit of both low and high hadron transverse momentum p t.The maximal polarization P max(x F)is reached at p t≈M and depends on the Feynman variable x F.Its magnitude(and in particular its mass dependence)is determined by two quark-gluon correlators which are not predicted by perturbative QCD.The polarization of charm baryons in hadronic reactions is still unmeasured due to the lack of sufficient statistics.Only some indications on a nonzero polarization were reported[16,17].For beauty physics the future experiments on LHC or HERA give an opportunity to obtain large statistical samples of beauty baryon(Λ0b,Ξ0b)decays intoΛ0J/ψ→pπ−l+l−,which is favorable mode to detect experimentally.Dedicated triggers for CP-violation effects in b-decays,like the high-p t one-muon trigger(LHC)[18] or the J/ψtrigger(HERA)[19]are selective also for this channel.Below we consider the possibility of polarization measurement of beauty baryonsΛ0b andΞ0b with the help of cascade decay angular distributions in the channelΛ0b(Ξ0b)→Λ0J/ψ→pπ−l+l−.1Polarization measurement method and an estimation of the statistical error.In the case of parity nonconserving beauty baryon(B b)decay the polarization causes the asymmetry of the distribution of the cosine of the angelθbetween the beauty baryon decay and production analyzers:w(cosθ)=1| p inc× p B b|,where p inc and p Bbare momenta of incident particle and B b in c.m.system.The asymmetry parameterαb characterizes parity nonconservation in a weak de-cay of B b and depends on the choice of the decay analyzer.In the two-body decay B b→Λ0J/ψit is natural to choose this analyzer oriented in the direction ofΛ0momentum pΛ0in the B b rest system.The considered decay can be described by4helicity ampli-tudes A(λ1,λ2)normalized to unity:a+=A(1/2,0),a−=A(−1/2,0),b+=A(−1/2,1) and b−=A(1/2,−1),|a+|2+|a−|2+|b+|2+|b−|2=1.(2) The difference ofΛ0and J/ψhelicitiesλ1-λ2is just a projection of B b spin onto the decay analyzer.The decay asymmetry parameterαb is expressed through these amplitudes in the formαb=|a+|2−|a−|2+|b+|2−|b−|2.(3) If P-parity in B b decay were conserved,then|a+|2=|a−|2,|b+|2=|b−|2so thatαb would be0.In the case of known and sufficiently nonzero value ofαb the beauty baryon polarization could be simply measured with the help of angular distribution(1)(see, e.g.,[20]).Due to lack of experimental information and rather uncertain theoretical estimates ofαb for the decayΛ0b→Λ0J/ψ[21]both the polarization andαb(or the decay amplitudes)should be determined simultaneously.This can be achieved with the2help of information onΛ0and J/ψdecays.Though it complicates the analysis,it shouldbe stressed that the measurement of the beauty baryon decay amplitudes could give valuable constrains on various theoretical models.Generally,such a measurement canbe done provided that at least one of the secondary decays is asymmetric and its decayasymmetry parameter is known[9].In our case it is the decayΛ0→pπ−with the asymmetry parameterαΛ=0.642.The angular distribution in the cascade decay B b→Λ0J/ψ→pπ−l+l−follows di-rectly from Eq.(6)of[9],taking into account that the only nonzero multipole parameters.It can be written in the formin the decay J/ψ→l+l−are T00=1and T20=110w(Ω,Ω1,Ω2)=1formula(4)integrated over the azimuthal anglesφ1,φ2would be in principle sufficient [9].In this case the number of free parameters is reduced to4(the phases don’t enter) and only a3-dimensionalfit is required.We will see,however,that the information on these angles may substantially increase the precision of the P b determination.To simplify the error analysis,we follow ref.[9]and consider here only the most unfavourable situation,when the parameters P2b,|a+|2−|a−|2and|b+|2−|b−|2are much smaller thanα2Λ.In this case the moments<F i>can be considered to be independent, having the diagonal error matrixW=13,19,115,145,16135,16135,2135,2135,245,245).(7)Here N is a number of B b events(assuming that the background can be neglected,see next section).The error matrix V of the vector a of the parameters a j,j=1,..7defined in (6)isV( a)=(A T W−1A)−1,(8) where the elements of the matrix A are A ij=d(f1i.f2i)V11=δ0N,(9)δ0=1α2Λ.[(2r0−1)2180+4r2015+(1−r0)(1+coshχ)10.(10)Hereδ0depends only on the relative contribution r0of the decay amplitudes with helicityλ2=0and on the relative phaseχ(Figs.1a,b).The maximal error on P b is δmax=4.7Nand it corresponds to the case when r0=1√√√,Σ∗b →Λ0b πand the electromagnetic decays Ξ0′b →Ξ0b γor Ξ0∗b →Ξ0b γ.The observable polarization P obs depends on the polarizations P B b of direct beauty baryons and their production fractions b B b (i.e.probabilities of the b-quark to hadronize to certain baryons B b ).In considered decays the beauty baryon Λ0b or Ξ0b retains −13)of the polarizationof a parent with spin 12+)(see Appendix).For P obs we get:P obs =b Λ0b P Λ0b + i (−13b Σ∗bi P Σ∗bi )3b Ξ0′b P Ξ0′b+1b Ξ0b +b Ξ0′b +b Ξ0∗b.(12)The summation goes over positive,negative and neutral Σb and Σ∗b .Assuming the polar-ization of the heavier states to be similar in magnitude to that of directly produced Λ0b or Ξ0b (P Λ0b or P Ξ0b )we may expect the observed polarization in an interval of (0.34−0.67)P Λ0bfor Λ0b and (0.69−0.84)P Ξ0b for Ξ0b.The polarization can be measured for Λ0b and Ξ0b baryons and for their antiparticles.Λ0b (Ξ0b )are unambigously distinquishable from their antiparticles by effective mass of pπ−system from Λ0→pπ−decay.The wrong assignment of antiproton and pion masses gives the kinematical reflection ofΞ0b is governed by b→dcc.HoweverΛ0fromΞ0b→Ξ0J/ψorΞ−b→Ξ−J/ψis produced in a weak hyperon decay,so this background can be efficiently reduced by the cut on the minimal distance d between J/ψandΛ0.A conservative cut d<1.5mm reduces this background by a factor≈0.05(Fig.3b).The background from B0d→J/ψK0when one ofπmesons is considered as a proton is negligible after the effective mass cuts on(pπ)and(pπJ/ψ)systems.Background from fake J/ψ′s,as it has been shown in[18],can be reduced to a low level by cuts on the distance between the primary vertex and the production point of the J/ψcandidate and the distance of closest approach between the two particles from the decay.These cuts also suppress the background from real J/ψ′s comming directly from hadronization.The number of producedΛ0b andΞ0b is calculated for the cross section of pp→busing the last segment of the hadron calorimeter by its minimum ionizing signature.-ForΛ0J/ψ→pπ−e+e−decay both electrons are required to have p e⊥>1GeV.The low threshold for electrons is possible,because of electron identification in the transition radiation tracker(TRT)[24].The events are required to contain one muon with a pµ⊥> 6GeV and|η|<1.6The second set of cuts corresponds to’offline’analysis cuts.The same cuts as for B0d→J/ψK0reconstruction[18]can be used(the only exception is the mass requirement forΛ0candidate,see the last of the next cuts):-The two charged hadrons fromΛ0decay are required to be within the tracking volume |η|<2.5,and transverse momenta of both to be greater than0.5GeV.-Λ0decay length in the transverse plane with respect to the beam axis was required to be greater than1cm and less than50cm.The upper limit ensures that the charged tracks fromΛ0decay start before the inner radius of TRT,and that there is a space point from the innermost layer of the outer silicon tracker.The lower limit reduces the combinatorial background from particles originating from the primary vertex.-The distance of closest approach between the two muon(electron)candidates forming the J/ψwas required to be less than320µm(450µm),giving an acceptance for signal of 0.94.-The proper time of theΛ0b decay,measured from the distance between the primary vertex and the production point of the J/ψin the transverse plane and the reconstructed p⊥ofΛ0b,was required to be greater than0.5ps.This cut is used to reduce the combina-torial background,giving the acceptance for signal events0.68.-The reconstructedΛ0and J/ψmasses were required to be within two standart de-viations of nominal values.The results on expectedΛ0b andΞ0b statistics and the errors of their polarization mea-surement are summarized in Table2.For both channels the statistics of reconstructed events at the luminosity104pb−1will be790000(220000)Λ0b and2600(720)Ξ0b,where the values are derived using UA1(CDF)results.For this statistics the maximal value of the statistical error on the polarization mea-surement,calculated from formulae(9)and(10),will be0.005(0.01)forΛ0b and0.09(0.17) forΞ0b.7ConclusionAt LHC luminosity104pb−1the beauty baryonsΛ0b andΞ0b polarizations can be measured with the help of angular distributions in the cascade decaysΛ0J/ψ→pπ−µ+µ−and Λ0J/ψ→pπ−e+e−with the statistical precision better than0.010forΛ0b and0.17for Ξ0b.AppendixThe polarization transfered toΛ0b,which was produced indirectly in strongΣb andΣ∗b decays,depends on the ratio∆| p inc× pΣb|,where p inc and pΣb are momenta of incident particle andΣb in c.m.system.Ω1=(θ1,φ1)are the polar and the azimuthal angles ofΛ0inΛ0b rest frame with the axes defined as z1↑↑ pΛ0b,y1↑↑ n× pΛ0b.After the transformation ofΩ1→Ω′1ofΛ0angles from the helicity frame x1,y1,z1to the canonical frame x,y,z with z↑↑ n and the integration over cosθandφ′1we get the distribution of the cosine of the angle between theΛ0momentum vector(Λ0b decay analyzer)and theΣb orΣ∗b production normal(which can be considered coinciding with theΛ0b production normal due to a small energy release in theΣb orΣ∗b decays):w(cosθ′1)∼1∓13(13(1References[1]K.Heller,Proceedings of the VII-th Int.Symp.on High Energy SpinPhysics,Protvino,1986vol.I,p.81.[2]L.Pondrom,Phys.Rep.122(19985)57.[3]B.Andersson et al.,Phys.Lett.85B(1979)417.[4]T.A.De Grand,H.I.Miettinen,Phys.Rev.D24(1981)2419.[5]B.V.Struminsky,Yad.Fiz.34(1981)1954.[6]R.Lednicky,Czech.J.Phys.B33(1983)1177;Z.Phys.C26(1985)531.[7]P.M.Ho et al.,Phys.Rev.Lett.65(1990)1713.[8]A.Morelos et al.,FERMILAB-Pub-93/167-E.[9]R.Lednicky,Yad.Fiz.43(1986)1275(Sov.J.Nucl.Phys.43(1986),817).[10]G.Kane,Y.P.Yao,Nucl.Phys.B137(1978)313.[11]J.Szwed,Phys.Lett.105B(1981)403.[12]W.G.D.Dharmaratna,Gary R.Goldstein,Phys.Rev.D41(1990)1731.[13]B.V.Batyunya et al.,Czech.J.Phys.B31(1981)11.[14]C.M.Troshin,H.E.Tyurin,Yad.Fiz.38(1983)1065.[15]A.V.Efremov,O.V.Teryaev,Phys.Lett.B150(1985)383.[16]A.N.Aleev et al.,Yad.Fiz.43(1986)619.[17]P.Chauvatet et al.,Phys.Lett.199B(1987)304.[18]The ATLAS Collaboration,CERN/LHCC/93-53,Oct.1993.[19]W.Hoffmann,DESY93-026(1993).[20]H.Albrecht et al.,DESY93-156(1993).9[21]A.H.Ball et al.,J.Phys.G:Nucl.Part.Phys.18(1992)1703.[22]UA1Collaboration,Phys.Lett.273B(1991)544.[23]CDF Collaboration,Phys.Rev.D47(1993)R2639.[24]I.Gavrilenko,ATLAS Internal Note INDET-NO-016,1992.[25]A.F.Falk and M.E.Peskin,SLAC-PUB-6311,1993.[26]R.Lednicky,DrSc Thesis,JINR-Dubna1990,p.174(in russian).10i f 2i011P ba +a ∗+−a −a ∗−−b +b ∗++b −b ∗−cos θ13P b αΛ−a +a ∗+−a −a ∗−+12b −b ∗−d 200(θ2)52b +b ∗+−1P b −a +a ∗++a −a ∗−−12b −b ∗−d 200(θ2)cos θ172b +b ∗+−1P b αΛ8P b αΛ3Im (a +a ∗−)sin θsin θ1sin 2θ2sin φ1102Re (b −b ∗+)sin θsin θ1sin 2θ2cos (φ1+2φ2)112Im (b −b ∗+)sin θsin θ1sin 2θ2sin (φ1+2φ2)−32Re (b −a ∗++a −b ∗+)sin θcos θ1sin θ2cos θ2cos φ213√P b αΛ−32Re (b −a ∗−+a +b ∗+)cos θsin θ1sin θ2cos θ2cos(φ1+φ2)15√P b αΛ16√P b−32Im (a −b ∗+−b −a ∗+)sin θsin θ2cos θ2sin φ218√αΛ−32Im (b −a ∗−−a +b ∗+)sin θ1sin θ2cos θ2sin(φ1+φ2)Table 1:The coefficients f 1i ,f 2i and angular functions F i in distribution (4).11Parameter Value forΛ0b CommentL[cm−2s−1]1033b(b→B b)0.08br(B b→Λ0J/ψ)2.210−2(0.610−2)J/ψ→µ+µ−0.06Λ0→pπ−0.641.110−4(0.310−3)0.060.64b)500µbN(µ+µ−pπ−)accepted1535000pµ⊥>6GeV,|η|<1.6(426000)pµ⊥>3GeV,|η|<2.5pπ,p⊥>0.5GeV,|η|<2.5740(210)2400(670)N(µeepπ−)reconstructed65000(18000)the maximum statistical error0.005on the polarization measurement(0.010)δ(P b)Table2:Summary on beauty baryon measurement possibilities for LHC experiment AT-LAS.The values in brackets correspond to the CDF result,while the analogical values without brackets to the UA1result.12Figure1:The maximal statistical error on the polarization measurementδ(P b)andδ0=N+1Figure 2:The Λ0J/ψeffective mass distribution:The peak at 5.62GeV is from Λ0b and background comes from J/ψfrom a b-hadron decay and Λ0either from the multiparticle production or from a b-hadron decay (a).The events that passed the cut on the transverse momenta (p T >0.5GeV )for p and π−from Λ0decay (b).14Figure 3:The Λ0J/ψeffective mass distribution:The peak at 5.84GeV is from Ξ0b →Λ0J/ψdecay.The background with the centre at ≈5.5GeV comes from Ξ0b →Ξ0J/ψ,Ξ0→Λ0π0and Ξ−b →Ξ−J/ψ,Ξ−→Λ0π−decays (a).The events that passed the cut on the minimal distance of J/ψand Λ0(d <1.5mm )(b).15。

a rXiv:as tr o-ph/11357v12Jan21**TITLE**ASP Conference Series,Vol.**VOLUME**,**PUBLICATION YEAR****EDITORS**Detectable Signals from Mergers of Compact Stars Hans-Thomas Janka Max-Planck-Institut f¨u r Astrophysik,Karl-Schwarzschild-Str.1,D-85741Garching,Germany Maximilian Ruffert Department of Mathematics &Statistics,University of Edinburgh,Edinburgh,EH93JZ,Scotland,U.K.Abstract.Merging neutron stars and neutron star–black holes binaries are powerful sources of gravitational waves.They have also been sug-gested as possible sources of cosmic gamma-ray bursts and are discussed as sites for the formation of r-process elements.Whereas the first aspect is undoubtable,the latter two are very uncertain.The current status of our knowledge and of the numerical modeling is briefly reviewed and the results of simulations are critically discussed concerning their significance and implications for potentially observable signals.1.Introduction Mergers of binary neutron stars (NS+NS)and neutron stars with companion black holes (NS+BH)are known to occur,because the emission of gravitational waves leads to a shrinking orbital separation and does not allow the systems to live forever.The Hulse-Taylor binary pulsar PSR B1913+16(Hulse &Taylor 1975)is the most famous example among a handful of known systems (Thorsett &Chakrabarty 1999).Eventually,after 108–1010years of evolution,these dou-ble stars will approach the final catastrophic plunge and will become powerfulsources of gravitational radiation.This makes them one of the most promising targets for the upcoming huge interferometer experiments,being currently un-der construction in Europe (GEO600,VIRGO),Japan (TAMA)and the United States (LIGO).The expected rate of merging events is of the order of 10−5per year per galaxy,estimated,with significant uncertainties,from population synthesis models and empirical data (see,e.g.,Bulik,Belczynski,&Zbijew-ski 1999;Fryer,Woosley,&Hartmann 1999;Kalogera,&Lorimer 2000;and references therein).Templates are urgently needed to be able to extract the expected signals from background noise and to interpret the meaning of possible measurements.Merging compact objects were suggested as possible sources of gamma-ray bursts (Blinnikov et al.1984;Eichler et al.1989;Paczy´n ski 1991;Narayan,Piran,&Shemi 1991),but so far observations cannot provide convincing ar-guments for this hypothesis.In fact,the exciting detection of afterglows at other wavelengths of the electromagnetic spectrum by the Dutch-Italian Bep-12Janka&RuffertpoSAX satellite and the discovery of lines in the afterglow spectra support the association of gamma-ray bursts(GRBs)with explosions of massive stars in star-forming regions of galaxies at high redshifts(e.g.,Klose2000,Lamb2000). However,so far afterglows were discoverd only for GRBs with durations longer than a few seconds.Similar observations for the class of short,hard bursts (∆t GRB<∼2seconds;Mao,Narayan,&Piran1994)are still missing,and we have neither a proof of their origin from cosmological distances,nor any hint on the astrophysical object they are produced by.NS+NS and NS+BH mergers are therefore still viable candidates for at least the subclass of short-duration bursts,in particular because such collisions of compact objects must occur at interesting rates,can release huge amounts of energy,and the energy is set free in a volume small enough to account for intensityfluctuations on a millisecond timescale.The ultimate proof of such an association would certainly be the coincidence of a GRB with a characteristic gravitational-wave signal.Lattimer&Schramm(1974,1976),Lattimer et al.(1977),and Meyer(1989)first considered merging neutron stars as a possible source of r-process nuclei. Their idea was that some matter might get unbound during the dynamical in-teraction.Since in neutron star matter neutrons are present with high densities and heavy nuclei can exist in surface-near layers,the environment might be suit-able for the formation of neutron-rich,very massive nuclei.Detailed models are needed to answer the question how much matter can possibly be ejected during the merging,and what the nuclear composition of this material is.Doing hydrodynamic simulations of the merging of compact stars is a chal-lenging problem.Besides three-dimensional hydrodynamics,preferably includ-ing general relativity,the microphysics within the neutron stars is very complex.A nuclear equation of state(EoS)has to be used,neutrino physics may be im-portant when the neutron stars heat up,magneticfields might cause interesting effects,etc.Also high numerical resolution is required to account for the steep density gradient at the neutron star surface,and a large computational volume is needed if the ejection of matter is to be followed.Different approximations and simplifications are possible when different as-pects are in the main focus of interest.For example,the surface layers are unim-portant and a simple ideal gas EoS[P=(γ−1)ε]and polytropic structure of the neutron stars can be used for parametric studies,when the gravitational-wave emission is to be calculated.This is an absolutely unacceptable simplification, however,when mass ejection and nucleosynthesis shall be investigated.For the latter problem as well as for the GRB topic,on the other hand,it is not clear whether general relativistic physics is essential.Nevertheless,of course,general relativity may be important to get quantitatively meaningful results.In partic-ular,Newtonian models cannot answer the question whether and when a black hole is going to form after two neutron stars have merged.Remarkable progress has been achieved during the past years.General rel-ativistic simulations are in reach now,at least with simple input physics and for certain phases of the evolution of the binary systems,especially for the in-spiraling phase(Duez,Baumgarte,&Shapiro2000)and for thefirst stage of thefinal dynamical plunge(Shibata&Ury¯u1999;Ury¯u,Shibata,&Eriguchi 2000),although the simulations become problematic once an apparent horizon begins to form.On the other hand,first attempts have been made to addDetectable Signals from Mergers of Compact Stars3Figure1.Possible evolution and observable signatures of NS+NS andBH+NS mergers.The evolution depends on the properties of the mergingobjects and the unknown conditions in the supranuclear interior of the neu-tron stars.detailed microphysics,i.e.,a physical nuclear EoS and neutrino processes,into Newtonian or to some degree post-Newtonian models(Ruffert,Janka,&Sch¨a fer 1996;(Ruffert et al.1997;Ruffert&Janka1998,1999;Rosswog et al.1999, 2000).The hydrodynamics results were used in post-processing steps to draw conclusions on implications for GRB scenarios(Janka&Ruffert1996,Ruffert& Janka1999,Janka et al.1999)and to evaluate the density-temperature trajec-tories of ejected matter for nucleosynthesis processes(Freiburghaus,Rosswog, &Thielemann1999).The results,however,are notfinally conclusive,because important effects have so far not been fully included and the simulations have to be improved concerning the resolution of the neutron star surfaces to reduce associated numerical artifacts.2.Importance of the Nuclear Equation of StateIndependent of numerical aspects and the unsatisfactory status of including gen-eral relativistic effects,one of the handicaps of merger models of compact stars are the unknown properties of the nuclear EoS.The compressibility of the nuclear EoS determines the neutron star structure and mass-radius relation.Therefore on the one hand,the evolution right before the merging,Roche-lobe overflow and the details of thefinal orbital instability and plunge of NS+NS binaries are4Janka&Ruffertaffected by the neutron star EoS.On the other hand the possibility of episodic mass transfer and of cycles of orbital decay and widening in BH+NS systems de-pend on the EoS properties(Lattimer&Prakash2000a;Lee&Klu´z niak1999a,b; Lee2001).The nuclear EoS thus determines the dynamics and gravitational-wave emission during these phases.But also the post-merging evolution is very sensitive to the EoS properties.The NS+NS merger remnant can be a rapidly and differentially rotating,protoneutron-star like object.If the EoS is stiff,this object might not collapse to a black hole on a dynamical timescale,but its mass may be below the maximum stable mass of such a configuration.In this case the merger remnant might approach thefinal instability only on a much longer secular timescale,driven by neutrino emission,viscous and magnetic braking (Lipunova&Lipunov1998)and angular momentum loss due to gravitational wave emission or mass shedding(Baumgarte&Shapiro1998).This could have very important implications for the gravitational-wave sig-nal,because the emission might not be shut offimmediately after the collision of the binary components,but—in particular in case of possibly developing triaxiality of the post-merger object—might continue for a considerable pe-riod of time.On the other hand,a hot,stable,massive protoneutron star-like object would cool by neutrino emission.Since neutrino energy deposition in the surface layers of the remnant causes a baryonic outflow of matter,similar to what happens in case of a protoneutron star at the center of a supernova, the merger site would be“polluted”by baryons and gamma-ray burst models might have a problem to explain the ejection of a baryon-poor pair-plasmafire-ball with highly relativistic velocities(Lorentz factors of several100),which is needed to produce the visible gamma-ray emission.Of course,ways have been suggested to reduce or circumvent this problem,for example by considering a non-standard composition(strange quark matter)and structure(no baryonic crust,extreme compactness)of the merger remnant(e.g.,Bombaci&Datta 2000;Mitra2000;and references therein),or by invoking magneticfields at the engine of the gamma-ray burst(e.g.,Klu´z niak&Ruderman1998;Usov1994).The currently discussed EoSs for the supranuclear regime do not set strict limits for the maximum mass of neutron stars.Whereas the nuclear EoS around saturation density is expected to be rather stiffwith a neutron star mass limit around2M⊙or larger,there is very little known about the EoS properties above twice nuclear matter density.A possible phase transition to a hyperonic state, kaon or pion condensates or a quark phase might soften the EoS and reduce the mass limit to significantly lower values(Lattimer&Prakash2000b;Heiselberg &Pandharipande2000;Balberg&Shapiro2000;Weber1999).Rigid rotation, on the other hand,can increase this limit by at most∼20%,but differential rotation,which is likely to be present in the post-merger configuration,can stabilize compact stars with much larger masses(Baumgarte,Shapiro,&Shibata 2000).The degree of differential rotation in the remnant of a neutron star merger can only be determined by hydrodynamic simulations.On the other hand,the post-merging object is very massive(its baryonic mass is roughly twice the mass of a single neutron star),and due to the violence of the dynamical interaction of the merging stars the equilibrium density is overshot,thus favoring the gravitational collapse on a dynamical timescale even when an object with the combined mass of both neutron stars couldfind a stable rotational equilibrium.Detectable Signals from Mergers of Compact Stars5 Thereforefirm theoretical predictions about the post-merging evolution of NS+NS systems are not possible at the moment,not even in case of a fully general relativistic treatment.Figure1summarizes possible evolutionary sce-narios in dependence of the EoS properties and the masses and spins of the merging compact stars.When NS+NS and BH+NS merging leads to a(rapidly rotating)black hole surrounded by an accretion torus,energy can be released on the accretion timescale of the gas,which is100to1000times longer than the dynamical timescale of the system.In this case thermal energy of the hot, dense torus can be radiated through neutrinos,or rotational energy of the disk or black hole might be extracted by magneticfields(Blandford&Znajek1971; Li&Paczy´n ski2000).The efficiency of energy conversion to neutrinos or Poynt-ingflux can be very high:A fair fraction of the rest mass energy of the accreted gas(up to several10per cent)or the rotational energy of the black hole can be released that way.Numerical values depend on the mass and the spin angular momentum of the black hole,on the strength of the magneticfields and on the physical conditions in the accretion torus.Since the outflows occur preferentially along the rotational axis of the system,one hopes that the baryon loading prob-lem can be avoided by the lower baryon densities above the poles of the black hole.Detailed models are necessary to make quantitative statements.Hydro-dynamical simulations have been performed for the neutrino-powered scenario, although problems like the baryonic entrainment of the outflow and the lifetime of the accretion torus are still unanswered and require further refinement of the models.In case of the magnetohydrodynamic mechanism time-dependent simulations are extremely challenging and are currently not at the horizon.Once bipolar outflow has formed,the question remains to be answered,what fraction of the kinetic or magnetic energy canfinally be converted into gamma rays.Pessimistic estimates quote a few per cent or less,optimistic values claim several10%(Kobayashi&Sari2000).Provided the remnant of a neutron star merger does not collapse to a black hole on a dynamical timescale,the baryon pollution problem is serious and probably hampers or destroys the chance for a gamma-ray burst.A massive,hot neutron star,which cools by neutrino emission,will poison its surroundings with a considerable amount of baryonic matter.Neutrino-driven winds might carry off∼10−2M⊙or more in all directions.This could have observable implications of such events,possibly generating a UV outburst of radiation when the expanding cloud dilutes to reach transparency(Li&Paczy´n ski1998).Concerning baryon pollution BH+NS mergers certainly offer an advantage because of the permanent presence of the black hole.The baryon pollution problem,however,might also occur there if the system runs into a state of long lasting,episodic mass transfer and the donating neutron star is not quickly disrupted into a disk.Circling around the black hole on a close orbit will lead to tidal heating of the neutron star and corresponding neutrino emission.In case of mass stripping of the neutron star down to its minimum mass,an explosion may occur,again possibly associated with the emission of radiation which is powered by radioactive decays in the expanding neutron star matter(Sumiyoshi et al.1998).6Janka&RuffertFigure2.Phases of mass ejection from NS+NS and BH+NS mergers.Detectable Signals from Mergers of Compact Stars7 3.Nucleosynthesis in the EjectaApart from a possible explosion of a neutron star at its minimum stable mass, two distinct processes can contribute to mass ejection from merger systems (Fig.2).On the one hand,there is a phase of dynamical ejection of cold, low-entropy surface material of the merging neutron stars,which starts shortly after the two stars have fallen into each other and the neutron stars develop long tidal arms by centrifugal forces.The amount of mass loss depends very sensi-tively on the total angular momentum of the system and is largest when the neutron star spin(s)is(are)aligned with the orbital angular momentum vector. Decompressed neutron star surface matter expands and releases energy through radioactive decays or nucleon recombination to nuclei,which might help theflow to become gravitationally unbound.On the other hand,general relativity,if it leads to a quick collapse of the merger remnant to a black hole,might suppress the possibility of ejecting material from the tips of the tidal tails.A second phase of mass loss is connected with the neutrino-driven wind. The matter will be neutrino-heated and hot,i.e.,the entropies are very high. This plasma outflow is unavoidable when the neutron stars heat up and emit neutrinos,or when a hot,radiating accretion torus is swallowed by a black hole after the merging.Nucleosynthesis calculations for the dynamically ejected matter of NS+NS mergers have been attempted(Freiburghaus et al.1999),but the problem is not easy to attack.Not only the disregard of general relativity implies uncer-tainties,also the numerical resolution of the surface layers of the neutron stars has to be good enough to calculate the thermodynamical conditions precisely. The current models yield sufficiently large temperatures(T>∼5×109K)that the matter reaches nuclear statistical equilibrium(NSE)and the r-processing proceeds very similar to the classical r-process in the shock-heated,expanding matter of supernovae.With a suitable superposition of conditions(degree of neutronization for certain amounts of ejected matter)the integral abundance curve can be forced tofit the solar r-process abundance distribution.But what is the“actual”(model determined)neutronization of the ejecta?The effects of radioactive decays and neutrino production on the neutron density have to be taken into account to answer this question.And,also crucial,what are the “true”temperatures in the ejected gas?It is not certain that the gas in the very extended spiral arms ever reaches NSE temperature.If it stays cooler, the neutron-rich isotopes which are present in the outer layers of the neutron star,would be swept into the interstellar medium and after decaying,would reflect an abundance distribution different from the solar r-process pattern.In hydrodynamical models one has tofight against numerical viscosity when the temperature is to be calculated very accurately.There is plenty of space for future work.Due to the lack of detailed models, nucleosynthesis has not been studied for the anisotropic neutrino-driven outflow of accreting black holes as merger remnants.In fact,the environment might pro-vide suitable conditions of entropy and neutronization to obtain high-entropy r-processing.Such conditions can be verified in neutrino-driven winds of pro-toneutron stars in supernovae only in extreme corners of the parameter space for neutron star masses and radii(Qian&Woosley1996;Otsuki et al.2000; Sumiyoshi et al.2000).8Janka&RuffertTable1.NS+NS and BH+NS merger simulations∆M ej Model Type Masses Spin t sim t ns d ns M minnsM⊙ms ms km M⊙M⊙/100 TN10BH/AD 2.9+0.26solid15............putational Methods and Computed ModelsHere we will only summarise the computational procedures that we used for doing NS+NS and BH+NS merger simulations.Details of the hydrodynamic method as well as the neutrino relevant algorithms can be found in Ruffert et al.(1996,1997).The nested grid was described by Ruffert&Janka(1998)and Ruffert(1992).The implementation of the black hole was explained in Ruffert &Janka(1999),Eberl(1998)and Janka et al.(1999).4.1.MethodsThe hydrodynamical simulations were done with a code based on the Piecewise Parabolic Method(PPM)developed by Colella&Woodward(1984).The code is basically Newtonian,but contains the terms necessary to describe gravitational-wave emission and the corresponding back-reaction on the hydrodynamicalflow (Blanchet et al1990).The modifications that follow from the gravitational potential are implemented as source terms in the PPM algorithm,which are obtained from fast Fourier transforms.The necessary spatial derivatives are evaluated as standard centered differences on the grid.In order to describe the thermodynamics of the neutron star matter,we use the EoS of Lattimer&Swesty(1991)for a compressibility modulus of bulk nuclear matter of K=180MeV in tabular e of a physical EoS instead of a simple ideal gas law implies that the adiabatic index is a function of space (i.e.,of density,temperature and composition in the star)as well as of time (i.e.,the value of the adiabatic index of the bulk of the matter changes when the neutron star strips some of its mass).This is an important difference compared to the polytropic neutron stars considered by Lee(2001).As mentioned above,Detectable Signals from Mergers of Compact Stars9 the effective adiabatic index influences the dynamics of NS+NS and BH+NS mergers.An ideal gas description defines an idealized,“pure”condition,the “realistic”case is more complex.Energy loss and changes of the electron abundance due to the emission of neutrinos and antineutrinos are taken into account by an elaborate“neutrino leakage scheme”.The energy source terms contain the production of all types of neutrino pairs by thermal processes and additionally of electron neutrinos and antineutrinos by lepton captures onto baryons.The latter reactions act as sources or sinks of lepton number,too,and are included as source terms in a continuity equation for the electron lepton number.Matter is rendered optically thick to neutrinos due to the main opacity producing reactions,which are neutrino-nucleon scattering and absorption of electron-type neutrinos onto nucleons.The presented simulations were done on multiply nested and refined grids. With an only modest increase in CPU time,the nested grids allow one to sim-ulate a substantially larger computational volume while at the same time they permit a higher local spatial resolution of the merging stars.The former is im-portant to follow the fate of matter that isflung out to distances far away from the merger site either to become unbound or to eventually fall back.The latter is necessary to adequately resolve the strong shock fronts and steep discontinu-ities of the plasmaflow that develop during the collision.The procedures used here are based on the algorithms that can be found in Berger&Colella(1989), Berger(1987)and Berger&Oliger(1984).Each grid had643zones,the size of the smallest zone was∆x=∆y=∆z=0.64or0.78km in case of binary neutron stars and1.25or1.5km for BH+NS mergers.The zone sizes of the next coarser grid levels were doubled to cover a volume of328or400km side length for NS+NS and640or768km for BH+NS simulations.In a post-processing step,performed after the hydrodynamical evolution had been calculated,we evaluated our models for neutrino-antineutrino(ν¯ν) annihilation in the surroundings of the merged stars in order to construct maps showing the local energy deposition rates per unit volume.Spatial integration finally yields the total rate of energy deposition outside the neutrino emitting high-density regions.4.2.Models and Initial ConditionsNS+NS merger simulations were started with two identical Newtonian neutron stars,or with neutron stars of different mass.In case of BH+NS coalescence we only considered neutron stars with a baryonic mass of about1.63M⊙so far,and varied the black hole mass.The distributions of densityρand elec-tron fraction Y e≡n e/n b(with n e being the number density of electrons minus that of positrons,and n b the baryon number density)were taken from a one-dimensional model of a cold,deleptonized(neutrino-less)neutron star in hy-drostatic equilibrium and were the same as in Ruffert et al.(1996).The initial central temperature was set to a value of typically a few MeV,the temperature profile decreasing towards the surface such that the thermal energy was much smaller than the degeneracy energy of the matter.For numerical reasons the surroundings of the neutron stars cannot be treated as completely evacuated.The density of the ambient medium was set to10Janka&Ruffertless than108g/cm3,more than six orders of magnitude smaller than the central densities of the stars.The total mass on the whole grid,associated with this finite background density is less than10−3M⊙.We prescribed the orbital velocities of the coalescing neutron stars according to the motions of point masses,as computed from the quadrupole formula.The tangential components of the velocities of the neutron star centers correspond to Kepler velocities on circular orbits,while the radial velocity components reflect the emission of gravitational waves leading to the inspiral of the orbiting bodies.A spin of the neutron stars around their respective centers was added to the orbital motion and was varied from model to model:The“A”models do not have any additional spins added on top of their orbital velocity.In this case all parts of the neutron stars start out with the same absolute value of the velocity, also called‘irrotational’motion.The angular velocity(both magnitude and direction)of the spin in“B”models is equal to the angular velocity of the orbit. This results in a‘solid body’type(tidally locked)motion,also called‘corotating’.“C”models represent counter-rotating cases where spins and orbital angular momentum vectors have opposite directions.We do not relax the neutron stars to equilibrium states before we start the simulations.Instead,we set the initial distances to sufficiently large values that most of the induced oscillations have been damped away by numerical viscosity before the dynamical phase of the merging is reached.Test calculations with even larger distances do not exhibit significant differences of the evolution.We are therefore pretty confident that thefinal orbital instability due to thefinite size of the objects was not affected by the non-equilibrated initial state.Table1gives a list of computed NS+NS and BH+NS merger models,for which the physical parameters of the systems were varied(t sim is the time in-terval covered by the simulation,t ns the time when the two density maxima of the neutron stars are a stellar radius,i.e.,d ns=15km,apart,and∆M ej the amount of dynamically ejected matter).Besides the baryonic mass of the neu-tron star(s)and the mass of the black hole,the spins of the neutron stars were varied.“Solid”means synchronously rotating stars,“none”irrotational cases, and“anti”counter-rotation,i.e.,spin angular momenta opposite to the direction of the orbital angular momentum vector.In addition to the listed models,we performed simulations where we changed numerical aspects,for example the res-olution,initial distances and initial temperatures,or used an entropy equation to follow the temperature evolution.In particular,the stability of the numer-ical handling,the smallness of the effects due to numerical viscosity andfinite resolution,and the quality of energy and angular momentum conservation were confirmed by BH+NS runs,where we followed the decay of the orbit for a large number(∼10)of full revolutions.The cool neutron stars with baryonic masses of1.2,1.63or1.8M⊙have a radius around15km(Ruffert et al.1996),the radius increasing weakly with the stellar mass.The runs in Table1were started with a center-to-center distance of42–46km for NS+NS and with47km in case of BH+NS for M BH=2.5M⊙, 57km for M BH=5M⊙and72km for M BH=10M⊙.The simulations were stopped at a time t sim between10ms and20ms.The black hole was treated as a point mass at the center of a sphere with radius R s=2GM BH/c2which gasDetectable Signals from Mergers of Compact Stars11 could enter unhindered.Its mass and momentum were updated along with the accretion of matter.Model TN10,which is added for comparison,is a continuation of the NS+NS merger Model B64,where at time t=10ms the formation of a black hole with a mass of2.5M⊙was assumed,and the accretion of the remaining gas on the grid(∼0.7M⊙)was followed for another5ms until a quasi-steady state was reached.Different times for the black hole collapse were tested(∼1,2,3and 9ms after the neutron stars had merged to one body),and gave qualitatively similar results(Ruffert&Janka1999).5.ResultsIn this proceedings contribution the results of our models can only be outlined. The results pertaining to the accretion of a high-density torus by a black hole can be found in Ruffert&Janka(1999).Latest models for neutron star merger simu-lations(the ones discussed here)will be published fully in Ruffert&Janka(2001, in preparation),older calculations(obtained with a less complete version of our code)were published by Ruffert et al.1996,1997),while the BH+NS simula-tions are summarized in Janka et al.(1999)and were reported in detail by Eberl (1998).5.1.Dynamical Evolution and Mass EjectionFigure3shows the temporal evolution of the density distribution in the orbital plane for both Models A64and B64.Initially,the orbits of the two neutron stars decay due to gravitational radiation,and the stars approach each other slowly.Once the distance decreases below the instability limit,thefinal plunge occurs(upper panels).Note that Model B64,which has more total angular momentum,develops prominent spiral arms in which matter isflung out to large distances(right middle panel)and a considerable amount of matter carries enough mechanical energy to escape the system(∆M ej∼2×10−2M⊙,see Table1).Model A64,on the contrary,remains more compact during this very early post-merging phase(left middle panel).The amount of ejected mass for the A-model is therefore about one order of magnitude lower.After10ms a very dense,nearly spherical central object has formed,surrounded by a less dense, extended,thick equatorial torus.When a steady state has been reached,the densities in this cloud are rather similar in both simulations(lower panels).With a Newtonian code we cannot determine whether and if,when,the merger remnant collapses to a black hole.In order to investigate what happens in such a case,we continued Run B64,assuming that such a catastrophe happens to the central,dense body on a dynamical timescale of a few milliseconds after the merger.In the simulation listed in Table1(Model TN10)this gravitational instability was assumed to occur at the end of the computed merger evolution at t=10ms.A Newtonian potential,GM BHΦN=−。

Modern Physics Letters B,Vol.21,No.5(2007)237–248c World Scientific Publishing CompanyLIOUVILLE AND BOGOLIUBOV EQUATIONS WITHFRACTIONAL DERIV ATIVESV ASILY E.TARASOVSkobeltsyn Institute of Nuclear Physics,Moscow State University,Moscow119992,Russiatarasov@theory.sinp.msu.ruReceived14March2006The Liouville equation,first Bogoliubov hierarchy and Vlasov equations with derivativesof non-integer order are derived.Liouville equation with fractional derivatives is obtainedfrom the conservation of probability in a fractional volume element.This equation is usedto obtain Bogoliubov hierarchy and fractional kinetic equations with fractional deriva-tives.Statistical mechanics of fractional generalization of the Hamiltonian systems isdiscussed.Fractional kinetic equation for the system of charged particles are considered.Keywords:Liouville equation;Bogoliubov equation;fractional derivatives;fractionalkinetics.PACS Number(s):05.20.-y,05.20.Dd,45.10.Hj1.IntroductionFractional equations1are equations that contain derivatives of non-integer order.2,3 The theory of derivatives of non-integer order goes back to Leibniz,Liouville,Rie-mann,and Letnikov.3Derivatives and integrals of fractional order have found many applications in recent studies in mechanics and physics.In a short period of time the list of applications have become long.For example,it includes chaotic dynamics,4,5 mechanics of fractal media,6–8physical kinetics,4,9–12plasma physics,13–15astro-physics,16long-range dissipation,17–19mechanics of non-Hamiltonian systems,20,21 theory of long-range interaction,22–24and many others physical topics.In this paper,we derive Liouville equation with fractional derivatives with re-spect to coordinates and momenta.To derive the fractional Liouville equation,we consider the conservation of probability tofind a system in the fractional differential volume ing the fractional Liouville equation,we derive the fractional generalization of the Bogoliubov hierarchy equations.These equations can be used to derive fractional kinetic equations.4,9,10,12A linear fractional kinetic equation for the system of charged particles is suggested.In Sec.2,we derive the Liouville equation with fractional derivatives from the conservation of probability tofind a system in the fractional volume element of the phase space.In Sec.3,we obtain thefirst Bogoliubov hierarchy equation with237238V.E.Tarasovfractional derivatives in the phase space from the fractional Liouville equation.In Sec.4,the Vlasov equation with fractional derivatives in phase space is considered. In Sec.5,a linear fractional kinetic equation for the system of charged particles is suggested.Finally,a short conclusion is given in Sec.6.2.Liouville Equation with Fractional DerivativesA basic principle of statistical mechanics is the conservation of probability.The Liouville equation is an expression of this basic principle in a convenient form for the analysis.In this section,we derive the Liouville equation with fractional derivatives from the conservation of probability in a fractional volume element.For the phase space R2n with coordinates(x1,...,x2n)=(q1,...,q n, p1,...,p n),we consider a fractional differential volume elementdαV=dαx1···dαx2n.(1) Here,dαis a fractional differential.25For the function f(x),dαf(x)=2nk=1Dαxkf(x)(dx k)α,(2)where Dαxkis a fractional derivative3of orderαwith respect to x k.The Caputo derivative6,26is defined byDαx f(x)=1(x−z)α+1−mdz,(3)where m−1<α<m,f(m)(τ)=d m f(τ)/dτm,andΓ(z)is the Euler gamma-function.For Caputo and Riesz3fractional derivatives,we have Dαxk1=0,and D xkx l=0(k=l).Using Eq.(2),we obtaindαx k=Dαxkx k(dx k)α,(α>0).(4) Then(dx k)α=(Dαxkx k)−1dαx k.(5) For Caputo derivatives,Dαxk xβk=Γ(β+1)∂t=d[ρ(t,x)(u,d S)],(8)Liouville and Bogoliubov Equations with Fractional Derivatives239 for the usual volume element(α=1),and−dαV∂ρ(t,x)∂t =dαV2nk=1(Dαxkx k)−1Dαxk[ρu k].(14)As the result,we obtain∂ρ∂t =−Γ(2−α)2nk=1xα−1kDαxk[ρu k].(16)Equation(15)is a Liouville equation that contains the derivatives of fractional orderα.Fractional Liouville equation(15)describes the probability conservation tofind a system in the fractional volume element(1)of the phase space.For the coordinates(x1,...,x2n)=(q1,...,q n,p1,...,p n),Eq.(15)is ∂ρ240V.E.Tarasovwhere V k=u k,and F k=u k+n(k=1,...,n).The functions V k=V k(t,q,p)are the components of velocityfield,and F k=F k(t,q,p)are the components of force field.In general,Dαpk [ρF k]=ρDαpkF k+F k Dαpkρ.(18)Suppose that F k does not depend on p k,and the k th component V k of the velocity field does not depend on k th component q k of coordinates.In this case,Eq.(17) gives∂ρdt =V k(t,q,p),dq k∂p j−∂V j∂q i+∂F i∂q j−∂F j dt=∂Hdt=−∂H∂t +nk=1(Dαqkq k)−1DαpkHDαqkρ−nk=1(Dαpkp k)−1DαqkHDαpkρ=0.(24)We can define{A,B}α=nk=1((Dαqkq k)−1DαqkADαpkB−(Dαpkp k)−1DαqkBDαpkA).(25)Forα=1,Eq.(25)defines Poisson brackets.Note that the brackets(25)satisfy the relations{A,B}α=−{B,A}α,{1,A}α=0.Liouville and Bogoliubov Equations with Fractional Derivatives241 In general,the Jacoby identity cannot be satisfied.The property{1,A}α=0is sat-isfied only for Caputo and Riesz fractional derivatives(Dαx1=0).For the Riemann–Liouville derivative,Dαx1=ing Eq.(25),we get Eq.(24)in the form∂ρ∂t =−Nk=1(Dαqk(V kρN)+Dαpk(F kρN)),(27)whereDαqk V k=(Dαqkq k)−1DαqkV k=ms=1(Dαqksq ks)−1DαqksV ks,(28)Dαpk F k=(Dαpkp k)−1DαpkF k=ms=1(Dαqksp ks)−1DαpksF ks.(29)The one-particle reduced density of probabilityρ1can be defined byρ1(q,p,t)=ρ(q1,p1,t)=ˆI[2,...,N]ρN(q,p,t),(30) whereˆI[2,...,N]is an integration with respect to variables q2,...,q N,p2,...,p N. Obviously,that one-particle density of probability satisfies the normalization con-ditionˆI[1]ρ1(q,p,t)=1.(31) The Bogoliubov hierarchy equations27–30describe the evolution of the reduced density of probability.They can be derived from the Liouville equation.Let us derive thefirst Bogoliubov equation with fractional derivatives from the fractional Liouville equation(27).Differentiation of Eq.(30)with respect to time gives∂ρ1∂t ˆI[2,...,N]ρN=ˆI[2,...,N]∂ρN242V.E.Tarasov Using Eq.(27),we get∂ρ1∂t =−ˆI[2,...,N](Dαq1(V1ρN)+Dαp1(F1ρN)).(35)Since the variable q1is an independent of q2,...,q N and p2,...,p N,thefirst term in Eq.(35)can be written asˆI[2,...,N]Dαq k (V1ρN)=Dαq1V1ˆI[2,...,N]ρN=Dαq1(V1ρ1).The force F1acts on thefirst particle.It is a sum of the internal forcesF1k=F(q1,p1,q k,p k,t),and the external force F e1=F e(q1,p1,t).In the case of binary interaction,we haveF1=F e1+Nk=2F1k.(36)Using Eq.(36),the second term in Eq.(35)can be rewritten in the formˆI[2,...,N]Dαp1(F1ρN)=ˆI[2,...,N](Dαp1(F e1ρN)+Nk=2Dαp1(F1kρN))=Dαp1(F e1ρ1)+Nk=2Dαp1ˆI[2,...,N](F1kρN).(37)We assume thatρN is invariant under the permutations of identical particles:31ρN(...,q k,p k,...,q l,p l,...,t)=ρN(...,q l,p l,...,q k,p k,...,t).Liouville and Bogoliubov Equations with Fractional Derivatives243 In this case,ρN is a symmetric function,and all(N−1)terms of sum(37)are identical.Therefore the sum can be replaced by one term with the multiplier(N−1):N k=2ˆI[2,...,N]Dαp1s(F1kρN)=(N−1)ˆI[2,...,N]Dαp1(F12ρN).(38)UsingˆI[2,...,N]=ˆI[2]ˆI[3,...,N],we rewrite the right-hand side of(38)in the formˆI[2]Dαp1(F12ˆI[3,...,N]ρN)=Dαp1ˆI[2](F12ρ2),(39)whereρ2is two-particle density of probability that is defined by the fractional integration of the N-particle density of probability over all q k and p k,except k= 1,2:ρ2=ρ(q1,p1,q2,p2,t)=ˆI[3,...,N]ρN(q,p,t).(40) Since p1is independent of q2,p2,we can change the order of the integrations and the differentiations:ˆI[2]Dαp1(F12ρ2)=Dαp1ˆI[2]F12ρ2.Finally,we obtain∂ρ1∂q1,V1=∂H(q1,p1)244V.E.TarasovIn this case,we can rewrite Eq.(45)in the formI(ρ2)=−Dαp1(ρ1F eff).(46) Substituting of Eq.(46)into Eq.(41),we obtain∂ρ1∂t+(v,Dαq f)+e(E,Dαp f)=0,(48) where f=ρ1is the one-particle density of probability,and(v,Dαq f)=ms=1(v s,Dαqsf).(49)If we take into account the magneticfield(B=0),then we must use the fractional generalization of Leibnitz rules:Dαp(fg)=∞r=0Γ(α+1)mc Dαp([p,B]f)=emc klmεklm B m Dαp k(p l f) =eΓ(r+1)Γ(α−r+1)[Dα−ip kf]δkl p r l =emc klmεklm B m p l[Dαp k f]=eLiouville and Bogoliubov Equations with Fractional Derivatives245 where f0is a homogeneous stationary density of probability that satisfies Eq.(48) for E=0.Substituting of Eq.(52)into(48),we get for thefirst perturbation∂δf2π +∞−∞dke−ikx e−|k|α(55)is the Levy stable density of probability.40Forα=1,the function(55)gives the Cauchy distributionL1[x]=1x2+1,(56)and Eq.(54)is1q2s(g s t)−2+1.(57) Forα=2,we get the Gauss distribution:L2[x]=1πe−x2/4,(58)and the function(54)is(g s t)−1/21πe−q2s/(4g s t).(59)For1<α≤2,the function Lα[x]can be presented as the expansionLα[x]=−1n!sin(nπ/2).(60)The asymptotic(x→∞,1<α<2)is given byLα[x]∼−1n!sin(nπ/2).(61)As the result,we arrive at the asymptotic of the solution,which exhibits power-like tails for x→∞.The tail is the important property for the solutions of fractional equations.246V.E.Tarasov6.ConclusionIn this paper,we consider equations with derivatives of non-integer order that can be used in statistical mechanics and kinetic theory.We derive the Liouville,Bogoli-ubov and Vlasov equations with fractional derivatives with respect to coordinates and momenta.To derive the fractional Liouville equation,we consider the conser-vation of probability tofind a system in the fractional differential volume element. Using the fractional Liouville equation,we obtain the fractional generalization of the Bogoliubov hierarchy equations.Fractional Bogoliubov equations can be used to derive fractional kinetic equations.4,9,10The fractional kinetic is related to equa-tions that have derivatives of non-integer orders.Fractional equations appear in the description of chaotic dynamics and fractal media.For the fractional linear oscil-lator,the physical meaning of the derivative of orderα<2is dissipation.Note that fractional derivatives with respect to coordinates can be connected with the long-range power-law interaction of the systems.22–24References1.I.Podlubny,Fractional Differential Equations(Academic Press,New York,1999).2.K.B.Oldham and J.Spanier,The Fractional Calculus(Academic Press,New York,1974).3.S.G.Samko,A.A.Kilbas and O.I.Marichev,Fractional Integrals and DerivativesTheory and Applications(Gordon and Breach,New York,1993).4.G.M.Zaslavsky,Chaos,fractional kinetics,and anomalous transport,Phys.Rep.371(2002)461–580.5.G.M.Zaslavsky,Hamiltonian Chaos and Fractional Dynamics(Oxford UniversityPress,Oxford,2005).6. A.Carpinteri and F.Mainardi,Fractals and Fractional Calculus in ContinuumMechanics(Springer,New York,1997).7.R.Hilfer,Applications of Fractional Calculus in Physics(World Scientific,Singapore,2000).8.V.E.Tarasov,Continuous medium model for fractal media,Phys.Lett.A336(2005)167–174;Fractional Fokker–Planck equation for fractal media,Chaos15 (2005)023102;Possible experimental test of continuous medium model for fractal media,Phys.Lett.A341(2005)467–472;Fractional hydrodynamic equations for fractal media,Ann.Phys.318(2005)286–307;Wave equation for fractal solid string, Mod.Phys.Lett.B19(2005)721–728;Dynamics of fractal solid,Int.J.Mod.Phys.B19(2005)4103–4114;Fractional Chapman–Kolmogorov equation,Mod.Phys.Lett.B21(4)(2007).9.G.M.Zaslavsky,Fractional kinetic equation for Hamiltonian chaos,Physica D76(1994)110–122.10. A.I.Saichev and G.M.Zaslavsky,Fractional kinetic equations:Solutions andapplications,Chaos7(1997)753–764.11.G.M.Zaslavsky and M. A.Edelman,Fractional kinetics:From pseudochaoticdynamics to Maxwell’s demon,Physica D193(2004)128–147.12.R.R.Nigmatullin,Fractional kinetic equations and universal decoupling of a memoryfunction in mesoscale region,Physica A363(2006)282–298;A.V.Chechkin,V.Yu.Gonchar and M.Szydlowsky,Fractional kinetics for relaxation and superdiffusion in magneticfield,Physics of Plasmas9(2002)78–88;R.K.Saxena,A.M.Mathai andLiouville and Bogoliubov Equations with Fractional Derivatives247H.J.Haubold,On fractional kinetic equations,Astrophysics and Space Science282(2002)281–287.13. B.A.Carreras,V.E.Lynch and G.M.Zaslavsky,Anomalous diffusion and exit timedistribution of particle tracers in plasma turbulence model,Physics of Plasmas8 (2001)5096–5103.14.L.M.Zelenyi and ovanov,Fractal topology and strange kinetics:Frompercolation theory to problems in cosmic electrodynamics,Physics Uspekhi47(2004) 749–788.15.V. E.Tarasov,Electromagneticfield of fractal distribution of charged particles,Physics of Plasmas12(2005)082106;Multipole moments of fractal distribution of charges,Mod.Phys.Lett.B19(2005)1107–1118;Magnetohydrodynamics of fractal media,Physics of Plasmas13(2006)052107;Electromagneticfields on fractals,Mod.Phys.Lett.A21(2006)1587–1600.16.V.E.Tarasov,Gravitationalfield of fractal distribution of particles,Celes.Mech.Dynam.Astron.19(2006)1–15.17. F.Mainardi and R.Gorenflo,On Mittag-Leffler-type functions in fractional evolutionprocesses,put.Appl.Math.118(2000)283–299.18. F.Mainardi,Fractional relaxation-oscillation and fractional diffusion-wavephenomena,Chaos,Solitons and Fractals7(1996)1461–1477.19.V.E.Tarasov and G.M.Zaslavsky,Dynamics with low-level fractionality,Physica A368(2006)399–415.20.V.E.Tarasov,Fractional generalization of Liouville Equation,Chaos14(2004)123–127;Fractional systems and fractional Bogoliubov hierarchy equations,Phys.Rev.E71(2005)011102;Fractional Liouville and BBGKI equations,J.Phys.Conf.Ser.7(2005)17–33;Transport equations from Liouville equations for fractional systems,Int.J.Mod.Phys.B20(2006)341–354;Fokker–Planck equation for frac-tional systems,Int.J.Mod.Phys.B21(2007)accepted.21.V. E.Tarasov,Fractional generalization of gradient and Hamiltonian systems,J.Phys.A38(2005)5929–5943;Fractional generalization of gradient systems,Lett.Math.Phys.73(2005)49–58;Fractional variations for dynamical systems:Hamilton and Lagrange approaches,J.Phys.A39(2006)8409–8425.skin and G.M.Zaslavsky,Nonlinear fractional dynamics on a lattice with long-range interactions,Physica A368(2006)38–54.23.V. E.Tarasov and G.M.Zaslavsky,Fractional dynamics of coupled oscillatorswith long-range interaction,Chaos16(2006)023110;Fractional dynamics of sys-tems with long-range interaction,Commun.Nonlin.Sci.Numer.Simul.11(2006) 885–898.24.N.Korabel,G.M.Zaslavsky and V.E.Tarasov,Coupled oscillators with power-lawinteraction and their fractional dynamics analogues,Commun.Nonlin.Sci.Numer.Simul.11(2006)accepted(math-ph/0603074).25.K.Cottrill-Shepherd and M.Naber,Fractional differential forms,J.Math.Phys.42(2001)2203–2212; F.B.Adda,The differentiability in the fractional calculus, Nonlinear Analysis47(2001)5423–5428.26.M.Caputo and F.Mainardi,A new dissipation model based on memory mechanism,Pure and Appl.Geophys.91(1971)134–147.27.N.N.Bogoliubov,Kinetic equations,Zh.Exper.Teor.Fiz.16(1946)691;J.Phys.USSR10(1946)265.28.N.N.Bogoliubov,in Studies in Statistical Mechanics,Vol.1,eds.J.de Boer and G.E.Uhlenbeck(North-Holland,Amsterdam,1962),p.1.29.K.P.Gurov,Foundation of Kinetic Theory:Method of N.N.Bogoliubov(Nauka,Moscow,1966)in Russian.248V.E.Tarasov30. D.Ya.Petrina,V.I.Gerasimenko and P.V.Malishev,Mathematical Basis of ClassicalStatistical Mechanics(Naukova dumka,Kiev,1985)in Russian.31.N.N.Bogoliubov and N.N.Bogoliubov,Jr.,Introduction to Quantum StatisticalMechanics(World Scientific,Singapore,1982),Sec.1.4.32. A.A.Vlasov,Vibrating properties of electronic gas,Zh.Exper.Teor.Fiz.8(1938)291;On the kinetic theory of an assembly of particles with collective interaction, SR9(1945)25.33. A.A.Vlasov,Many-particle Theory and its Application to Plasma(Gordon andBreach,New York,1961).34.G.Ecker,Theory of Fully Ionized Plasmas(Academic Press,New York,1972).35.N.A.Krall and A.W.Trivelpiece,Principles of Plasma Physics(McGraw-Hill,NewYork,1973).36.V. E.Tarasov,Stationary solution of Liouville equation for non-Hamiltoniansystems,Ann.Phys.316(2005)393–413;Classical canonical distribution for dis-sipative systems,Mod.Phys.Lett.B17(2003)1219–1226.37. A.Isihara,Statistical Physics(Academic Press,New York,London,1971),Appendix IV,Sec.7.5.38.P.Resibois and M.De Leener,Classical Kinetic Theory of Fluids(John Wiley andSons,New York,London,1977),Sec.IX.4.39. D.Forster,Hydrodynamics Fluctuations,Broken Symmetry,and CorrelationFunctions(W.E.Benjamin Inc.,London,Amsterdam,1975),Sec.6.4.40.V.Feller,An Introduction to Probability Theory and its Applications,Vol.2(Wiley,New York,1971).。

p I(J P)=12+)Status:∗∗∗∗938.27231±0.000281COHEN87RVUE1986CODATA value•••We do not use the following data for averages,fits,limits,etc.•••938.2796±0.0027COHEN73RVUE1973CODATA value1The mass is known much more precisely in u:m=1.007276470±0.000000012u.p MASSp mass much more precisely.VALUE(MeV)DOCUMENT ID TECN COMMENTp p /(qpp/p CHARGE-TO-MASS RATIO, qm m p)/(A test of CPT invariance.Listed here are measurements involving theinertial masses.For a discussion of what may be inferred about the ratioofp’s.VALUE DOCUMENT ID TECN COMMENTp)/M(p)=0.999999998p/m e−=1836.152660±0.000083and m p/m e−=1836.152680±0.000088.Both are completely consistent with the1986CODATA(COHEN87)value for m p/me−of1836.152701±0.000037.We use the CODATAvalues of the masses(they come from an overallfit to a variety of data on the fundamental constants)and don’t try to take into account more recent measurements involving the masses.( q m m p)/ q p p –q p maverageA test of CPT invariance.Taken from the(1.5±1.1)×10−9OUR EVALUATIONp eq p +q p /p charge-to-mass ratio,givenabove,is much better determined.See also a similar test involving the electron.VALUEDOCUMENT IDTECN<2×105−4HUGHES 92RVUE4HUGHES 92uses recent measurements of Rydberg-energy and cyclotron-frequency ra-tios.p MAGNETIC MOMENTSee the “Note on Baryon Magnetic Moments”in the ΛListings.VALUE (µN )DOCUMENT IDTECNCOMMENTp MAGNETIC MOMENTA few early results have been omitted.VALUE (µN )DOCUMENT IDTECNCOMMENTp 208Pb 11→10X-ray−2.817±0.048ROBERTS 78CNTR−2.791±0.021HU75CNTR Exotic atoms(µp+µp)µaverageA test of CPT invariance.Calculated from the p and(−2.6±2.9)×10−3OUR EVALUATIONp ELECTRIC POLARIZABILITYαpVALUE(10−4fm3)DOCUMENT ID TECN COMMENTp MAGNETIC POLARIZABILITYβpThe electric and magnetic polarizabilities are subject to a dispersion sum-rule constraintβ=(14.2±0.5)×10−4fm3.Errors here areanticorrelated with those on2.1±0.8±0.510MACGIBBON95RVUE global average Page3Created:6/29/199812:15•••We do not use the following data for averages,fits,limits,etc.•••1.7±0.6±0.9MACGIBBON95CNTRγp Compton scattering 4.4±0.4±1.1HALLIN93CNTRγp Compton scattering3.58+1.19−1.25+1.03−1.07ZIEGER92CNTRγp Compton scattering3.3±2.2±1.3FEDERSPIEL91CNTRγp Compton scattering 10MACGIBBON95combine the results of ZIEGER92,FEDERSPIEL91,and their own experiment to get a“global average”in which model errors and systematic errors are treated in a consistent way.See MACGIBBON95for a discussion.>1.6×1025p,n11,12EVANS77•••We do not use the following data for averages,fits,limits,etc.•••>3×1023p12DIX70CNTR>3×1023p,n12,13FLEROV5811Mean lifetime of nucleons in130Te nuclei.12Converted to mean life by dividing half-life by ln(2)=0.693.13Mean lifetime of nucleons in232Th nuclei.p MEAN LIFEp Partial Mean Lives”after“p Partial Mean Lives,”below.LIMIT(years)CL%EVTS DOCUMENT ID TECN COMMENTp/p,cosmic rays>3.7×10−3BREGMAN78CNTR Storage ring Page4Created:6/29/199812:15Antilepton+mesonτ1N→e+π>130(n),>550(p)90%τ2N→µ+π>100(n),>270(p)90%τ3N→νπ>100(n),>25(p)90%τ4p→e+η>14090%τ5p→µ+η>6990%τ6n→νη>5490%τ7N→e+ρ>58(n),>75(p)90%τ8N→µ+ρ>23(n),>110(p)90%τ9N→νρ>19(n),>27(p)90%τ10p→e+ω>4590%τ11p→µ+ω>5790%τ12n→νω>4390%τ13N→e+K>1.3(n),>150(p)90%τ14p→e+K0S>7690%τ15p→e+K0L>4490%τ16N→µ+K>1.1(n),>120(p)90%τ17p→µ+K0S>6490%τ18p→µ+K0L>4490%τ19N→νK>86(n),>100(p)90%τ20p→e+K∗(892)0>5290%τ21N→νK∗(892)>22(n),>20(p)90%Antilepton+mesonsτ22p→e+π+π−>2190%τ23p→e+π0π0>3890%τ24n→e+π−π0>3290%τ25p→µ+π+π−>1790%τ26p→µ+π0π0>3390%τ27n→µ+π−π0>3390%τ28n→e+K0π−>1890%Lepton+mesonτ29n→e−π+>6590%τ30n→µ−π+>4990%τ31n→e−ρ+>6290%τ32n→µ−ρ+>790%τ33n→e−K+>3290%τ34n→µ−K+>5790% Page5Created:6/29/199812:15Lepton+mesonsτ35p→e−π+π+>3090%τ36n→e−π+π0>2990%τ37p→µ−π+π+>1790%τ38n→µ−π+π0>3490%τ39p→e−π+K+>2090%τ40p→µ−π+K+>590%Antilepton+photon(s)τ41p→e+γ>46090%τ42p→µ+γ>38090%τ43n→νγ>2490%τ44p→e+γγ>10090%Three(or more)leptonsτ45p→e+e+e−>51090%τ46p→e+µ+µ−>8190%τ47p→e+νν>1190%τ48n→e+e−ν>7490%τ49n→µ+e−ν>4790%τ50n→µ+µ−ν>4290%τ51p→µ+e+e−>9190%τ52p→µ+µ+µ−>19090%τ53p→µ+νν>2190%τ54p→e−µ+µ+>690%τ55n→3ν>0.000590%τ56n→5νInclusive modesτ57N→e+anything>0.6(n,p)90%τ58N→µ+anything>12(n,p)90%τ59N→νanythingτ60N→e+π0anything>0.6(n,p)90%τ61N→2bodies,ν-free∆B=2dinucleon modesThe following are lifetime limits per iron nucleus.τ62p p→π+π+>0.790%τ63p n→π+π0>290%τ64n n→π+π−>0.790%τ65n n→π0π0>3.490%τ66p p→e+e+>5.890% Page6Created:6/29/199812:15τ67p p→e+µ+>3.690%τ68p p→µ+µ+>1.790%τ69p n→e+ν>1.690%τ71n n→νeνµ>0.00000690%p DECAY MODESPartial mean lifeMode(years)Confidence level p→e−γ>184895%τ74p→e−η>17195%τ76p→e−K0L>995%>26n 900<0.7ARISAKA 85KAMI >82p (free)9000.2BLEWITT 85IMB >250p 9000.2BLEWITT 85IMB >25n 9044PARK 85IMB >15p ,n 900BATTISTONI84NUSX >0.5p 9010.315BARTELT 83SOUD >0.5n 9010.315BARTELT 83SOUD >5.8p 90216KRISHNA...82KOLR >5.8n 90216KRISHNA...82KOLR >0.1n 9017GURR 67CNTR14This BECKER-SZENDY 90result includes data from SEIDEL 88.15Limit based on zero events.16We have calculated 90%CL limit from 1confined event.17We have converted half-life to 90%CL mean life.µτ N →+πτ2LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>100n 900<0.2HIRATA 89C KAMI >270p 9000.5SEIDEL 88IMB •••We do not use the following data for averages,fits,limits,etc.•••>81p 9000.2BERGER 91FREJ >35n 9011.0BERGER 91FREJ >230p 900<0.07HIRATA 89C KAMI >63n 9000.5SEIDEL 88IMB >76p 9021HAINES 86IMB >23n 9087HAINES 86IMB >46p 900<0.7ARISAKA 85KAMI >20n900<0.4ARISAKA 85KAMI >59p (free)9000.2BLEWITT 85IMB >100p 9010.4BLEWITT 85IMB >38n 9014PARK85IMB >10p ,n 900BATTISTONI 84NUSX >1.3p ,n900ALEKSEEV81BAKSN τ →νπτ3LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>25p 903232.8HIRATA 89C KAMI >100n 9013HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>13n 9011.2BERGER 89FREJ >10p 901114BERGER 89FREJ >6n 907360HAINES 86IMB >2p 901613KAJITA 86KAMI >40n 9001KAJITA 86KAMI >7n 902819PARK85IMB >7n 900BATTISTONI 84NUSX >2p 90≤3BATTISTONI 84NUSX >5.8p 90118KRISHNA...82KOLR >0.3p 90219CHERRY 81HOME >0.1p9020GURR67CNTR Page 8Created:6/29/199812:1518We have calculated 90%CL limit from 1confined event.19We have converted 2possible events to 90%CL limit.20We have converted half-life to 90%CL mean life.e τ p →+ητ4LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>140p 900<0.04HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>44p 9000.1BERGER 91FREJ >100p 9000.6SEIDEL 88IMB >200p 9053.3HAINES 86IMB >64p 900<0.8ARISAKA 85KAMI >64p (free)9056.5BLEWITT 85IMB >200p 9054.7BLEWITT 85IMB >1.2p90221CHERRY81HOME21We have converted 2possible events to 90%CL limit.µτ p →+ητ5LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>69p 901<0.08HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>26p 9010.8BERGER 91FREJ >1.3p 9000.7PHILLIPS 89HPW >34p 9011.5SEIDEL 88IMB >46p 9076HAINES 86IMB >26p901<0.8ARISAKA 85KAMI >17p (free)9066BLEWITT 85IMB >46p9078BLEWITT85IMBn τ →νητ6LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>54n 9020.9HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>29n 9000.9BERGER 89FREJ >16n 9032.1SEIDEL 88IMB >25n 9076HAINES 86IMB >30n 9000.4KAJITA 86KAMI >18n 9043PARK 85IMB >0.6n90222CHERRY81HOME22We have converted 2possible events to 90%CL limit.e τ N →+ρτ7LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>75p 9022.7HIRATA 89C KAMI >58n 9001.9HIRATA 89C KAMI Page 9Created:6/29/199812:15•••We do not use the following data for averages,fits,limits,etc.•••>29p 9002.2BERGER 91FREJ >41n 9001.4BERGER 91FREJ >38n 9024.1SEIDEL 88IMB >1.2p 900BARTELT 87SOUD >1.5n 900BARTELT 87SOUD >17p 9077HAINES 86IMB >14n 9094HAINES 86IMB >12p 900<1.2ARISAKA 85KAMI >6n 902<1ARISAKA 85KAMI >6.7p (free)9066BLEWITT 85IMB >17p 9077BLEWITT 85IMB >12n 9042PARK85IMB >0.6n 9010.323BARTELT 83SOUD >0.5p 9010.323BARTELT 83SOUD >9.8p 90124KRISHNA...82KOLR >0.8p 90225CHERRY 81HOME23Limit based on zero events.24We have calculated 90%CL limit from 0confined events.25We have converted 2possible events to 90%CL limit.µτ N →+ρτ8LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>110p 9001.7HIRATA 89C KAMI >23n 9011.8HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>12p 9000.5BERGER 91FREJ >22n 9001.1BERGER 91FREJ >4.3p 9000.7PHILLIPS 89HPW >30p 9000.5SEIDEL 88IMB >11n 9011.1SEIDEL 88IMB >16p 9044.5HAINES 86IMB >7n 9065HAINES 86IMB >12p 900<0.7ARISAKA 85KAMI >5n901<1.2ARISAKA 85KAMI >5.5p (free)9045BLEWITT 85IMB >16p 9045BLEWITT 85IMB >9n9012PARK85IMBτ N →νρτ9LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>27p 9051.5HIRATA 89C KAMI >19n9000.5SEIDEL88IMB Page 10Created:6/29/199812:15•••We do not use the following data for averages,fits,limits,etc.•••>9n 9042.4BERGER 89FREJ >24p 9000.9BERGER 89FREJ >13n 9043.6HIRATA 89C KAMI >13p 9011.1SEIDEL 88IMB >8p 9065HAINES 86IMB >2n 901510HAINES 86IMB >11p 9021KAJITA 86KAMI >4n 9022KAJITA 86KAMI >4.1p (free)9067BLEWITT 85IMB >8.4p 9065BLEWITT 85IMB >2n 9073PARK 85IMB >0.9p 90226CHERRY 81HOME >0.6n90226CHERRY81HOME26We have converted 2possible events to 90%CL limit.e τ p →+ωτ10LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>45p 9021.45HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>17p 9001.1BERGER 91FREJ >26p 9011.0SEIDEL 88IMB >1.5p 900BARTELT 87SOUD >37p 9065.3HAINES 86IMB >25p 901<1.4ARISAKA 85KAMI >12p (free)9067.5BLEWITT 85IMB >37p 9065.7BLEWITT85IMB >0.6p 9010.327BARTELT 83SOUD >9.8p 90128KRISHNA...82KOLR >2.8p 90229CHERRY 81HOME27Limit based on zero events.28We have calculated 90%CL limit from 0confined events.29We have converted 2possible events to 90%CL limit.µτ p →+ωτ11LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>57p 9021.9HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>11p 9001.0BERGER 91FREJ >4.4p 9000.7PHILLIPS 89HPW >10p 9021.3SEIDEL 88IMB >23p9021HAINES 86IMB >6.5p (free)9098.7BLEWITT 85IMB >23p9087BLEWITT85IMB Page 11Created:6/29/199812:15ττ n →νω 12LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>43n 9032.7HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>17n 9010.7BERGER 89FREJ >6n 9021.3SEIDEL 88IMB >12n 9066HAINES 86IMB >18n 9022KAJITA 86KAMI >16n 9012PARK 85IMB >2.0n90230CHERRY81HOME30We have converted 2possible events to 90%CL limit.e τ N →+Kτ13LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>150p 900<0.27HIRATA 89C KAMI >1.3n 900ALEKSEEV 81BAKS •••We do not use the following data for averages,fits,limits,etc.•••>60p 900BERGER 91FREJ >70p 9001.8SEIDEL 88IMB >77p 9054.5HAINES 86IMB >38p900<0.8ARISAKA 85KAMI >24p (free)9078.5BLEWITT 85IMB >77p 9054BLEWITT 85IMB >1.3p900ALEKSEEV81BAKSe τ p →+K 0Sτ14LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>76p9000.5BERGER91FREJp e τ→+K 0Lτ15LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>44p90≤0.1BERGER91FREJτ N →µ+Kτ16LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>120p 9010.4HIRATA 89C KAMI >1.1n900BARTELT 87SOUD •••We do not use the following data for averages,fits,limits,etc.•••>54p 900BERGER 91FREJ >3.0p 9000.7PHILLIPS 89HPW >19p 9032.5SEIDEL 88IMB >1.5p 90031BARTELT 87SOUD >40p 9076HAINES 86IMB >19p901<1.1ARISAKA 85KAMI >6.7p (free)901113BLEWITT85IMB Page 12Created:6/29/199812:15>40p 9078BLEWITT 85IMB >6p 901BATTISTONI84NUSX >0.6p 90032BARTELT 83SOUD >0.4n 90032BARTELT 83SOUD >5.8p 90233KRISHNA...82KOLR >2.0p 900CHERRY81HOME >0.2n 9034GURR67CNTR31BARTELT 87limit applies to p →µ+K 0S.32Limit based on zero events.33We have calculated 90%CL limit from 1confined event.34We have converted half-life to 90%CL mean life.p µτ →+K 0Sτ17LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>64p 9001.2BERGER 91FREJτ p →µ+K 0Lτ18LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>44p90≤0.1BERGER91FREJτ N →νKτ19LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>100p 9097.3HIRATA 89C KAMI >86n 9002.4HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>15n 9011.8BERGER 89FREJ >15p 9011.8BERGER 89FREJ >0.28p 9000.7PHILLIPS 89HPW >0.3p 900BARTELT87SOUD >0.75n 90035BARTELT 87SOUD >10p 9065HAINES 86IMB >15n 9035HAINES 86IMB >28p 9033KAJITA 86KAMI >32n 9001.4KAJITA 86KAMI >1.8p (free)90611BLEWITT 85IMB >9.6p 9065BLEWITT 85IMB >10n 9022PARK 85IMB >5n 900BATTISTONI 84NUSX >2p 900BATTISTONI84NUSX >0.3n 90036BARTELT 83SOUD >0.1p 90036BARTELT 83SOUD >5.8p 90137KRISHNA...82KOLR >0.3n 90238CHERRY 81HOME35BARTELT 87limit applies to n →νK 0S.36Limit based on zero events.37We have calculated 90%CL limit from 1confined event.38We have converted 2possible events to 90%CL limit. Page 13Created:6/29/199812:15ττ p →e +K ∗(892)0 20LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>52p 9021.55HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>10p 9000.8BERGER 91FREJ >10p901<1ARISAKA85KAMIτ N →νK ∗(892)τ21LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>22n 9002.1BERGER 89FREJ >20p9052.1HIRATA 89C KAMI •••We do not use the following data for averages,fits,limits,etc.•••>17p 9002.4BERGER 89FREJ >21n 9042.4HIRATA 89C KAMI >10p 9076HAINES 86IMB >5n 9087HAINES 86IMB >8p 9032KAJITA 86KAMI >6n 9021.6KAJITA 86KAMI >5.8p (free)901016BLEWITT 85IMB >9.6p 9076BLEWITT 85IMB >7n 9014PARK85IMB >2.1p90139BATTISTONI82NUSX39We have converted 1possible event to 90%CL limit.τ p →e +π+π−τ22LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>21p 9002.2BERGER 91FREJe τ p →+π0π0 τ23LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>38p9010.5BERGER91FREJe τ n →+π−π0 τ24LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>32n9010.8BERGER91FREJµτ p →+π+π− τ25LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>17p 9012.6BERGER 91FREJ •••We do not use the following data for averages,fits,limits,etc.•••>3.3p9000.7PHILLIPS89HPW Page 14Created:6/29/199812:15ττ p →µ+π0π0 26LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>33p 9010.9BERGER 91FREJµτ n →+π−π0 τ27LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>33n9001.1BERGER91FREJe τ n →+K 0π− τ28LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>18n9010.2BERGER91FREJn e τ →−π+τ29LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>65n 9001.6SEIDEL 88IMB •••We do not use the following data for averages,fits,limits,etc.•••>55n 9001.09BERGER 91B FREJ >16n 9097HAINES 86IMB >25n9024PARK85IMBµτ n →−π+ τ30LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>49n 9000.5SEIDEL 88IMB •••We do not use the following data for averages,fits,limits,etc.•••>33n 9001.40BERGER 91B FREJ >2.7n 9000.7PHILLIPS 89HPW >25n 9076HAINES 86IMB >27n9023PARK85IMBe τ n →−ρ+ τ31LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>62n 9024.1SEIDEL 88IMB •••We do not use the following data for averages,fits,limits,etc.•••>12n 90136HAINES 86IMB >12n9053PARK85IMBn µ τ →−ρ+τ32LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>7n 9011.1SEIDEL 88IMB •••We do not use the following data for averages,fits,limits,etc.•••>2.6n 9000.7PHILLIPS 89HPW >9n 9075HAINES 86IMB >9n9022PARK85IMB Page 15Created:6/29/199812:15>32n 9032.96BERGER 91B FREJ •••We do not use the following data for averages,fits,limits,etc.•••>0.23n9000.7PHILLIPS89HPWµτ n →−K+ τ34LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>57n 9002.18BERGER 91B FREJ •••We do not use the following data for averages,fits,limits,etc.•••>4.7n9000.7PHILLIPS89HPWe τ p →−π+π+τ35LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>30p 9012.50BERGER 91B FREJ •••We do not use the following data for averages,fits,limits,etc.•••>2.0p9000.7PHILLIPS89HPWe τ n →−π+π0 τ36LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>29n9010.78BERGER91B FREJµτ p →−π+π+ τ37LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>17p 9011.72BERGER 91B FREJ •••We do not use the following data for averages,fits,limits,etc.•••>7.8p9000.7PHILLIPS89HPWµτ n →−π+π0 τ38LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>34n9000.78BERGER91B FREJτ p →e −π+K +τ39LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>20p9032.50BERGER91B FREJτ p →µ−π+K +τ40LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>5p9020.78BERGER91B FREJ Page 16Created:6/29/199812:15>460p 9000.6SEIDEL 88IMB •••We do not use the following data for averages,fits,limits,etc.•••>133p 9000.3BERGER 91FREJ >360p 9000.3HAINES 86IMB >87p (free)9000.2BLEWITT 85IMB >360p 9000.2BLEWITT 85IMB >0.1p 9040GURR67CNTR40We have converted half-life to 90%CL mean life.µτ p →+γτ42LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>380p 9000.5SEIDEL 88IMB •••We do not use the following data for averages,fits,limits,etc.•••>155p 9000.1BERGER 91FREJ >97p 9032HAINES 86IMB >61p (free)9000.2BLEWITT 85IMB >280p 9000.6BLEWITT 85IMB >0.3p 9041GURR67CNTR41We have converted half-life to 90%CL mean life.τ n →νγτ43LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>24n 90106.86BERGER 91B FREJ •••We do not use the following data for averages,fits,limits,etc.•••>9n 907360HAINES 86IMB >11n902819PARK85IMBe τ p →+γγτ44LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>100p9010.8BERGER91FREJe τ p →+e +e− τ45LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>510p 9000.3HAINES 86IMB •••We do not use the following data for averages,fits,limits,etc.•••>147p9000.1BERGER 91FREJ >89p (free)9000.5BLEWITT 85IMB >510p9000.7BLEWITT85IMB Page 17Created:6/29/199812:15ττ p →e +µ+µ− 46LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>81p 9000.16BERGER 91FREJ •••We do not use the following data for averages,fits,limits,etc.•••>5.0p9000.7PHILLIPS89HPWe τ p →+νντ47LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>11p90116.08BERGER91B FREJe τ n →+e −ντ48LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>74n 900<0.1BERGER 91B FREJ •••We do not use the following data for averages,fits,limits,etc.•••>45n 9055HAINES 86IMB >26n9043PARK85IMBµτ n →+e −ντ49LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>47n900<0.1BERGER91B FREJµτ n →+µ−ντ50LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>42n 9001.4BERGER 91B FREJ •••We do not use the following data for averages,fits,limits,etc.•••>5.1n 9000.7PHILLIPS 89HPW >16n 90147HAINES 86IMB >19n9047PARK85IMBp µτ →+e +e −τ51LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>91p90≤0.1BERGER91FREJτ p →µ+µ+µ−τ52LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>190p 9010.1HAINES 86IMB •••We do not use the following data for averages,fits,limits,etc.•••>119p 9000.2BERGER 91FREJ >10.5p 9000.7PHILLIPS 89HPW >44p (free)9010.7BLEWITT 85IMB >190p 9010.9BLEWITT 85IMB >2.1p90142BATTISTONI82NUSX42We have converted 1possible event to 90%CL limit. Page 18Created:6/29/199812:15ττ p →µ+νν 53LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>21p 90711.23BERGER 91B FREJe τ p →−µ+µ+ τ54LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>6.0p9000.7PHILLIPS89HPWτ n →3ντ55LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>0.00049n 902243SUZUKI 93B KAMI•••We do not use the following data for averages,fits,limits,etc.•••ττ N →νanything 59Anything =π,ρ,K ,etc.LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN•••We do not use the following data for averages,fits,limits,etc.•••>0.0002p ,n90LEARNED79RVUEe τ N →+π0anythingτ60LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN>0.6p ,n 90LEARNED79RVUEτ N →2bodies,ν-freeτ61LIMIT(1030years)PARTICLECL %EVTS BKGD ESTDOCUMENT IDTECN•••We do not use the following data for averages,fits,limits,etc.•••>1.3p ,n90ALEKSEEV81BAKSπτ p p →+π+ τ62LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>0.7904.342BERGER91B FREJτper iron nucleusπτ p n →+π0 τ63LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>2.0900.310BERGER91B FREJτper iron nucleusπ τ n n →+π−τ64LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>0.7904.182BERGER91B FREJτper iron nucleusπτ n n →0π0 τ65LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>3.4900.780BERGER91B FREJτper iron nucleuse τ p p →+e+ τ66LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>5.8900<0.1BERGER91B FREJτper iron nucleuse τ p p →+µ+ τ67LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>3.6900<0.1BERGER91B FREJτper iron nucleusµτ p p →+µ+ τ68LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>1.7900.620BERGER91B FREJτper iron nucleus Page 20Created:6/29/199812:15τ p n →e +ντ69LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>2.8905.679BERGER 91B FREJ τper iron nucleusτ p n →µ+ντ70LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>1.6904.374BERGER91B FREJτper iron nucleusτn n →νe νeτ71LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>0.000012905.79BERGER91B FREJτper iron nucleusτn n →νµνµτ72LIMIT(1030years)CL %EVTS BKGD ESTDOCUMENT IDTECNCOMMENT>0.000006904.44BERGER91B FREJτper iron nucleusp PARTIAL MEAN LIVESτ/B i ,wherep →e −γ τ73τ p beamτ p →e −πτ74VALUE (years)CL%DOCUMENT IDTECNCOMMENT>55495GEER94CALO 8.9GeV/cp →e −η τ75τ p beamτ p →e −K 0Sτ76VALUE (years)CL%DOCUMENT IDTECNCOMMENT>2995GEER94CALO 8.9GeV/cp →e −K 0Lτ77τ p beamp REFERENCESGLICENSTEIN97PL B411326J.F.Glicenstein(SACL) GABRIELSE95PRL743544+Phillips,Quint+(HARV,MANZ,SEOUL) MACGIBBON95PR C522097+Garino,Lucas,Nathan+(ILL,SASK,INRM) GEER94PRL721596+Marriner,Ray+(FNAL,UCLA,PSU) HALLIN93PR C481497+Amendt,Bergstrom+(SASK,BOST,ILL) SUZUKI93B PL B311357+Fukuda,Hirata,Inoue+(KAMIOKANDE Collab.) HUGHES92PRL69578+Deutch(LANL,AARH) ZIEGER92PL B27834+Van de Vyver,Christmann,DeGraeve+(MPCM) Also92B PL B281417(erratum)Zieger,...,Van den Abeele,Ziegler(MPCM) BERGER91ZPHY C50385+Froehlich,Moench,Nisius+(FREJUS Collab.) BERGER91B PL B269227+Froehlich,Moench,Nisius+(FREJUS Collab.) FEDERSPIEL91PRL671511+Eisenstein,Lucas,MacGibbon+(ILL) BECKER-SZ...90PR D422974Becker-Szendy,Bratton,Cady,Casper+(IMB-3Collab.) ERICSON90EPL11295+Richter(CERN,DARM) GABRIELSE90PRL651317+Fei,Orozco,Tjoelker+(HARV,MANZ,WASH,IBS) BERGER89NP B313509+Froehlich,Moench+(FREJUS Collab.) CHO89PRL632559+Sangster,Hinds(YALE) HIRATA89C PL B220308+Kajita,Kifune,Kihara+(Kamiokande Collab.) PHILLIPS89PL B224348+Matthews,Aprile,Cline+(HPW Collab.) KREISSL88ZPHY C37557+Hancock,Koch,Koehler,Poth+(CERN PS176Collab.) SEIDEL88PRL612522+Bionta,Blewitt,Bratton+(IMB Collab.) BARTELT87PR D361990+Courant,Heller+(Soudan Collab.) Also89PR D401701erratum Bartelt,Courant,Heller+(Soudan Collab.) COHEN87RMP591121+Taylor(RISC,NBS) HAINES86PRL571986+Bionta,Blewitt,Bratton,Casper+(IMB Collab.) KAJITA86JPSJ55711+Arisaka,Koshiba,Nakahata+(Kamiokande Collab.) ARISAKA85JPSJ543213+Kajita,Koshiba,Nakahata+(Kamiokande Collab.) BLEWITT85PRL552114+LoSecco,Bionta,Bratton+(IMB Collab.) DZUBA85PL154B93+Flambaum,Silvestrov(NOVO) PARK85PRL5422+Blewitt,Cortez,Foster+(IMB Collab.) BATTISTONI84PL133B454+Bellotti,Bologna,Campana+(NUSEX Collab.) MARINELLI84PL137B439+Morpurgo(GENO) WILKENING84PR A29425+Ramsey,Larson(HARV,VIRG) BARTELT83PRL50651+Courant,Heller,Joyce,Marshak+(MINN,ANL) BATTISTONI82PL118B461+Bellotti,Bologna,Campana+(NUSEX Collab.) KRISHNA...82PL115B349Krishnaswamy,Menon+(TATA,OSKC,INUS) ALEKSEEV81JETPL33651+Bakatanov,Butkevich,Voevodskii+(PNPI) Translated from ZETFP33664.CHERRY81PRL471507+Deakyne,Lande,Lee,Steinberg+(PENN,BNL) COWSIK80PR D222204+Narasimhan(TATA) BELL79PL86B215+Calvetti,Carron,Chaney,Cittolin+(CERN) GOLDEN79PRL431196+Horan,Mauger,Badhwar,Lacy+(NASA,PSLL) LEARNED79PRL43907+Reines,Soni(UCI) BREGMAN78PL78B174+Calvetti,Carron,Cittolin,Hauer,Herr+(CERN) ROBERTS78PR D17358(WILL,RHEL) EVANS77Science197989+Steinberg(BNL,PENN) ROBERSON77PR C161945+King,Kunselman+(WYOM,CIT,CMU,VPI,WILL) HU75NP A254403+Asano,Chen,Cheng,Dugan+(COLU,YALE) COHEN73JPCRD2663+Taylor(RISC,NBS) DYLLA73PR A71224+King(MIT) BAMBERGER70PL33B233+Lynen,Piekarz+(MPIH,CERN,KARL) DIX70Thesis Case(CASE) HARRISON69PRL221263+Sandars,Wright(OXF) GURR67PR1581321+Kropp,Reines,Meyer(CASE,WITW) FLEROV58DOKL379+Klochkov,Skobkin,Terentev(ASCI)。

a rXiv:h e p-la t/931113v115Nov1993THE “SPIN”STRUCTURE OF THE NUCLEON ∗–A LATTICE INVESTIGATION R.ALTMEYER,G.SCHIERHOLZ DESY,Notkestraße 85,D-22603Hamburg,Germany M.G ¨OCKELER,R.HORSLEY HLRZ,c/o Forschungszentrum J¨u lich,D-52425J¨u lich,Germany and ERMANN Fakult¨a t f¨u r Physik,Universit¨a t Bielefeld,D-33501Bielefeld,Germany ABSTRACT We will discuss here an indirect lattice evaluation of the baryon axial singlet current matrix element.This quantity may be related to the fraction of nucleon spin carried by the quarks.The appropriate structure function has recently been measured (EMC experiment).As in this experiment,we find that the quarks do not appear to carry a large component of the nucleon spin.1.Introduction and Theoretical Discussion Hadrons appear to be far more complicated than the (rather successful)con-stituent quark model would suggest.For example an old result is from the πN sigma term,which seems to give a rather large strange component to the nucleon mass.A more recent result is from the EMC experiment 1which suggests that constituent quarks are responsible for very little of the nucleon spin.These are non-perturbative effects and so is an area where lattice calculations may be of some help.The EMC experiment measured deep inelastic scattering (DIS)using a µbeamon a proton target.The new element in the experiment was that both the inital µand proton were longitudinally polarised.A measurement of the difference in the cross sections for parallel and anti-parallel polarised protons enabled the structure function g 1(x,Q 2)to be found.Theoretically 2this is of interest as using the Wilson operator product expansion,moments of the structure function are related to certain matrix elements.In this case,defining s µ∆q = P s |¯q γµγ5q |P s for q =u ,d or s quarks (i.e.the expectation value of the axial singlet current)the lowest moment is then given byPreprint HLRZ 93-71November 1993 10dxg 1(x,Q 2)=19∆u +19∆s≈0.20+12T≫t≫τ≫0,where T is the time box size=24here.µN=+exp(−E N)and the parity partner(which always occurs when using staggered fermions)is denoted byΛ,so thatµΛ=−exp(−EΛ).The amplitudes Aαβand energies Eαare known from2-point correlation functions.Thus from eq.(3)we can extract P s|ˆΩ|P s .Simply settingΩto be¯χγiγ5χhas,however,a number of disadvantages:the correlation function in eq.(3)has connected and disconnected parts,in thefit there are cross terms present,and the operator must be renormalised.Another approach is to consider the divergence of the current6.This can be related to the QCD anomaly.In the chiral limit an equivalent formulation of the problem is thus (n f=4)n f∆Σ=limp′→ pd3 x Fµν˜Fµνe i p· x.(5)8π2(We shall take p′= 0, p=(0,0,p).)On the lattice we have used the L¨u scher7 construction for this charge.Although technically complicated it is integer valued and hence has no renormalisation.In the3-point correlation function there are no cross terms(Q has a definite parity).A disadvantage is that we need to take p→0. On our lattice the smallest momentum available is the rather large p≈500MeV.Nevertheless on attempting this measurement,wefind a reasonable signal with ∆Σ≈0.18(2).This is small and tends to support the EMC result,which would indicate a rather large sea contribution to the proton.The existence of the QCD anomaly also proved important.Further details of our calculation are given in8.3.AcknowledgementsThis work was supported in part by the DFG.The numerical simulations were performed on the HLRZ Cray Y-MP in J¨u lich.We wish to thank both institutions for their support.The configurations used were those generated for the MT c project.4.References1.J.Ashman et.al.,Nucl.Phys.B328(1989)1.2.Some reviews:S.D.Bass and A.W.Thomas,preprint,ADP-92-183/T115(Adelaide,1992),H.Fritzsch,Lecture at Schladming Winter School,Schladming,1991,A.V.Kiselev and V.A.Petrov,preprint CERN-TH.6355/91,E.Reya,QCD–20years later,Conference,Aachen,Germany,1992,eds.P.M.Zerwas and H.A.Kastrup,(World Scientific,1993).3.R.L.Jaffe and A.Manohar,Nucl.Phys.B337(1990)509.4.R.Altmeyer,K.D.Born,M.G¨o ckeler,R.Horsley,ermann and G.Schierholz,Nucl.Phys.B389(1993)445.5.L.Maiani et.al.,Nucl.Phys.B293(1987)420.6.J.E.Mandula,Nucl.Phys.B(Proc.Suppl.)26(1992)356.7.M.L¨u scher,Comm.Math.Phys.85(1982)39.8.R.Altmeyer,K.D.Born,M.G¨o ckeler,R.Horsley,ermann and G.Schierholz,Nucl.Phys.B(Proc.Suppl.)30(1993)483;preprint,DESY 92-186,HLRZ92-103,December1992.。