Effective lagrangians for QCD at high density
Euler–Lagrange_equation
Euler–Lagrange equationJump to: navigation, searchIn calculus of variations, the Euler–Lagrange equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph Louis Lagrange in the 1750s.Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing (or maximizing) it. This is analogous to Fermat's theorem in calculus, stating that where a differentiable function attains its local extrema, its derivative is zero.In Lagrangian mechanics, because of Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for the action of the system. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations (see, for example, the "Field theory" section below).HistoryThe Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.Lagrange solved this problem in 1755 and sent the solution to Euler. The two further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.[1]StatementThe Euler–Lagrange equation is an equation satisfied by a function q of a real argument t which is a stationary point of the functionalwhere:∙q is the function to be found:such that q is differentiable, q(a) = x a, and q(b) = x b;∙q′ is the derivative of q:TX being the tangent bundle of X (the space of possible values of derivatives of functions with values in X);∙L is a real-valued function with continuous first partial derivatives:The Euler–Lagrange equation, then, is the ordinary differential equationwhere L x and L v denote the partial derivatives of L with respect to the second and third arguments, respectively.If the dimension of the space X is greater than 1, this is a system of differential equations, one for each component:Derivation of one-dimensionalEuler-Lagrange equationAlternate derivation of one-dimensionalEuler-Lagrange equationExamplesA standard example is finding the real-valued function on the interval[a , b ], such that f (a ) = c and f (b ) = d , the length of whose graph is as short as possible. The length of the graph of f is:the integrand function being 2'1)',,(y y y x L += evax , y , y ′)= (x , f (x ), f ′(x )).The partial derivatives of L are:By substituting these into the Euler –Lagrange equation, we obtainthat is, the function must have constant first derivative, and thus its graph is a straight line .Classical mechanicsBasic methodTo find the equations of motions for a given system, one only has to follow these steps:∙From the kinetic energy T, and the potential energy V, compute the Lagrangian L = T−V.∙Compute .∙Compute and from it, . It is important that be treated as a complete variable in its own right, and not as a derivative.∙Equate . This is, of course, the Euler–Lagrange equation.∙Solve the differential equation obtained in the preceding step. At this point, is treated "normally". Note that the above might bea system of equations and not simply one equation.Particle in a conservative force fieldThe motion of a single particle in a conservative force field (for example, the gravitational force) can be determined by requiring the action to be stationary, by Hamilton's principle. The action for this system iswhere x(t) is the position of the particle at time t. The dot above is Newton's notation for the time derivative: thus ẋ(t) is the particle velocity, v(t). In the equation above, L is the Lagrangian (the kinetic energy minus the potential energy):where:∙m is the mass of the particle (assumed to be constant in classical physics);∙v i is the i-th component of the vector v in a Cartesian coordinate system (the same notation will be used for other vectors);U is the potential of the conservative force.In this case, the Lagrangian does not vary with its first argument t. (By Noether's theorem, such symmetries of the system correspond to conservation laws. In particular, the invariance of the Lagrangian with respect to time implies the conservation of energy.)By partial differentiation of the above Lagrangian, we find:where the force is F = −∇U (the negative gradient of the potential, by definition of conservative force), and p is the momentum. By substituting these into the Euler–Lagrange equation, we obtain a system of second-order differential equations for the coordinates on the particle's trajectory,which can be solved on the interval [t0, t1], given the boundary values x(t0) and x i(t1). In vector notation, this system readsior, using the momentum,which is Newton's second law.Field theoryThis section contains too much jargon and may need simplification or further explanation. Please discuss this issue on the talk page, and/or remove or explain jargon terms used in the article. Editing help is available. (December 2009)Field theories, both classical field theory and quantum field theory, deal with continuous coordinates, and like classical mechanics, has its own Euler–Lagrange equation of motion for a field,where∙is the field, and∙is a vector differential operator:Note: Not all classical fields are assumed commuting/bosonic variables, (like the Dirac field, the Weyl field, the Rarita-Schwinger field) are fermionic and so, when trying to get the field equations from the Lagrangian density, one must choose whether to use the right or the left derivative of the Lagrangian density (which is a boson) with respect to the fields and their first space-time derivatives which arefermionic/anticommuting objects.There are several examples of applying the Euler–Lagrange equation to various Lagrangians:∙Dirac equation;∙Proca equation;∙electromagnetic tensor;∙Korteweg–de Vries equation;∙quantum electrodynamics.Variations for several functions, several variables, and higher derivativesSingle function of single variable with higher derivatives The stationary values of the functionalcan be obtained from the Euler-Lagrange equation[2]Several functions of one variableIf the problem involves finding several functions () of a single independent variable (x) that define an extremum of the functionalthen the corresponding Euler-Lagrange equations are[2]Single function of several variablesA multi-dimensional generalization comes from considering a function on n variables. If Ω is some surface, thenis extremized only if f satisfies the partial differential equationWhen n = 2 and is the energy functional, this leads to the soap-film minimal surface problem.Several functions of several variablesIf there are several unknown functions to be determined and several variables suchtthe system of Euler-Lagrange equations is[2]Single function of two variableswith higher derivativesIf there is a single unknown functionto be determined that is dependent ontwo variables and their higherderivatives such thatthe Euler-Lagrange equation is[2]Notes1.^ A short biography of Lagrange2.^ a b c d Courant, R. and Hilbert, D., 1953, Methods of Mathematical Physics:Vol I, Interscience Publishers, New York.References∙Weisstein, Eric W., "Euler-Lagrange Differential Equation" from MathWorld.∙Calculus of Variations on PlanetMath∙Izrail Moiseevish Gelfand (1963). Calculus of Variations. Dover.ISBN0-486-41448-5.∙Calculus of Variations at Example (Provides examples of problems from the calculus of variations that involve theEuler–Lagrange equations.)。
The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules
a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we find that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark effective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coefficients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suffices to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the effective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures thefinite mass difference between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)defined by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus effectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the effective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have refined the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such effects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only briefly sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”off-shell momenta of the heavy quarks.Due to the phase redefinition of the effective heavy quark fields in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coefficient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which define the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the effective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coefficient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one first writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coefficient function.possible subtraction terms.Because of theflavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The effective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suffices to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonfield,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usfirst present the result for the nonperturbative power corrections.WefindK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its effect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we find that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our final result,there is one more effect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the effective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the effective theory[16,18].Onefinds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,wefind stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourfinal result forχ3(y)is not very different from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisfigure one can also deduce how changes in the values of the vacuum condensates would affect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Wefind that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonfield and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesfinancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientific,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisfigure and are both represented by the narrow dots.11。
AGA 9 - 多路超声流量计测量技术（第2版）说明书
AN OVERVIEW AND UPDATE OF AGA 9 – ULTRASONIC METERSJim BowenSICK, Inc.A BSTRACTThe American Gas Association published Report No. 9, Measurement of Gas by Multipath Ultrasonic Meters 2nd Edition [Ref 1] in April 2007. Report 9 details recommended practice for using multipath gas ultrasonic meters (USMs) in fiscal (custody) measurement applications. This paper reviews some of history behind the development of AGA Report No. 9 (often referred to as AGA 9), key Report contents, which includes information on meter performance requirements, design features, testing procedures, and installation criteria. This paper also discusses changes that will be incorporated in the next revision. At the time of this paper the expected publication date is spring of 2017.I NTRODUCTION/B ACKGROUNDMembers of the AGA Transmission Measurement Committee (TMC) wrote AGA 9. It started in 1994 with the development of Technical Note M-96-2-3, Ultrasonic Flow Measurement for Natural Gas Applications [Ref 2]. This technical note was a compilation of the technology and discussed how the USMs worked, and is included as Appendix C in the 2007 edition, but it is deleted from the upcoming revision since most of the principles described in it have been adapted to the Report’s text. After competition of the Technical Note, the AGA TMC began the developing a new Report for custody application of gas USM technology. More than 50 contributors participated in its development, and included participants from the USA, Canada, The Netherlands, and Norway, representing a broad cross-section of measurement personnel in the natural gas industry at the time (circa 1997).AGA 9 incorporates many of the recommendations in the GERG Technical Monograph 8 [Ref 3] and certain related OIML [Ref 4 & 5] recommendations. Since USM’s are linear meters, much of the document was patterned around AGA 7, Measurement of Gas by Turbine Meters [Ref 6], another linear device. Most of the performance requirements in revision 1 of AGA 9 were based on the limited test data available at the time, and absent high rate test labs. Where no data was available to support a specific requirement, AGA 9 was silent, or left it up to the manufacturer to specify.R EVIEW OF AGA9§1 I NTRODUCTION (S COPE OF R EPORT)Section 1 of AGA 9 provides information on the scope of the document. It states that it’s for multipath ultrasonic transit-time flow meters that are used for the measurement of natural gas. A multipath meter is defined as having two or more independent acoustic paths used to measure transit time difference of sound pulses traveling upstream and downstream at an angle to the gas flow. Today most users require a minimum of 3 acoustic paths for fiscal measurement. The scope goes on to state “Typical applications include measuring the flow of large volumes of gas through production facilities, transmission pipelines, storage facilities, distribution systems and large end-use customer meter sets.”AGA 9 provides information to meter manufacturers that are more performance-based than manufacturing-based. Unlike orifice meters that basically are all designed the same, USM manufacturers have developed their products somewhat differently from one another: most notably, in path configurations. Thus, AGA 9 does not tell the manufacturers how to build their meter, but rather provides information on the performance the product must meet.§2 T ERMINOLOGYSection 2 discusses terminology and definitions that are used throughout the document. Terms like auditor, designer, inspector, manufacturer, etc. are defined there.§3 O PERATING C ONDITIONSSection 3 discusses operating range conditions over which the USM shall perform with specified accuracy. This includes sub-sections on gas quality, pressures, temperatures (both gas and ambient), gas flow considerations, and upstream piping and flow profiles. The gas quality specifications were based upon typical pipeline quality gas and no discussion was included for sour gas applications other than to consult with the manufacturer. It is important to note that these requirements were based upon the current manufacturer’s specifications in order to not exclude anyone. Based on the current state of AGA 9 revision, there are no significant user impact issues, yet on the table, regarding Section 3.§4 M ETER R EQUIREMENTSSection 4, titled “Meter Requirements”, addresses specific mechanical and electrical reuqirments manufacturers need meet to operate the devices in a hazardous environment. Sub-sections on codes and regulations, meter body markings, ultrasonic transducers, electronics, computer programs, and supporting documentation are included.§4.3 Meter BodyThe section on meter body discusses items such as operating pressure, corrosion resistance, mechanical issues relative to the meter body, and markings. Here is says manufacturers should publish the overall lengths of their ultrasonic meter bodies for the different ANSI flange ratings.External and Internal Corrosion resistance and compatibility with gas mixtures commonly found in today’s pipeline is stipulated.The inside diameter of the ultrasonic meter shall have the same inside diameter as the upstream tube’s diameter and must be within 1%. The value of 1% was based mainly on early European studies and also work performed at the Southwest Research Institute’s MRF (Metering Research Facility) in San Antonio, Texas.AGA 9 discusses the ability to remove transducers under pressure. Experience shows that transducers are rarely extracted under pressure, since many meter stations employ multiple runs, making short term outages possible. This allows de-pressurization and transducer removal after blow-down. Additionally, once the meter run is de-pressurized, the internal condition of the meter and associated piping can be inspected if boroscope and inspection ports are available.Proposed Changes:•Reference to use of extraction tools is eliminated.§4.4 TransducersThe section on transducers discusses a variety of issues including specifications, rate of pressure change, and transducer tests. The intent was to insure the manufacturer provided sufficient information to the end user in order to insure reliable and accurate operation in the field, and also to insure accurate operation should one or more pairs need replacement in the field. Subsections include basic specifications, rate of pressure change, exchange and transducer tests.Proposed Changes:•Hydrostatic test of meter bodies shall not be done with transducers installed if test pressure is above ANSI rating.For higher pressures, transducers need be removed.•§4.4.3, “Exchange”, in reference to transducers, is deleted and replaced with existing §4.4.4, “Transducer Tests” , which itself remains unchanged.§4.5 ElectronicsThe electronics section includes two suggested types of flow output communicated to flow computers: serial and frequency. Serial communication (digital using either RS-232 or RS-485) is suggested because the ultrasonic meter is clearly a very “smart” instrument and much of its usefulness relies on the internal information contained in the meter. The frequency output is a not required but standard on all USM, and is needed in applications where flow computers and Remote Terminal Units (RTUs) do not have the necessary software application to poll the USM using the serial port.A majority of users apply the frequency output as input to flow computers. Most USM manufacturers employ standard Modbus collection of measurement information via a serial link or Ethernet adapter. Additional serial or Ethernet ports are used for local interrogation using the manufacturer’s software.AGA 9 requires the manufacturer to also provide digital outputs (DOs) for flow direction and data valid. A digital out is used for monitoring by the flow computer to determine direction of flow (when a single frequency is used for both forward and reverse flow). Data valid is an indicator that the meter has an alarm condition that may impact its accuracy.AGA 9 requires the meter be electrically rated for a hazardous environment as defined by the National Electrical Code [Ref 12]. The minimum rating for a USM is for Class 1, Division 2, Group D environments. Some users specify a rating of Division 1, and, for the most part, all manufacturers design for the more stringent Division 1 requirement.Proposed Changes:•Elimination of the requirement for a scaled 4-20 ma output, now making it an option.•Added statement that a scaled 4-20 ma output is not to be used for custody measurement.•Delete of the paragraph specifying “significant change” as +/- 0.2% shift in the meter’s output, and new stipulation that Manufacturer’s replacement of cables, electronics, transducers, etc won’t shift meter output more that Manufacturer stated repeatability (0.2%).§4.6 Computer Programs Meter Firmware and SoftwareUSMs typically do not provide a local display or keyboard for communicating with the meter as is traditional with some flow computers. USM Manufacturers provide their meter specific software for meter configuration and diagnostic interrogation. There are few specific diagnostic data requirement in manufacturer’s to have similar looking and functioning software, but Report 9 does cite some in §4.6.4, “Meter Diagnostics”.The velocity data is used to indicate flow profile irregularities and to calculate volume rate from average velocity. The flow rate is determined from by multiplying velocity times the meter’s cross-sectional area of the meter. The speed-of-sound data is used as a diagnostic tool to check for erroneous transit time measurement errors. Other information is required to judge the quality of the data such as percent of accepted ultrasonic pulses, signal to noise ratio and transducer gains. A discussion on these is documented in several papers [Ref 13 & 14].Other meter requirements in this section include anti-roll devices (feet), pressure tap design and location on the meter, and standard meter markings. Many of these requirements are based on field experience and the lessons learned from other metering technologies.§4.7 §6 Individual Meter Testing Requirements Moved to “dimensional measurements” et alSection’s 4.7 discusses how the manufacturer will perform tests on the USM prior to shipment. Many also call this testing dry calibration. In reality dry calibration is simply an assembly process to help verify proper meter operation prior being installed in the field. Since there were no calibration facilities in North America until the late 1990’s, it was felt that if a manufacturer could precisely control the assembly process, flow calibration would not be required. Hence the term dry calibration has often been used to describe this section.AGA 9 requires the manufacturer to document the internal diameter of the meter to the nearest 0.0001 inches. This is to be determined from 12 separate inside diameter measurements. This dimension is to be adjusted back to 68 °F and reported on the documents. Measurements should be traceable to a national standard such as NIST, the National Institute for Standards and Technology.Individual meters are to be tested to strict tolerances for leaks and imperfections. AGA 9 also specifies a Zero-Flow Verification Test and a Flow-Calibration Test procedure (although a flow-calibration is not required).No significant changes have been proposed to §4.7 at this time.§4.8 DocumentationSection 4.7 of Report 9’s 2nd edition discusses the manufacturing and certification documents required to be delivered by USM builders. The section has been significantly reworked to add description of required dimensional measurements, leakage tests, etc.§5.0 Performance Requirements Installation (was §7 in second edition of Report 9)Many of the variables the designer should take into consideration when using USMs are addressed, such as operating conditions, use of flow conditioners and meter tube configurations etc.§5.2 Piping Configuration Metering Package Design CriteriaReport 9’s 1st Edition was developed with only limited empirical data. The 2nd Edition, published in 2007 was not much of an improvement. For instance, Section 5.2 of AGA 9 discusses upstream piping issues. The intent here is to provide the designer with some basic designs that will provide accurate measurement. It states “Recommend upstream and downstream piping configuration in minimum length — one without a flow conditioner and one with a flow conditioner — that will not create an additional flow-rate measurement error of more than +0.3% due to the installation configuration. This error limit should apply for any gas flow rate between q min and q max. The recommendation should be supported by test data.” In other words, the manufacturer is required to let the designer know what type of piping is permitted upstream so that the impact on accuracy will not be greater than 0.3%.In reality this is difficult given the variety of upstream pipe configurations operators use, so the so-called end-treatments (or entrance/exit piping) are often included in the calibration of the “metering package”.Much data has been taken and published since 1998, and, as a consequence of this data, and the desire to provide the highest level of accuracy, most users have elected to use a high-performance flow conditioner with their USM. Testing has shown that the use of a 19-tube bundle, typical with turbine and orifice metering, will not improve the USM performance, and inmost cases actually will degrade accuracy [Ref 7]. However, high performance flow conditioners, usually specified in meter package design, help to obtain a uniform and repeatable flow profile.Proposed Changes:•Process section added to indicate guidance for operating conditions such as pulsating flow and a more detailed discussion of potential noise impacts on USM operation.•Recommended Default Meter Run Configurations (§5.2.3) are modified to eliminate the optional entrance to the meter run through either a tee or elbow, leaving users to their design preference so long a meter performance meets §6 requirements.Corrected flow rate output requires USM metering packages include temperature measurement on the outlet leg of the meter run. AGA 9 recommends the thermowell be installed between 2D and 5D downstream of the USM on a uni-directional installation. It states the thermowell should be at least 3D from the meter on a bi-directional installation. This was based on research done at SwRI sponsored by GRI in the 1990’s. These studies found measurable influence onset at 2D downstream of USMs during and the committee settled on 3D as a reasonable distance.§6.0 Performance RequirementsSince Report 9 is a performance based document, “Performance Requirements” are core to its scope, and it addresses minimum performance requirements the USM’s must meet. It requires flow calibration for fiscal use.AGA 9’s second edition (2007) separates ultrasonic meters into two categories; smaller than 12” and meters that are 12” and larger. This division was created to allow relaxed accuracy requirements for smaller meters where tolerances are more difficult to maintain. All other requirements, including repeatability, resolution, velocity sampling interval, peak-to-peak error and zero-flow readings are the same, regardless of meter size.The edition preparing for release in 2017 will have 3 size ranges and new accuracy limits:•>10” Max Error: +/-0.7% q t≤q i≤q max; +/-1.4% for q min≤q i≤q t with linearity of+/-0.2% for q t≤q i≤q max•3”-8”: Max Error: +/-1.0% q t≤q i≤q max; +/-1.4% for q min≤q i≤q t with linearity of+/-0.2% for q t≤q i≤q max•<3”: Max Error: +/-2.0% q t≤q i≤q max; +/-3.0% for q min≤q i≤q t with linearity of+/-0.2% for q t≤q i≤q maxPer the second edition, the maximum error allowable for a 12-inch and larger ultrasonic flow meter is ±0.7%, and ±1.0% for small meters. This error expands to ±1.4% below Q t, the transition flowrate. Within the error bands, the error peak-to-peak error (also thought of as linearity) must be less than 0.7%. The repeatability of the meters must be with ±0.2% for the higher velocity range, and is permitted to be ±0.4 below Q t. Figure 1 is a graphical representation of these performance requirements as shown in AGA 9.Figure 1 – Performance Specification Summary, 2nd EditionSection 5 also discusses the potential effects of pressure, temperature and gas composition on the USM. Here is states “The UM shall meet the above flow-measurement accuracy requirements over the full operating pressure, temperature and gas composition ranges without the need for manual adjustment, unless otherwise stated by the manufacturer.” There has been some concern about calibrating a USM at one pressure and then operating at a different pressure. Although there are a variety of opinions on this, most believe the meter’s accuracy is not significantly impacted by pressure [Ref 13] since there has not been correlation with significant per pat Speed of Sound measurements (one would expect if the meter were changing dimensions due to pressure change that a path length change would result and turn up in a path SoS deviation).§7 C OMMISSIONING &F IELD V ERIFICATIONSection 7 briefly discusses field verification requirements. Since each USM provides somewhat different software to interface with the meter, AGA 9 was not too specific about how to verify field performance. Rather they left it up to the manufacturer to provide a written field verification procedure that the operator could follow. The 2nd edition made limited progress on Field Verification, but it is intended the 3rd edition will be much more expansive regarding Field Verification practice. Much debate is occurring on these topics now to insure recommended practice matches Operator requirements, current Field practice and Manufacturer capability, and since this discussion is on-going, it is difficult to state proposed changes due to their number and variety.Typically today the operator would check the basic diagnostic features including velocity profile, speed-of-sound by path, transducer performance, signal to noise ratios and gain. One additional test is to compare the meter’s reported SOS with that computed by a program based upon AGA 8 [Ref 12].At the time of the first release there was no universally excepted document that discussed how to compute SOS. However, in 2003 AGA published AGA Report No. 10, Speed of Sound in Natural Gas and Other Related Hydrocarbon Gases [Ref 13]. This document, based upon AGA 8, provides the foundation for computing SOS that most software uses today, and is incorporated by reference into the 2nd Edition. It will pass that AGA 10 collapses into AGA 8 and that the SoS calculation will be harmonized, to some extent with SGERG, the European consortium’s method to calculate SoS in hydrocarbon mixtures. This effort is nearly complete now and will result in an AGA 8 Equation of State document that includes AGA 10 Detailed and SGERG.R EFERENCES1.AGA Report No. 9, Measurement of Gas by Multipath Ultrasonic Meters, 2nd Edition, April 2007, American GasAssociation, 1515 Wilson Boulevard, Arlington, VA 222092.AGA Engineering Technical Note M-96-2-3, Ultrasonic Flow Measurement for Natural Gas Applications,AmericanGas Association, 1515 Wilson Boulevard, Arlington, VA 222093.GERG Technical Monograph 8 (1995), Present Status and Future Research on Multi-path Ultrasonic Gas Flow Meters,Christian Michelsen Research AS, the GERG Project Group and Programme Committee No. 2 - Transmission and Storage, Groupe Européen De Recherches Gazières4.OIML R 6General provisions for gas volume meters,1989 (E), International Recommendation, OrganizationInternationale de Métrologie Légale, Bureau International de Métrologie Légale, 11, rue Turgot - 75009 Paris - France 5.OIML D 11General requirements for electronic measuring instruments,1994 (E), International Document,Organization Internationale de Métrologie Légale, Bureau International de Métrologie Légale, 11, rue Turgot - 75009 Paris - France6.AGA Transmission Measurement Committee Report No. 7, Measurement of Gas by Turbine Meters, American GasAssociation, 1515 Wilson Boulevard, Arlington, VA 222097.T. A. Grimley, Ultrasonic Meter Installation Configuration Testing, AGA Operations Conference, 2000, Denver, CO8.John Stuart, Rick Wilsack, Re-Calibration of a 3-Year Old, Dirty, Ultrasonic Meter, AGA Operations Conference, 2001,Dallas, TX9.Code of Federal Regulations, Title 49 —Transportation, Part 192, (49 CFR 192), Transportation of Natural Gas andOther Gas by Pipeline: Minimum Federal Safety Standards, U.S. Government Printing Office, Washington, DC 20402 10.NFPA 70, National Electrical Code,1996 Edition, National Fire Protection Association, Batterymarch Park, Quincy,MA 0226911.Umesh Karnik & John Geerlings, Effect of Steps and Roughness on Multi-Path Ultrasonic Meters, ISFFM 2002,Washington, DC12.AGA Transmission Measurement Committee Report No. 8, Compressibility Factors of Natural Gas and Other RelatedHydrocarbon Gases, American Gas Association, 1515 Wilson Boulevard, Arlington, VA 2220913.AGA Report No 10, Speed of Sound in Natural Gas and Other Related Hydrocarbon Gases, July 2002, American GasAssociation, 1515 Wilson Boulevard, Arlington, VA 22209。
SAE J17112010
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SAE reviews each technical report at least every five years at which time it may be reaffes your written comments and suggestions.
SURFACE VEHICLE RECOMMENDED PRACTICE
J1711 JUN2010
Issued Revised
1999-03 2010-06
Superseding J1711 MAR1999
(R) Recommended Practice for Measuring the Exhaust Emissions and Fuel Economy of Hybrid-Electric Vehicles, Including Plug-in Hybrid Vehicles
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Screamin' Blues
Owner’s ManualScreamin’ Blues ™Overdrive/DistortionWARRANTY:We at DigiT ech are proud of our products and back-up each one with the following warranty:1.The warranty registration card must be mailed within ten days after purchase date to validate this war-ranty.2.DigiT ech warrants this product,when used solely within the U.S.,to be free from defects in materialsand workmanship under normal use and service.3.DigiT ech liability under this warranty is limited to repairing or replacing defective materials that showevidence of defect,provided the product is returned to DigiT ech WITH RETURN AUTHORIZATION, where all parts and labor will be covered up to a period of one year (this warranty is extended to a period of six years when the product has been properly registered by mail or through our website).A Return Authorization number may be obtained from DigiT ech by telephone.The company shall notbe liable for any consequential damage as a result of the product's use in any circuit or assembly.4.Proof-of-purchase is considered to be the burden of the consumer.5.DigiT ech reserves the right to make changes in design,or make additions to,or improvements uponthis product without incurring any obligation to install the same on products previously manufactured.6.The consumer forfeits the benefits of this warranty if the product's main assembly is opened and tam-pered with by anyone other than a certified DigiT ech technician or,if the product is used with AC volt-ages outside of the range suggested by the manufacturer.7.The foregoing is in lieu of all other warranties,expressed or implied,and DigiT ech neither assumes norauthorizes any person to assume any obligation or liability in connection with the sale of this product.In no event shall DigiT ech or its dealers be liable for special or consequential damages or from any delay in the performance of this warranty due to causes beyond their control.NOTE:The information contained in this manual is subject to change at any time without notification.Some information contained in this manual may also be inaccurate due to undocumented changes in the product or operating system since this version of the manual was completed.The information contained in this version of the owner's manual supersedes all previous versions.The Screamin’ Blues Overdrive / Distortion is designed for guitarists who need a pedal that responds to playing dynamics.Blues players in particular will love how the Screamin’ Blues gives them total control of their tone.Playing lightly gives you a mild overdrive,but dialing up the gain and digging in hard will make the Screamin’ Blues sing with ultra-rich harmonics and sustain.1.Level KnobControls the effect output level.T urn this knob clockwise to increase the output level,and counter-clockwise to decrease output level. When using an amplifier with moderate gain,the Level knob can be turned up to drive the input harder and increase your amp's own natural distortion.Be sure to turn down the Gain knob when using it in this manner.2.Low KnobControls the boost or cut of low frequencies.T urn this knob clockwise to increase the low frequency content,and counter-clockwise to decrease the low frequency content.3.AC Adapter JackConnect a DigiT ech PS200R power supply to this e the proper supply for your area’s Mains line voltage.4.High KnobControls the boost or cut of high frequencies.T urn this knob clockwise to increase the high frequency content,and counter-clockwise to decrease the high frequency content.5.Gain KnobControls the amount of distortion.T urn this knob clockwise to increase the amount of distortion,and counter-clockwise to decrease the amount of distortion.6.Input JackConnect your instrument to this jack.Connecting a guitar cable to this jack engages battery power even though the Indicator LED may not be lit.T o prolong battery life,disconnect all cables from the pedal when not in use.7.Pedal Release PinsThese two pins are spring-loaded hinges that hold the pedal in place. Push these pins in to release the pedal from the chassis exposing the battery compartment.(See battery replacement diagram.)8.PedalPress the pedal to turn the effect on and off.9.Out 1(Amp) JackConnect this output to your guitar amplifier.10.Indicator LEDThis LED indicates when the effect is turned on.If the LED becomes dim or does not light when the cables are connected,the battery needs to be replaced.11.Out 2 (Mixer) JackThis output features DigiT ech’s speaker cabinet compensation,that lets you run the pedal directly into a mixer or recorder input without having to use a guitar amplifier.ConnectionsThe Screamin’ Blues has a single input (INPUT) and a pair of outputs (AMP and MIXER).The two outputs give you the following options: running into a guitar amplifier,running directly to a mixing console or recording device,or both.The MIXER output uses DigiT ech’s speaker cabinet compensation for an authentic guitar amplifier sound without using an amp.Follow these setup directions before use:1.T urn the volume down on the amplifier/mixer you are connecting to.2.Connect the pedal output to the amplifier/mixer input.3.Connect the guitar to the pedal input.4.Connect the power supply to the pedal (optional).5.T urn the level control on the pedal to the minimum position.6.T urn up your amplifier/mixer level to the desired listening level.7.T urn on the pedal by pressing the pedal switch and gradually turn up the level to the desired listening level.Battery Replacementing the tip of a 1/4” guitar cable,push one of the release pins in on either side of the pedal,and remove the pedal from the pedal chassis.2.Remove the battery from the battery compartment and disconnect the battery cable.3.Connect a new battery to the battery cable and put it back in the battery compartment.Make sure the battery cable does not interfere with the spring or pedal switch arm.4.Place one hole of the pedal over its corresponding pin.5.Push the opposite pin in and lower the other side of the pedal into place over the depressed pin.Release the pin.When the pedal is properly fastened,both released pins are flush with the outer side of the pedal.DigiT ech 8760 South Sandy Parkway Sandy,Utah 84070PH (801) 566-8800 FAX (801) 566-7005DigiT ech is a registered trademark of the Harman Music Group Inc.Copyright - Harman Music Group Printed 5/2004Manufactured in the USA Screamin’ Blues Owners Manual 18-1865-A Please visit DigiT ech on the World Wide Web at: SPECIFICATIONS:• Controls - Level,Low,High,Gain,On/Off Pedal• Jacks - Input,Amp Out,Mixer Out • Input Impedance - 1 MOhms • Output Impedance - 100 Ohms• Power Supply - 9V Alkaline Dry Battery • Current Draw ~ 23mA (at 9VDC)• Power Consumption - 4.8Watts (w/optional PS200R power supply)• Battery Life - Approximately 18 hours (with continuous usage)• Dimensions - 4 15/16”(L) x 3 1/8”(W)x 2 1/8”(H)• Weight - 1.38 lbs.• Optional Power SupplyPS200R - 100 (100V ~ 50/60Hz)PS200R - 120 (120V ~ 60Hz)PS200R - 230 (230V ~ 50Hz)PS200R - 240 (240V ~ 50Hz)。
Fate of the Born-Infeld solution in string theory
a r X i v :h e p -t h /0211033v 5 1 F eb 2003Fate of the Born-Infeld solution in string theoryG.Karatheodoris,A.Pinzul and A.SternDepartment of Physics,University of Alabama,Tuscaloosa,Alabama 35487,USA ABSTRACT We argue that the Born-Infeld solution on the D9−brane is unstable under inclusion of derivative corrections to Born-Infeld theory coming from string theory.More specifically,we find no electrostatic solutions to the first order corrected Born-Infeld theory on the D9−brane which give a finite value for the Lagrangian.The Born-Infeld(BI)nonlinear description of electrodynamics is of current interest due to its role as an effective action for Dp-branes.[1],[2]One of its main features is the existence of a maximum allowed value|E|max for electrostaticfields.∗In the30′s,Born and Infeld promoted this theory because it has a spherically symmetric solution withfinite classical self-energy.[3]It agrees with the Coulumb solution at large distances,but has afinite limit for the radial component of the electricfield at the origin.This limit is just|E|max.As a result,the vectorfield is singular at the origin.Equivalently,the solution is thus defined on a Euclidean manifold with one point(corresponding to the‘source’)removed,despite the fact that the energy density is well behaved at the point.The theory contains one dimensionful parameter (namely,|E|max),which was determined by Born and Infeld afterfixing the classical self-energy with the electron mass.For the Dp-brane the dimensionful parameter is the string tension,and so any BI-type solution(or BIon[4])appearing there should correspond to a charged object with characteristic mass at the string tension.The fact that energetics admits vectorfields that are not everywhere defined indicates that BI theory cannot be a complete description.[5]Moreover, in string theory,the BI effective action is only valid for slowly varyingfields,i.e.it is the lowest order term in the derivative expansion for the full effective Dp-brane action.Since derivatives of thefields are not‘small’in the interior of the BI solution,the validity of such a solution in string theory can be questioned.It is therefore of interest to know whether or not analogues of the BI solutions survive for the full effective Dp-brane action.A derivative expansion has been recently carried out to obtain lowest order string corrections to the BI action for the space-filling D9-brane.[6],[7]With the restriction to electrostatic configurations,wefind that the BI solution on the D9-brane is unstable with the inclusion of such corrections,indicating that such singularfield configurations may not follow from the full effective action.The reason is basically due to the result that derivative corrections make it difficult for the electrostaticfield to attain its maximum value|E|max.Here we only allow forfield configurations that lead to a finite value for the Lagrangian.†This is a reasonable requirement from the point of view of the path integral,where one expects to recover the classical solutions in the WKB approximation. Moreover,wefind no nontrivial electrostatic solutions on the D9-brane D9-brane associated with afinite Lagrangian.The situation here is in contrast to skyrmion physics,where there are no nontrivial solutions to the zeroth order effective action for QCD.Higher order derivative corrections,like the Skyrme term,are necessary to stabilize the skyrmion.On the other hand,BIons appear at lowest order,but become unstable upon including the next order electrostatic corrections. There remains the possibility,however,of stabilizing the BIon with the inclusion of other degrees of freedom.For example,if we allow for magnetic effects one mightfind that the higher order corrections give a magnetic dipole moment to the BIon.Another possibility which is of current interest,concerns BI-type solutions that appear after dimensional reduction.In thiscase,the BI action is replaced by the Dirac-Born-Infeld action(DBI),containing degrees of freedom associated with the transverse modes of the brane.Classical solutions to the DBI action were found in[4],[8],[9],and they represent fundamental strings attached to branes.‡It is of interest to examine how such solutions are affected by derivative corrections[6].(One set of solutions(the BPS solutions)were found to be unaffected to all orders[10].)We hope to address these issues in future works.We begin with a review of the Born-Infeld electrostatics.The BI action is expressed in terms of the determinant of the matrix with elementshµν=ηµν+(2πα′)Fµν,(1)where Fµν=∂µAν−∂νAµis thefield strength and one assumes theflat metricηµν.On the D9−brane it is given byS(0)BI=1−det[hµν],(2)where d is9.In the absence of magneticfields L(0)BI simplifies toL(0)BI=1−r d−1.(5) The integration constant Q is the charge.(5)approaches the Coulomb solution when r→∞. When r→0,θ→π‡As no such interpretation is possible for the BIon on the D9-brane,it is convenient that wefind it to be unstable.=Πi F0i+−det[hµν](h i0−h0i),(7)are the momenta conjugate to A i and hµνhνρ=δµρ.As usual,the momentum conjugate to A0 is constrained to be zero.In the absence of magneticfields,(7)reduces toπ= Π1+ π2−1−(2πα′)∂iπi A0=11− f2−1−(2πα′)∂iπi A0,(9)where we integrated by parts.The coefficient of A0gives the Gauss law constraint.The remaining terms can be used to identify the self-energy of the BI solutionE(0)BI=Ωd−1(4π2α′)5g s d10x 1− 4∆ ,∆=hµνhρσhαβhγδ(SνραβSσµγδ−2SβγνρSδασµ),(12) whereκ=(2πα′)2under such diffeos.§We avoid this complication and work in Cartesian coordinates x i.For electrostaticfields E( x)= f( x)/(2πα′),S ijk0=−S ij0k=∂i∂j f k+fℓ1− f2(∂i f k∂j fℓ−∂i fℓ∂j f k)(13) Then exploiting the symmetry of the indices we can write−12S ijkℓS ijkℓ+1r4tanθcos4θ H′2+d−1cos3θ H′2+d−1κ(Q−r d−1tanθ)cos2θ=2 r d−1H′cos3θ−d−12r d−5sinθ 126−53d+85r2H2+4cos2θ(5+d−3r2H2)+cos4θ(d−2−r2H2) (17) We note that by rescaling r and Q we can setκequal to unity.Also(17)is invariant under Q→−Q andθ(r)→−θ(r),and this gives a prescription for mapping any possible chargedsolution to the anti-solution.The left hand side of (17)vanishes for the original BI solution(5),while the right hand side represents derivative corrections.Substituting (5)into the left hand side gives a vanishingly small correction as r →∞,but it is singular for r →0.The BI solution therefore cannot be trusted near the origin.Due to the presence of high order time derivatives in the correction terms in the action(12),the computation of the Hamiltonian for the system,and consequently the correction to the electrostatic energy (10),is problematic.For this reason we only require solutions to (17)to be associated with a finite value for the Lagrangian (15)instead of the energy.Since the BI (actually,Coulumb)solution is valid at large r ,θ(r )→Qǫ3 ′′+3r d −1H ′2−(d −1) 8(r d −3H )′+r d −5[13−7d +12r 2H 2] ,(19)where H ′→−(log ǫ)′′and we again set κequal to one.Taking d =9and Q >0,one has the following solution near the originǫ(r )→76432r 45,as r →0(20)It is easy to check that it gives a finite contribution to the Lagrangian as r →0.(For this one only needs that ǫ(r )goes to zero slower than r 5/3.)On the other hand,after numerically integrating this solution starting from some initial value r 0to increasing values of r we find that for any value of Q ,ǫgoes quickly to π/2or −π/2,and so the Lagrangian density is poorly behaved for large values of r .Below we plot the results for Q =1:0.000050.00010.000150.00021.56851.5691.56951.57050.000050.00010.000150.000220040060080010001200fig .3θ(r )vs.r fig .4r 8(L (0)BI +L (0)BI )vs.r From fig.3,θ′>0to the right of the minimum,and so no match with fig.1is possible there.To the left of the minimum,θ′tends to −∞as r →0according to (20),while from fig.1it appears to vanish in the limit.To make matters worse,the graphs we obtain for θnear the origin are highly sensitive to the initial value r 0of the integration,and this persists for all values of Q .We have checked that this is not due to the neglect of higher order terms in (20).So not only is there no agreement between the two numerical integration procedures,the validity of the solution near the origin is questionable.We are thus unable to find any spherically symmetric charged solutions to (17)consistent with the requirement of a finite Lagrangian.We also find no axially symmetric charged solutions with finite Lagrangian density every-where.Again,they correspond to d <9,and at zeroth order were characterized by a (9−d )-dimensional Euclidean surface where fis ing (18)we can once again numerically integrate to small values of r .The conclusions are the same as for d =9.For Q >0,θ(r )tends to the limit of π/2as r →0,and the corresponding radial Lagrangian density diverges as r →0(faster than 1/r ).For Q >0and d ≥6,the solution to (19)near the origin has the form ǫ(r )→ Cr d −55,as r →0,(21)where C =20008,33324,51836for d =6,7,8,respectively.As with the case d =9,it is consistent with the requirement of a finite Lagrangian for small r ,but the Lagrangian density diverges after numerically integrating to increasing large r ,and the problem of the sensitivity to the initial value r 0persists.So again we find no agreement between the two numerical integration procedures.For d <6,we find no solutions to (19)near the origin satisfying the requirement of a finite Lagrangian.Above we found negative results for all configurations associated with some nonzero value for the charge.Although there are no Q=0soliton solutions of the zeroth order Born-Infeld system,this need not be a priori true at higher orders.In this case,θ(r)for any spherically symmetric solution would not fall offas a power as r→∞,but ratherθ(r)→ A cosαr+B sinαr e−αr[8]C.G.Callan and J.M.Maldacena,Nucl.Phys.B513,198(1998).[9]P.S.Howe,mbert and P.C.West,Nucl.Phys.B515,203(1998).[10]L.Thorlacius,Phys.Rev.Lett.80,1588(1998).。
Power corrections in heavy-to-light decays at large recoil energy
a r X iv:h e p -p h /0210040v 1 2 O c t 20021Power corrections in heavy-to-light decays at large recoil energy ∗M.Diehl aa Institut f¨u r Theoretische Physik E,RWTH Aachen,52056Aachen,GermanyI briefly present recent work on QCD power corrections in heavy-to-light meson decays,using an effective field theory approach.1.INTRODUCTION The strong interaction physics in B decays is characterized by the presence of several distance scales.The general idea of factorization in QCD is to separate the dynamics at short distances (which can be treated perturbatively)from the long-distance physics (encoded in hadronic ma-trix elements like decay constants or form fac-tors).There are different techniques to achieve this separation.One is the detailed analysis of Feynman graphs,used for instance in [1]to es-tablish QCD factorization for exclusive B decays.Here I focus on an effective field theory approach,whose principal objects are operators and fields.This framework is very general and well suited to be applied to a wide class of processes.As an explicit application I consider exclu-sive B decays to light mesons (e.g.B →πℓν,B →K ∗ℓ+ℓ−)in kinematics where the recoil en-ergy of the light meson is large,E ∼m b .In the limit m b →∞symmetry relations betweenthe various form factors describing the B →πorB →K ∗transition emerge [2],which greatly re-duce hadronic uncertainties when analyzing thesedecays in the Standard Model and beyond.Theserelations receive corrections in inverse powers ofm b ,which we have systematically studied in ef-fective field theory [3].They also receive correc-tions in αs from the heavy quark decay vertex(which are part of the effective theory)and fromhard interactions of the spectator quark in the Bmeson (which have so far only been treated by di-rect calculation of the relevant Feynman graphsin QCD [4]).m b ΛQCD .The heavy quark and its interactions with ultrasoft gluons is de-scribed in conventional heavy-quark effective the-ory (HQET),and the dynamics of collinear and ultrasoft modes in the soft-collinear effective the-ory (SCET)pioneered by Bauer et al.[5].For each momentum mode a separate field is introduced,e.g.A µc (x )and A µus (x )for collinear and ultrasoft gluons.For these fields one then must find the Lagrangian and current operators which reproduce the results of QCD to a given accuracy in a power-counting parameter λ,in our case given by2(1+v /)Q v (x ),where Q v (x )=e im b (vx )Q (x )has2Figure 1.Heavy-quark decay into a cluster of collinear and ultrasoft particles.the large phase of the full heavy quark field re-moved.For collinear quarks the large componentisξ(x )=1in +∂W †iD /⊥ n /+3one obtains Q effasWZ†Q v−1qΓQ→e−im b(vx)ΛQCD/m b,which can be sys-tematically described in the effective theory of Sec.2by matching the QCD matrix elements L|q effΓQ eff|B .Projectingq effor Q eff,and one re-mains with2(5)independent form factors.3.the O(λ2)corrections come from the smallcomponents of bothfields.Hence no sym-metry relations are left to this accuracy,andone has3(7)independent form factors as infull QCD.There is a caveat concerning the corrections at orderλ.The corresponding operators in the ef-fective theory involve different parton configura-tions in the light meson than the leading-order operators,for instance with an additional trans-versely polarized gluon.There may well be fur-ther dynamical suppression of their matrix ele-ments,bringing them down from O(λ)to O(λ2).Whether this happens requires knowledge of the light meson wave functions which goes beyond the scope of SCET.We remark that these puta-tive O(λ)corrections affect the form factor rela-tions used in the analysis of the forward-backward asymmetry in B→K∗ℓ+ℓ−,see[6].4.SUMMARYWe have performed a systematic analysis of power corrections to soft-collinear factorization in the effectivefield theory formalism.The in-teractions between collinear and ultrasoft quarks and gluons are described in SCET,whose effec-tive Lagrangian was obtained including the power corrections offirst and second order inλ.Heavy-to-light meson decays at large recoil energy can be described by the effective actions of SCET and HQET plus the effective heavy-to-light current, which was also obtained including O(λ2)correc-tions.Aspects of the O(λ)corrections were pre-viously investigated in[7].Among the2(5)leading-order symmetry rela-tions between the transition form factors for B to pseudoscalar(vector)mesons,1(3)receive power corrections at order4C.W.Bauer,S.Fleming,D.Pirjol andI.W.Stewart,Phys.Rev.D63(2001)114020[hep-ph/0011336];C.W.Bauer,D.Pirjol and I.W.Stew-art,Phys.Rev.D65(2002)054022[hep-ph/0109045].6.G.Burdman,Phys.Rev.D57(1998)4254[hep-ph/9710550].7.J.Chay and C.Kim,Phys.Rev.D65(2002)114016[hep-ph/0201197];C.W.Bauer,D.Pirjol and I.W.Stewart,hep-ph/0205289.。
开启片剂完整性的窗户(中英文对照)
开启片剂完整性的窗户日本东芝公司，剑桥大学摘要：由日本东芝公司和剑桥大学合作成立的公司向《医药技术》解释了FDA支持的技术如何在不损坏片剂的情况下测定其完整性。
太赫脉冲成像的一个应用是检查肠溶制剂的完整性，以确保它们在到达肠溶之前不会溶解。
关键词：片剂完整性，太赫脉冲成像。
能够检测片剂的结构完整性和化学成分而无需将它们打碎的一种技术，已经通过了概念验证阶段，正在进行法规申请。
由英国私募Teraview公司研发并且以太赫光（介于无线电波和光波之间）为基础。
该成像技术为配方研发和质量控制中的湿溶出试验提供了一个更好的选择。
该技术还可以缩短新产品的研发时间，并且根据厂商的情况，随时间推移甚至可能发展成为一个用于制药生产线的实时片剂检测系统。
TPI技术通过发射太赫射线绘制出片剂和涂层厚度的三维差异图谱，在有结构或化学变化时太赫射线被反射回。
反射脉冲的时间延迟累加成该片剂的三维图像。
该系统使用太赫发射极，采用一个机器臂捡起片剂并且使其通过太赫光束，用一个扫描仪收集反射光并且建成三维图像(见图)。
技术研发太赫技术发源于二十世纪九十年代中期13本东芝公司位于英国的东芝欧洲研究中心，该中心与剑桥大学的物理学系有着密切的联系。
日本东芝公司当时正在研究新一代的半导体，研究的副产品是发现了这些半导体实际上是太赫光非常好的发射源和检测器。
二十世纪九十年代后期，日本东芝公司授权研究小组寻求该技术可能的应用，包括成像和化学传感光谱学，并与葛兰素史克和辉瑞以及其它公司建立了关系，以探讨其在制药业的应用。
虽然早期的结果表明该技术有前景，但日本东芝公司却不愿深入研究下去，原因是此应用与日本东芝公司在消费电子行业的任何业务兴趣都没有交叉。
这一决定的结果是研究中心的首席执行官DonArnone和剑桥桥大学物理学系的教授Michael Pepper先生于2001年成立了Teraview公司一作为研究中心的子公司。
TPI imaga 2000是第一个商品化太赫成像系统，该系统经优化用于成品片剂及其核心完整性和性能的无破坏检测。
New Terms for the Compact Form of Electroweak Chiral Lagrangian
a r X i v :0704.1075v 2 [h e p -p h ] 3 M a y 2007Preprint typeset in JHEP style -HYPER VERSIONHong-Hao Zhang,Wen-Bin Yan,and J.K.Parry Center for High Energy Physics &Department of Physics,Tsinghua University,Beijing 100084,China E-mail:zhanghonghao@ ,wenbin.yan@ ,jkparry@ Xue-Song Li Science College,Hunan Agricultural University,Changsha 410128,China E-mail:lixuesong@ Abstract:The compact form of the electroweak chiral Lagrangian is a reformulation of its original form and is expressed in terms of chiral rotated electroweak gauge fields,which is crucial for relating the information of underlying theories to the coefficients of the low-energy effective Lagrangian.However the compact form obtained in previous works is not complete.In this letter we add several new chiral invariant terms to it and discuss the contributions of these terms to the original electroweak chiral Lagrangian.Keywords:Electroweak Chiral Lagrangian,Beyond Standard Model.So far the postulated Higgs particle of the standard model has not been observed in ex-periments.We do not know what the electroweak symmetry breaking mechanism in nature is.The electroweak chiral Lagrangian is a general low-energy description for electroweak symmetry breaking patterns[1,2],especially for those strong dynamical electroweak sym-metry breaking mechanisms[3].All the coefficients of the electroweak chiral Lagrangian, in principle,can befixed by experiments[4,5,6,7,8,9,10,11,12,13].Due to the non-perturbative property of the possible strong dynamics,it is difficult to relate the measured coefficients of the electroweak chiral Lagrangian to underlying theories.Recently,the series of work of Ref.[14,15,16,17,18,19,20,21]successfully produced the predictions for the coefficients of the chiral Lagrangian from the underlying theory of QCD,which lights up the hope of building up the relationship between the coefficients of the electroweak chiral Lagrangian and underlying strong dynamical models.As seen in Ref.[14],a crucialfirst step of the derivation of the chiral Lagrangian from QCD is to reformulate the original chiral Lagrangian in terms of chiral rotated external sourcefields.For the case of the elec-troweak chiral Lagrangian,we need to similarly reformulate it in terms of chiral rotated electroweak gaugefields in order to deduce the information of underlying theories.An attempt of this line of thought is the work of Ref.[22],which tries to give a so-called com-pact form of the electroweak chiral Lagrangian in terms of chiral rotated electroweak gauge fields.However,there are still several relevant terms not included in the reformulation of Ref.[22].In this letter,we shall present3new terms for this reformulation and give their contributions to the original electroweak chiral Lagrangian.We refer the interested reader to Ref.[22]for all the details on the compact form of the electroweak chiral Lagrangian and the relations between the compact form and the original form.Besides the inner-product terms c2g42V a′µV a′νV b′µV b′ν,c8g22(dµV a′ν−dνV a′µ)V a′µAν, and c9g22(dµV a′µ)V a′νAνalready included in the compact form of Ref.[22],there should be 3new cross-product terms which are given by,∆L=c′2g42(ǫa′b′V a′µV b′ν)2+c′8g22ǫa′b′(dµV a′ν−dνV a′µ)V b′µAν+c′9g22ǫa′b′(dµV a′µ)V b′νAν,(1) where a′,b′run from1to2,andǫ12=−ǫ21=1.Here and henceforth V a′µand Aµand Aµξin Ref.[22]respectively.There are short for the chiral rotated gaugefields V a′µξare also3cross-product terms corresponding to the c15,17,25-terms in that paper,but they vanish.Let us consider Eq.(1)term by term.From the definitions of Ref.[22],it is straightforward to obtain the relation between thefirst term of Eq.(1)and the ordinary terms of the electroweak chiral Lagrangian as follows,c′2g42(ǫa′b′V a′µV b′ν)2=−4c′2 [tr(XµXν)]2−[tr(XµXµ)]2−tr(XµXν)tr(τ3Xµ)tr(τ3Xν)+tr(XµXµ)[tr(τ3Xν)]2 ,(2) where Xµ≡U†(DµU).And the last two terms of Eq.(1)are respectively given by, c′8g22ǫa′b′(dµV a′ν−dνV a′µ)V b′µAν=c ′8 −1W µνX µX ν)−i W µν)tr(τ3X µX ν) ,(3)with 2W a µνU ,and c ′9g 22ǫa ′b ′(d µV a ′µ)V b ′νA ν=12,∆α4=−4c ′2,∆α5=4c ′2,∆α6=4c ′2−c ′82(1−4β1)c ′9,∆α7=−4c ′2+c ′82,∆α10=(1−4β1)c ′9.(5)The coefficients c ′2,8,9and c i (i =1,2,...,25)in this compact form of the electroweak chiral Lagrangian are determined by the underlying ultraviolet theories.For example,if we take the one-doublet technicolor model as the underlying theory,it can be shown that these 3new coefficients c ′2,8,9are all non-vanishing,and full details will be presented in forthcoming publications[23].In summary,we have provided 3new terms to the compact form of the electroweak chiral Lagrangian introduced in Ref.[22].These additional terms were not considered in the previous work.In this letter we have related these new terms to the original electroweak chiral Lagrangian,which will be crucial in forthcoming studies of strong dynamical models.AcknowledgmentsWe are indebted to Qing Wang for all our knowledge about the chiral Lagrangian and his helps and supports for this work.This work is supported in part by the National Natural Science Foundation of China.References[1]T.Appelquist and C.W.Bernard,Phys.Rev.D 22,200(1980).T.Appelquist and G.H.Wu,Phys.Rev.D 48,3235(1993)[arXiv:hep-ph/9304240].[2]A.C.Longhitano,Phys.Rev.D 22,1166(1980);Nucl.Phys.B 188,118(1981).[3]C.T.Hill and E.H.Simmons,Phys.Rept.381,235(2003)[Erratum-ibid.390,553(2004)][arXiv:hep-ph/0203079].[4]J.F.Donoghue,C.Ramirez and G.Valencia,Phys.Rev.D39,1947(1989);J.F.Donoghueand C.Ramirez,Phys.Lett.B234,361(1990).A.Dobado and M.T.Urdiales,Z.Phys.C 71,659(1996)[arXiv:hep-ph/9502255].[5]H.J.He,Y.P.Kuang and C.P.Yuan,Phys.Lett.B382,149(1996)[arXiv:hep-ph/9604309];Phys.Rev.D55,3038(1997)[arXiv:hep-ph/9611316];“Global analysis for probing electroweak symmetry breaking mechanism at high energy colliders,”arXiv:hep-ph/9704276.[6]H.J.He,“Quartic gauge boson couplings,”arXiv:hep-ph/9804210,Invited overview talkpublished in the Proceedings of Workshop on Physics at the First Muon Collider,pp.685-700,Fermilab,Batavia,USA,Nov.6-9,1997;T.Han,H.J.He and C.P.Yuan,Phys.Lett.B422,294(1998)[arXiv:hep-ph/9711429].[7]S.Alam,S.Dawson and R.Szalapski,Phys.Rev.D57,1577(1998)[arXiv:hep-ph/9706542].[8]H.J.He,Y.P.Kuang,C.P.Yuan and B.Zhang,Phys.Lett.B554,64(2003)[arXiv:hep-ph/0211229].[9]B.Zhang,Y.P.Kuang,H.J.He and C.P.Yuan,Phys.Rev.D67,114024(2003)[arXiv:hep-ph/0303048].[10]H.J.He,Y.P.Kuang,C.P.Yuan and B.Zhang,arXiv:hep-ph/0401209,Published in theProceedings of the3rd Les Houches Workshop:Physics at TeV Colliders,Les Houches,France,May26-Jun6,2003.[11]R.S.Chivukula,E.H.Simmons,H.J.He,M.Kurachi and M.Tanabashi,Phys.Rev.D72,075012(2005)[arXiv:hep-ph/0508147].[12]T.Han,Y.P.Kuang and B.Zhang,Phys.Rev.D73,055010(2006)[arXiv:hep-ph/0512193].[13]E.Boos,H.J.He,W.Kilian,A.Pukhov,C.P.Yuan and P.M.Zerwas,Phys.Rev.D57,1553(1998)[arXiv:hep-ph/9708310];Phys.Rev.D61,077901(2000)[arXiv:hep-ph/9908409].M.Beyer,W.Kilian,P.Krstonosic,K.Monig,J.Reuter,E.Schmidt and H.Schroder,Eur.Phys.J.C48,353(2006)[arXiv:hep-ph/0604048].[14]Q.Wang,Y.P.Kuang,M.Xiao and X.L.Wang,Phys.Rev.D61,054011(2000)[arXiv:hep-ph/9903201].[15]Q.Wang,Y.P.Kuang,X.L.Wang and M.Xiao,arXiv:hep-ph/9910289.[16]X.L.Wang,Z.M.Wang and Q.Wang,Commun.Theor.Phys.34,683(2000).[17]X.L.Wang and Q.Wang,Commun.Theor.Phys.34,519(2000).[18]H.Yang,Q.Wang,Y.P.Kuang and Q.Lu,Phys.Rev.D66,014019(2002)[arXiv:hep-ph/0203040].[19]Q.Wang,Y.P.Kuang,H.Yang and Q.Lu,J.Phys.G28,L55(2002)[arXiv:hep-ph/0209201].[20]Q.Wang,Int.J.Mod.Phys.A20,1627(2005).[21]Y.L.Ma and Q.Wang,Phys.Lett.B560,188(2003)[arXiv:hep-ph/0302143].[22]Z.M.Wang and Q.Wang,Commun.Theor.Phys.36,417(2001).[23]H.H.Zhang,S.Z.Jiang and Q.Wang,“Dynamical Computation on Coefficients ofElectroweak Chiral Lagrangian from One-doublet and Topcolor-assisted TechnicolorModels,”arXiv:0705.0115[hep-ph].。
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基于成本—效益分析的海平面上升背景下基础设施适应性规划策略——以美国佛罗里达为例
32 | 安全韧性城市Assessing Adaptation Planning Strategies of Interconnected Infrastructure under Sea-level Rise by Cost-benefit Analysis: A Case Study of Florida基于成本—效益分析的海平面上升背景下基础设施适应性规划策略——以美国佛罗里达为例翟 炜 黄文亮 ZHAI Wei, HUANG Wenliang基于成本—效益分析，评估海平面上升背景下佛罗里达州西北部的基础设施适应性规划策略。
通过分析基础设施之间的相互依赖性来考虑海平面上升对该区域的直接和间接影响。
从研究结果来看，虽然全面的海岸线保护能够带来更多的效益，但最有效的策略还是保护部分土地使其免受海水淹没，同时升级可能被淹没的交通网络。
基于总效益和成本效益两个标准对目标规划年份的适应性规划策略进行对比分析，结果表明：将2080年定为长期基础设施规划的目标年份将是最为经济有效的。
由于总成本会分摊到多个年份，基础设施的经济效益会随着时间的推移越来越大，但这并不表明规划目标年限越长，适应性规划策略就越经济有效。
This study assesses the adaptation planning strategies of infrastructures for the Northwest Florida region in the context of sea-level rise based on cost-benefit analysis. Specifically, this paper considers both direct and indirect impacts of sea-level rise on the region by deploying the interdependence of infrastructure. Based on the results, we recommend that the most effective strategy is partial protection of land use plus inundated transportation network upgrade, even though the total shoreline protection can make more benefits. Furthermore, we compare the adaptive planning strategies of the objective planning year from two criteria: total benefits and cost-benefit. The result indicates that the year 2080 would be the most economical if it is set as the objective year of the long-term infrastructure planning. The result also highlights that the economic benefits of infrastructure should be greater over time since the total costs are distributed over many years. It is not to say that the farther the year is, the more effective the strategy would be.海平面上升；适应性规划；成本—效益分析；规划策略sea level rise; adaptation planning; cost-benefit analysis; planning strategies文章编号 1673-8985（2022）06-0032-08 中图分类号 TU984 文献标志码 A DOI 10.11982/j.supr.20220605摘 要Abstract 关 键 词Key words 作者简介翟 炜美国德克萨斯大学圣安东尼奥分校助理教授，博士生导师，*****************黄文亮南京大学建筑与城市规划学院硕士研究生0 引言已有研究表明，如果海平面上升1 m ，全球会有500万km²的土地被淹没，将影响世界10多亿人口和1/3的耕地[1]。
阿尔法拉瓦尔 SB Carlsberg 蓄氧茬说明书
Operation:
Never dismantle the Carlsberg Flask when under pressure. Never dismantle the Carlsberg Flask when it is hot. Never operate the Carlsberg Flask without bursting disc. Never sterilise the Carlsberg Flask without bursting disc. Always read the technical data thoroughly (see chapter 6 Technical Data)
1. Declarations of Conformity ................................................................. 4 2. Safety .......................................................................................... 6
2.2 Warning signs
General warning: Caustic agents:
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All warnings in the manual are summarised on this page. Pay special attention to the instructions below to avoid serious personal injury and damage to the valve.
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1 Declarations of Conformity
best practices for radiative cooling
best practices for radiative cooling1. Choose the right materials: Opt for materials with high emissivity and low reflectivity to enhance radiative cooling. Materials such as ceramics, polymers, and certain metals can be effective in radiating heat away.2. Ensure proper insulation: Proper insulation is essential to prevent the interference of external heat sources. Insulating the radiative cooling surface can prevent heat from being conducted or convected into the system, allowing for more efficient cooling.3. Optimize surface geometry: The surface geometry plays a crucial role in radiative cooling. Designing the surface with features like nanoscale structures, microgrooves, or pyramid-like patterns can enhance radiative heat transfer by increasing the surface area and promoting better light absorption.4. Enhance thermal emissivity: Applying special coatings or pigments with high thermal emissivity on the cooling surface can boost its cooling performance. These coatings enhance the material's ability to emit thermal radiation, increasing the cooling efficiency.5. Maximize heat exchange: Design the cooling device to maximize heat exchange with the surroundings. This can be achieved by using finned heat sinks or incorporating heat pipes to enhance heat transfer from the cooling surface.6. Utilize selective radiative cooling: Selective radiative cooling involves selectively emitting thermal radiation in the atmospherictransparency window (around 8-13 μm). Incorporating selective radiators and filters can enhance the radiative cooling efficiency by allowing the surface to radiate heat effectively while minimizing absorption of solar radiation.7. Implement active cooling methods: Combining radiative cooling with active cooling methods, such as using fans or liquid cooling, can further enhance cooling efficiency. Active cooling can remove any excess heat absorbed by the cooling surface and maintain lower temperatures.8. Consider environmental factors: Factors like ambient conditions, humidity, and air quality can impact radiative cooling efficiency. Shielding the cooling surface from direct sunlight, ensuring proper ventilation, and keeping the surface clean from dust or contaminants are important considerations.9. Optimize system design: Consider the specific application and optimize the overall system design accordingly. Factors such as the size, shape, and positioning of the radiative cooling surface, as well as any additional components like heat sinks or thermal insulation, should be carefully considered to maximize cooling performance.10. Conduct regular maintenance and monitoring: Regular inspection and maintenance of the radiative cooling system can help identify any issues or performance degradation. Monitoring the system's cooling performance and making adjustments as needed will ensure its continued effectiveness.。
IntegrallyGeared...
Mitsubishi Integrally Geared compressor has found widespread application principally as air andgas booster compressors, nitrogen-gas compressor, CO 2 compressors, natural gas compressors and vacuum pumps, and have gained a superior reputation among every user.The latest Mitsubishi Integrally Geared Compressor has been designed to offer “even higher levels of performance” and “greater user-friendliness”, thus giving users greater satisfaction in operation and maintenance.Application Line upAir compressor and Nitrogen compressor are used in several applications, such as Air Separation, PTA, Ammonia and other chemical plants. Application range is as follows;- Flow volume 20,000 to 75,000Am /h with pressure up to 100barA - Flow volume 75,000 to 450,000Am /h with pressure up to 50barA - Flow volume up to 1,000,000Am /h with pressure up to 8barAIntegrally geared fuel gas compressor is fully standardized to be suitable for low/middle pressure fuel gas condition. Compressor core unit, motor driver, lube oil system, dry gas seal control unit and process gas piping are completely packaged oncommon base skid to minimize site installation work. Flow volume is up to 200,000Am 3/h with discharge pressure up to 60barA.Fields of application are CCS, EOR and CO 2 separation in IGCC, oxy-fuel, pre combustion and post combustion processes. Flow volume is up to 200,000Am 3/h with discharge pressure up to 200barA.Expander, Integrally geared radial type, is designed based on proven integrally geared compressor concept as power recovery system. Fields of application are PTA, Nitric Acid and others. Gas inlet temperature is upto 550C, discharge volume flow is up to 550,000Am 3/h.Air and N 2 compressorFuel gas compressorTypical 6-stage integrally geared compressorCarbon dioxide (CO 2) compressorExpanderTotal gear casing system (casing, gear, shaft, bearing) is designed by Mitsubishi own technology.With the impellers, gears, and pinion shafts thoroughly dynamically balanced during manufacture, compressor vibration is reduced to an extremely low level.Our highly skilled engineers and technicians are ready to perform installation, test runs, and after-sales servicing to meet user demands for quick and efficient attention.Also, we can positively respond to users requests for modification, etc. as Mitsubishi integrally geared Compressor is of our own development and manufacture.Standardized component designOn Mitsubishi standard control system for compressor, depressing merely a single push-button results in both the lubricating oil (L.O.) pump and compressor to automatically commence constant blow-pressure operation. The highly reliable control system and protection devices employed for the fully automated operation continuously monitor the entire compressor plant during operation.Also, customer’s requirement control system is available to be included in Mitsubishi standard control system.Simplicity of operation and optimum controlComplete freedom of air from L.O. contaminationQuick and excellent after-sales serviceHighly reliable continuous operation withminimum maintenanceThe Mitsubishi integrally geared compressor is built by combining proven standardized components for the required specification. And complete packing design including coolers, lubrication system and driver can facilitate shipment, site installation and start-up.High efficiency and low running costThe three-dimensional impeller design employed smoother internal fluid flow than the conventional impellers and an intercooler is provided at each stage of the multi-staged compression to deliver a very high level of efficiency.Both simple and rigid, Mitsubishi integrally geared Compressor is manufactured under strict quality control promising highly reliable and continuous long-term operation. Its simple structure also facilitates easy maintenance.The space between the gear casing where each shaft enters and the speed-increasing-gear compartment is open to the atmosphere to completely prevent L.O. c ontamination into the discharged air. Additionally, pulsation-free air can be supplied as it is forced by the rotary motion of the impellers.ExpanderCompressor1 or 2-stage compressorThis map valid for Air Compressor3 or 4-stage compressor4-stage and 2-stage combined compressor6-stage compressor with drive pinion2-stage compressor with drive pinionBull GearBull GearBull GearBull GearBull GearDriverDriverDriverDriverDriverDriverDriverDrive GearDrive GearPinion GearPinion GearPinion GearPinion GearPinion GearPinion GearPinion GearPinion GearPinion Gear Pinion Gear2C-3C 3C-4C 4C-6C 4E 6E 8E 12E14E16E8C-12C12C-14C16C100101100000100001000000Package TypeStandardized Module TypeOptimized TypeDouble Flow TypeI n l e t P r e s s u r e (b a r A )Exhaust Volume Flow (Am 3/h)No. of impellerModel Size110100D i s c h a r g e P r e s s u r e (b a r A )1000100001000001000000Suction Flow (Am 3/h)12C-16CApplication RangeHow to Indicate ModelsMitsubishi Integrally Geared Compressor Models are generally indicated by a three-digit code.Gear ArrangementApplication models of Mitsubishi Integrally Geared Compressor are shown in the following chart.12 C 6CompressorExpanderCEInlet guide vaneD i s c h a r g e p r e s s u r e (%)Flow rate (%)I n l e t g u i d e v a n e s u r g i n g l i n eInlet guide vaneSuction valve surging lineSuction valve 35%40%Flow rate (%)S h a f t i n p u t (%)ImpellerShaft sealPinion ShaftThrust bearingBull gear shaft journal bearingPinion shaft journal bearingI m p e l l e r s a r e e q u i p p e d w i t h t h r e e -dimensional-backward-curved blading which was developed through computer analysis of air flow for an ideal flow pattern, guaranteeing maximum compressor efficiency. Impellers are also high in strength and resistant to corrosion and abrasive wear.Each pinion shaft rotates at high speed enable the impellers fitted on each end to efficiently compress the gas. The shafting with the two impellers is nearly symmetrical for well-balanced aerodynamic thrust.Pinion shaft assembly (with Impeller) is removable design, for easy maintenance.An air seal is integral constructed with the oil baffle as one labyrinth seal. It is free from abrasive wear and corrosion as well as easily and securely adjustable for optimum clearance when reinstalling the seal following overhaul.The space open to the atmosphere between the air seal and oil baffle prevents the lubricating oil from entering the compressed air as well as foreign matter from entering the L.O.Gear casingA two-part casing with its upper and lower halves joined together by means of their horizontal flanges permits the gears and bearings to be readily checked without removing the scroll casing.A combined wheel-pinion-t h r u s t -c o l l a r -bearing system is employed for the thrust bearing of each pinion shaft. In this system, the thrust force of the compressor is transmitted to the thrust bearing of the low-speed wheel by the pinion shaft thrust collar, resulting in very low mechanical loss.A plain bearing is used as the Bull gear shaft journal bearing.A tilting pad type bearing is adopted as the pinion shaft journal bearing. With the tilting-pad type journal bearing, increase or decrease in shaft load is automatic through pad tilting. Consequently, the properlubricating oil film can be maintained even if a sharp change in load occurs.Inlet Guide Vane (IGV)Movable inlet guide vanes enable a wide operating range and excellent part load performance. The inlet guide vanes are installed ahead of first stage or all stages as option, and close to the impeller to achieve maximum efficiency .Speed-increasing-gearA single helical gear is employed to raise thedrive unit output shaft rotary speed to the specified impeller speed using a single-stage gear train. The gear is made of special heat-treated steel, which is then polished into a high precision, high strength gear capable of stable operation low in vibration and noise.Speed-increasing gear is designed by Mitsubishi own criteria based on research and development, minimum gear quality corresponds to AGMA standard.In standard unit, labyrinth seal is assembled in gear casing.Depend on the actual design and operating condition (Gas pressure level, Gas handled, etc.), optional seal systems are selected.Type AOptional seal systemsThe standard flow rate adjustment resorts to throttling the suction valve, for constant pressure operation. If high-efficiency operation over a wide range of flow rates without blow-off is required the movable inlet guide v a n e c o n t r o l m e t h o d i s a v a i l a b l e. T h e installation of suction guide vane ahead of first stage realizes stable operation with flow rate between 60-100% of design capacity.Adjustment of flow rateDry gas seal systemType BAbradable seal systemBooster Air Compressor Module for Air separation plantLube Oil System DesignAs lubrication system is essential for reliable operation of compressor train, MCO makes special efforts to design on our long accumulated experience with or without API 614 design specification, considering customer’s required operating condition.Each oil systems component is subjected to strict inspections and they are assembled on base plate as one console. Console tests are then performed at verify the performance of the complete system before shipping.Ergonomics DesignErgonomics design by using three dimensional simulations can realize the optimum arrangement design of overall compressor train system, for each stage as assembling, installation, operating and maintenance.Impeller flow analysisGas cooler analysisAero dynamic Design (Impeller & Gas Cooler)Always in pursuit of maximum performance and reliabilityThe optimum blade root profile is determined based on the result of three dimensional flow analysis.The inside operating condition of gas cooler (heat transfer performance, velocity, pressure loss) is confirmed by three dimensional flow analysis to ensure maximum heat exchange with lowest possible reduction in air pressure.T heco ns t r uc t e d c om p one nts a r e designed as optimized and reliable for realizing the long time safety operation by FE and dynamic analysis.Construction DesignTotal PressureGas VelocityVisualization of Flow pattern &Total pressure lossStress and Deformation analysis for Large casingStress and Vibration analysis for Steel structureThermal analysis for Hot gas expanderMitsubishi Own Design Analysis Flow Chart for Geared CompressorIncreased Reliability of RotorsDesign TargetFeatureDesign CodeApplied TechniqueRotor Dynamics Torsional Vibration AnalysisStress Optimization of Gear Intensity &Speed Optimization of Bearing Low Shaft VibrationAvoid the Resonance of Torsional VibrationDynamic Gear Stress Analysis 3-D Bearing Hydrodynamic Analysis Rotor Dynamics Unbalance Response Analysis High EfficiencyVerified the Stress Value by Test Rig3-D Stress Analysis NASTRANIncreased ReliabilityImproved Efficiency of Compressor StageReduced LossIncreased Reliability of Impellers High Efficiency 3-D Impellers Verified Performance byTest Rig and Actual Job CompressorReduce the Labyrinth Leak Applied the Abradable Seal Stress Optimization of Impeller 3-D Viscous Fluid Computer AnalysisR&D for Minimized LeakGas blower for Acrylic Acid plantNitrogen gas compressor (Combined type) for Air separation plantAir compressor (6-stage) and gas expander for P.T.A. plantBooster air compressor for Power Station IGCCFuel gas compressor for Power plantGas and Air compressor (Combined type) for Styrene plantAir compressor and gas expander (Combined type) for P.T.A. plantAir compressor and gas expander for P.T.A. plant。
Arnold Engineering - Alnico Permanent Magnets说明书
p. 19.1February 2003 Rev. CALNICO PERMANENT MAGNETSIntroductionFor over 50 years an acknowledged leader in the field of permanent magnets, Arnold Engineering continues its tradition of supe-rior quality and value with the full line of Alnico products presented in this catalog. One of the oldest members of our magnet family, Alnico remains the workhorse of the permanent magnet industry and can be relied upon to deliver impressive flux density at an economical price. Alnico also boasts the lowest temperature coefficient of any commercial magnet material (.02% per degree centigrade) allowing for excellent stability over a wide temperature range. Alnico has experienced renewed interest in recent years in such temperature-sensitive applications as hall-effect and MR-based automotive and electronic sensors, in addition to the traditional magnetron, TWT amplifiers, actuators, motors and instruments applica-tions that have long relied upon Alnico.Call Arnold Engineering today for applica-tions engineering assistance, samples or pricing.General characteristicsAlnico permanent magnets form a family of alloys whose major constituents are iron, aluminum, nickel and cobalt, though other elements (e.g., copper and titanium) may be included in minor amounts. Grades of Alnico vary in cobalt content, which ranges from zero (Alnico 3) to 40% (Alnico 8).Lower grades (2, 3 and 4) are unoriented—magnetic properties are isotropic and equal in all directions. Their magnetic output is relatively low. Anisotropic grades (Alnico 5, 6, 8 and 9) are designed to produce high magnetic output in a specified direction. Orientation is achieved during heat treatment, by cooling the product from a temperature of about 2000°F (1093°C) at a controlled rate, within a magnetic field which conforms to the preferred direction of magnetization.Alnico 5, the most widely used grade, can usually be depended upon to deliver energy to a magnetic circuit more economically than any of the other Alnico grades. Alnico 8 pos-sesses the highest coercive force of all Alnico grades, as well as excellent energy per unit volume and a low temperature coefficient. It is recommended for applications exposed to strong demagnetizing fields, or where space considerations dictate a short magnetic length.ArKomax®800 and Alnico 9 are premium grades of Alnico 5 and 8, respectively. The superior magnetic output of these grades is produced by crystal orientation from the molten state in the desired direction of magnetization.Manufacturing processesAlnico magnets can be produced in a wide variety of shapes and sizes, ranging in weight from less than one ounce (28 grams) to over 80 pounds (36 kilograms).The majority of magnets are cast in sand molds, following melting in a high-frequency induction furnace.Magnets weighing less than an ounce (28 grams) may be produced by sintering. In this process, the desired mix of metal powders is pressed to shape and size in a die, and then sintered at a temperature of approximately 2300°F (1260°C) in a hydrogen atmosphere. The sintering process is well suited to large volume production and results in parts which are structurally stronger than cast magnets. Relatively close tolerances can be achieved without grinding.Stock items and toolingA variety of shapes and sizes in selected grades of Alnico have become standard over the years. A partial list of these parts is provided later in this manual. Arnold maintains the required tooling for these parts and in many cases has quantities of them in stock, ready for immediate shipment.The use of stock items, especially on small quantity orders, can result in substantial savings to the user because the need for costly production tooling is eliminated.Sintered Alnico 2,5,6Sintered Alnico 2, 5, 6Sintered Alnico 8B, 8H Unoriented Sintered Alnico 8Sintered Alnico 8B, 8H& Unoriented Sintered Alnico 8Sintered Alnico Stock MagnetsAll Sintered Alnico stock magnets listed are furnished in Sintered Alnico 2 except those marked Alnico 5.Sintered Alnico 8Rectangles, Washers & DiscsDiscsRectanglesStandard Alnico T olerances Cast AlnicoDimension ToleranceUp to 1"Up to 25.4 mm±1/64".40 mm 1" to 3"25.4 to 76.2 mm±1/32".81 mm 3" to 5"76.2 to 127.0 mm±3/64" 1.19 mm 5" to 7"127.0 to 177.8 mm±1/16" 1.59 mm 7" to 9"177.8 to 228.6 mm±5/64" 1.98 mm Sintered AlnicoDimension ToleranceUp to .125"Up to 3.18 mm±.005".13 mm .126" to .625" 3.20 to 15.88 mm±.010".25 mm .626" to 1.250"15.90 to 31.75 mm±.015".38 mm ConcentricityConcentricity between inside and outside as cast diameters: .031" (.79 mm) F.I.R. when hole diameter is greater than hole length. 1.5 X total O.D. tolerance F.I.R. when hole diameter is less than hole length.SquarenessPerpendicularity between one ground surface and an “as cast” surface is ±1-1/2 degrees. FinishPermanent magnet materials can be finish ground when dimensional tolerances closer than the initial manufacturing tolerances are required.Magnets with surfaces marked ”f” will be furnished with these surfaces finish ground. All other surfaces will be “as cast” or “as sintered.”Ground SurfacesNormal tolerance between ground surface is ±.005" (.13 mm).Perpendicularity between two ground surfaces is ±1/2 degree.MagnetizationUnless otherwise specified, stock magnets will be furnished in a magnetized condition. The direction of magnetization is designated byM ➞. On some parts, polarity is shown as N-S. Caution: Care should be taken in handling magnetized magnets. Magnets should not be placed in repelling positions or exposed to stray ac fields as these conditions will affect their performance.Design ConsiderationsBecause Alnico magnets are coarse-grained, hard and brittle, they cannot be drilled or conventionally machined.Given these mechanical properties, Alnico parts are not used as structural members. Where a choice exists, select simple shapes. Slots are preferred over holes. Cross sections of less than .125" (3.18 mm) should be avoided. Finished surfaces, when required, may be produced by grinding; however, for many applications, the “as cast” or “as sintered” surface yields a satisfactory result at a significantly lower cost. To facilitate mounting and give adequate protection to the magnet, special assemblies such as rotors and magnetron magnets may be supplied with an aluminum jacket. Additionally, low carbon steel pole pieces may be attached to magnet pole faces by adhesive bonding or with studs or bolts. Plastisol or paint can be applied for improved appearance.MagnetizationThe most efficient use of Alnico requires that it be magnetized after the magnet has been assembled with its pole pieces into the final magnetic circuit. (Shipping the product non-magnetized also eliminates the need for special packaging, and prevents contamination by stray iron-chips, etc.)Magnetic saturation at the end-use location requires the application of a magnetizing force 4 to 5 times greater than the coercive force of the material. For Alnico 5, 3000 oersteds (240 KA/M) are recommended. For Alnico 8, the force should be at least 7000 oersteds (560 KA/M). The magnetizing force need be applied only momentarily. Thus, impulse magnetizers employing a capacitor discharge are commonly used. Direct current apparatus are also effective. Stability and temperature effectsAlnico magnets offer excellent stability with respect to temperature changes: reversible change is 0.02% per degree Centigrade. Heating may produce an irreversible loss of magnet strength. The magnitude of loss depends upon the dimensions of the magnet and its composition, but is usually less than 5%, and may be recovered by remagnetization.At temperatures exceeding 1000°F (538°C), a metallurgical change takes place which causes magnetic strength to reduce rapidly. This change will not be recovered by remagnetization.External magnetic fields can also induce partial demagnetization. In some critical applications, it is desirable to stabilize the magnet by intentionally reducing magnetic output by 5% to 10%. Such stabilization may reduce, or even eliminate, the effect of stray external fields.➞Alnico Ordering InformationPlease refer to Alnico Product Specification Sheets when ordering standard Alnico permanent magnets. The magnets listed on these specification sheets, however, represent only a small percentage of the wide variety of shapes and sizes available in the various grades of Alnico. We invite your inquiries concerning magnet designs other than those listed. These inquiries should be submitted to Arnold Engineering, Marengo, Illinois, or to the nearest Arnold sales office.It is our desire to respond to your inquiries rapidly and accurately. In order to do this, your requests should contain certain magnetic, mechanical, and general infor-mation to better serve you. Please use the Request for Quotation Form on the back of this sheet to speed the processing of your inquiry. Photocopy this form, fill it in with the necessary information, and then fax or mail to The Arnold Engineering Company.All inquiries should include:Inquiry reference number; part number; quantity of parts required; annual part usage; the address and person to whom the quotation should be submitted.If a design has already been determined, your inquiry should include:A sketch or print of the part or assembly; all dimensions and tolerances; material grade required; direction of orientation; whether the part is to be magnetized.If you require design assistance, our staff of engineers will be happy to assist you. In addition to all of the information mentioned previously, the following would be helpful in making recommendations:Operating point of the magnet; required flux density; physical stress placed on the magnet; demagnetizing forces; temper-ature range; a description of the magnetic circuit; proposed method of mounting; any application information available.In many instances, Arnold will already have adequate tooling available to make parts to your specifications. This could result in substantial savings. It is always our goal to provide the customer with the most efficient and economical part for a given application. Detailed inquiry information will help us to achieve this.Limited Warranty and Exclusive RemedyThe Arnold Engineering Company warrants that these products conform to industry standards specific herein and will be free from defects in material and workmanship. THIS WARRANTY IS EXPRESSLY GIVEN IN LIEU OF ANY AND ALL OTHER EXPRESS OR IMPLIED WARRANTIES, INCLUDING ANY IMPLIED WARRANTY OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE, AND IN LIEU OF ANY OTHER OBLIGATION ON THE PART OF THE ARNOLD ENGINEERING COMPANY. The Arnold Engineering Company will, at its option, repair or replace free of charge (excluding all shipping and handling costs) any products which have not been subject to misuse, abuse, or modification and which in its sole determination were not manufactured in compliance with the warranty given above.THE REMEDY PROVIDED FOR HEREIN SHALL BE THE EXCLUSIVE REMEDY FOR ANY BREACH OF WARRANTY OR ANY CLAIM ARISING IN ANY WAY OUT OF THE MANUFACTURE, SALE, OR USE OF THESE PRODUCTS. In no event shall The Arnold Engineering Company and its parent company, SPS Technologies, Inc., be liable for consequential, incidental or any other damages of any nature whatsoever except those specifically provided herein for any breach of warranty or any claim arising in any way out of the manufacture, sale, or use of these products. No other person is authorized by The Arnold Engineering Company to give any other warranty, written or oral, pertaining to the products.Company:Address:Phone No.:Fax No.:E-mail:Reply To:Application and General Information:p. 19.11February 2003 Rev. CInquiry No.:Reply Requested By:Fax Phone Mail E-mailArnold Part No.:Material:Magnetized:Direction of Orientation:Delivery Requested:Quantity/Estimated Annual Usage:Print or sketch — Please include description: all dimensions and tolerances; material; magnetic requirementsand material grade required; type of coating and color.For 24-hour turn-around on quotations:Request for Quotation770 Linden Ave. • Rochester, NY 14625-2764 USATel: 800-593-9127 • (+1) 585-385-9010 • Fax: (+1) 585-385-5625Email:************************©2003 Arnold Magnetic TechnologiesForm No. 613-00-125。
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a r X i v :h e p -t h /0108195v 1 27 A u g 2001Effective lagrangians for QCD at high densityRoberto CasalbuoniDipartimento di Fisica dell’Universita’di Firenze and Sezione INFN,L.go E.Fermi 2,50125Firenze,Italy.E-mail:casalbuoni@fi.infn.itAbstract.We describe low energy physics in the CFL and LOFF phases by means of effective lagrangians.In the CFL case we present also how to derive expressions for the parameters appearing in the lagrangian via weak coupling calculations taking advantage of the dimensional reduction of fermion physics around the Fermi surface.The Goldstone boson of the LOFF phase turns out to be a phonon satisfying an anisotropic dispersion relation.INTRODUCTION Ideas about color superconductivity go back to almost 25years ago [1],but only recently this phenomenon has received a lot of attention (for recent reviews see ref.[2]).The naive expectation is that at very high density,due to the asymptotic freedom,quarks would form a Fermi sphere of almost free fermions.However,Bardeen,Cooper and Schrieffer proved that the Fermi surface of free fermions is unstable in presence of an attractive,arbitrary small,interaction.Since in QCD the gluon exchange in the ¯3channel is attractive one expects the formation of a coherent state of particle/hole pairs (Cooper pairs).For a careful description of the formation of the condensates and of the approximations involved in going from asymptotic densities to finite ones it is useful to see the contribution of K.Rajagopal at this meeting [3].The phase structure of QCD at high density depends on the number of flavors and there are two very interesting cases,corresponding to two massless flavors (2SC)[1,4]and to three massless flavors (CFL)[5,6]respectively.In this talk we will be mainly concerned with the latter case.Thetwo cases correspond to very different patterns of symmetry breaking.If we denote left-and right-handed quark fields by ψαiL (R )with α=1,2,3,the SU (3)c color index,and i =1,···,N f the flavor index (N f is the number of massless flavors),in the 2SC phase we have the following structure for the condensate (C =i γ2γ0is the charge-conjugation matrix) q αiL (R )Cq βjL (R ) ∝εi j εαβ3.(1)The condensate breaks the color group SU (3)c down to the subgroup SU (2)c but it does not break any flavor symmetry.Although the baryon number,B ,is broken,there is a combination of B and of the broken color generator,T 8,which is unbroken in the 2SC phase.Therefore no massless Goldstone bosons are present in this phase.On the other hand,five gluon fields acquire mass whereas three are left massless.It is worth to notice that for the electric charge the situation is very similar to the one for the baryon number.Again a linear combination of the broken electric charge and of the broken generatorT8is unbroken in the2SC phase.The condensate(1)gives rise to a gap,∆,for quarksof color1and2,whereas the two quarks of color3remain un-gapped(massless).Theresulting effective low-energy theory has been described in[7].In this contribution wewill be mainly interested in the formulation of the effective theory for the three masslessquarks case.At high density it has been shown that the following condensate is formed[5,6]qαiL(R)CqβjL(R) ∝εi jXεαβX+κ(δiαδjβ+δiβδjα).(2) Due to the Fermi statistics,the condensate must be symmetric in color andflavor.Asa consequence the two terms appearing in eq.(2)correspond to the(¯3,¯3)and(6,6)channels of SU(3)c⊗SU(3)L(R).It turns out thatκis small[5,8,9]and therefore the condensation occurs mainly in the(¯3,¯3)channel.The expression(2)shows that theground state is left invariant by a simultaneous transformation of SU(3)c and SU(3)L(R).This is called Color Flavor Locking(CFL).The symmetry breaking pattern isSU(3)c⊗SU(3)L⊗SU(3)R⊗U(1)B⊗U(1)A↓(3)SU(3)c+L+R⊗Z2⊗Z2The U(1)A symmetry is broken at the quantum level by the anomaly,but it gets restoredat very high density since the instanton contribution is suppressed[10,8,11].TheZ2symmetries arise since the condensate is left invariant by a change of sign of theleft-and/or right-handedfields.As for the2SC case the electric charge is broken buta linear combination with the broken color generator T8annihilates the ground state.On the contrary the baryon number is broken.Therefore there are8+2broken globalsymmetries giving rise to10Goldstone bosons.The one associated to U(1)A getsmassless only at very high density.The color group is completely broken and all thegauge particles acquire mass.Also all the fermions are gapped.We will show in the following how to construct an effective lagrangian describing the Goldstone bosons,and how to compute their couplings in the high density limit where the QCD coupling gets weaker.Afinal problem we will discuss has to do with the fact that when quarks(in particular the strange quark)are massive,their chemical potentials cannot be all equal. This situation has been modeled out in[12].If the Fermi surfaces of differentflavors are too far apart,BCS pairing does not occur.However it might be favorable for different quarks to pair each of one lying at its own Fermi surface and originating a pair of non-zero total momentum.This is the LOFF statefirst studied by the authors of ref.[13]in the context of electron superconductivity in the presence of magnetic impurities.Since the Cooper pair has non-zero momentum the condensate breaks space symmetries and we will show that in the low-energy spectrum a massless particle,a phonon,the Goldstone boson of the broken translational symmetry,is present.We will construct the effective lagrangian also for this case.EFFECTIVE THEORY FOR THE CFL PHASEWe start introducing the Goldstonefields as the phases of the condensates in the(¯3,¯3) channel[14,15]X iα≈εi jkεαβγ q jβL q kγL ∗,Y iα≈εi jkεαβγ q jβR q kγR ∗.(4) Since quarks belong to the representation(3,3)of SU(3)c⊗SU(3)L(R)and transform under U(1)B⊗U(1)A according toq L→e i(α+β)q L,q R→e i(α−β)q R,e iα∈U(1)B,e iβ∈U(1)A,(5) the transformation properties of thefields X and Y under the total symmetry group G=SU(3)c⊗SU(3)L⊗SU(3)R⊗U(1)B⊗U(1)A are(g c∈SU(3)c,g L(R)∈SU(3)L(R)) X→g c Xg T L e−2i(α+β),Y→g c Y g T R e−2i(α−β).(6) Thefields X and Y are U(3)matrices and as such they describe9+9=18fields. Eight of thesefields are eaten up by the gauge bosons,producing eight massive gauge particles.Therefore we get the right number of Goldstone bosons,10=18−10.These fields correspond to the breaking of the global symmetries in G(18generators)to the symmetry group of the ground state H=SU(3)c+L+R⊗Z2⊗Z2(8generators).For the following it is convenient to separate the U(1)factors in X and Y definingfields belonging to SU(3)X=ˆXe2i(φ+θ),Y=ˆY e2i(φ−θ),ˆX,ˆY∈SU(3).(7) Thefieldsφandθcan also be described through the determinants of X and Yd X=det(X)=e6i(φ+θ),d Y=det(Y)=e6i(φ−θ),(8) The transformation properties under G areˆX→gc ˆXg TL,ˆY→g cˆY g T R,φ→φ−α,θ→θ−β.(9)The breaking of the global symmetry can be discussed in terms of gauge invariantfields given by d X,d Y andΣi j=∑α(ˆY jα)∗ˆX iα→Σ=ˆY†ˆX.(10)TheΣfield describes the8Goldstone bosons corresponding to the breaking of the chiral symmetry SU(3)L⊗SU(3)R,as it is made clear by the transformation properties ofΣT,ΣT→g LΣT g†R.That isΣT transforms exactly as the usual chiralfield.The other two fields d X and d Y provide the remaining two Goldstone bosons related to the breaking of the U(1)factors.In order to build up an invariant lagrangian,it is convenient to define the following currentsJµX=ˆXDµˆX†=ˆX(∂µˆX†+ˆX†gµ),JµY=ˆY DµˆY†=ˆY(∂µˆY†+ˆY†gµ),(11)with gµ=ig s g aµT a/2the gluonfield and T a=λa/2the SU(3)c generators.These currents have simple transformation properties under the full symmetry group G,JµX,Y→g c JµX,Y g†c.The most general lagrangian,up to two derivative terms,invariant under G,the rotation group O(3)(Lorentz invariance is broken by the chemical potential term)and the parity transformation defined asˆX↔ˆY,φ→φ,θ→−θ,is[14]L=−F2T4Tr J0X+J0Y)2 +12(∂0θ)2(12)+F2S4Tr J X+ J Y)2 −v2φ2| ∇θ|2.Using SU(3)c color gauge invariance we can chooseˆX=ˆY†,making8of the Goldstone bosons disappear and giving mass to the gluons.The properly normalized Goldstone bosons,Πa,are given in this gauge byˆX=ˆY†=e iΠa T a/F T,(13) and expanding eq.(12)at the lowest order in thefields we getL≈12(∂0φ)2+12|∇Πa|2−v2φ2|∇θ|2,(14)with v=F s/F T.The gluons g a0and g a i acquire Debye and Meissner masses given bym2D=αT g2s F2T,m2M=αS v2g2s F2T.(15) It should be stressed that these are not the true rest masses of the gluons,since there is a large wave function renormalization effect making the gluon masses of order of the gap∆,rather thanµ(see later)[16].Since this description is supposed to be valid at low energies(we expect much below the gap∆),we could also decouple the gluons solving their classical equations of motion neglecting the kinetic term.The result from eq.(12) isgµ=−14 Tr[˙Σ˙Σ†]−v2Tr[ ∇Σ· ∇Σ†] +12 ˙θ2−v2φ| ∇θ|2 .(17) Notice that thefirst term is nothing but the chiral lagrangian except for the breaking of the Lorentz invariance.This is a way of seeing the quark-hadron continuity,that is the continuity between the CFL and the nuclear matter phases in threeflavor QCD.The identification is perfect if one realizes that in nuclear matter the pairing may occur in such a way to give rise to a superfluid due to the breaking of the baryon number as it happens in the CFL phase[17].FERMIONS NEAR THE FERMI SURFACEWe will introduce now the formalism described in ref.[18]in order to evaluate theparameters appearing in the effective lagrangian.This formulation is based on theobservation that,at very high-density,the energy spectrum of a massless fermion isdescribed by states|± with energies E±=−µ±| p whereµis the quark number chemical potential.For energies much lower than the Fermi energyµ,only the states|+ close to the Fermi surface.i.e.with| p|≈µ,can be excited.On the contrary,the states|− have E−≈−2µand therefore decouple.This can be seen more formally by writing the four-momentum of the fermion aspµ=µvµ+ℓµ,(18)where vµ=(0, v F),and v F is the Fermi velocity defined as v F=∂E/∂ p| p= pF .Formassless fermions| v F|=1.Since the hamiltonian for a massless Dirac fermion in a chemical potentialµisH=−µ+ α· p, α=γ0 γ,(19) one hasH=−µ(1− α· v F)+ α· ℓ.(20) Then,it is convenient to introduce the projection operatorsP±=1± α· v F2 ψ(x)= |ℓ|<µd4ℓ2γν 2gµν−Vµ˜Vν−˜VµVν .(26)We notice that thefields appearing in this expression are evaluated at the same Fermi velocity because off-diagonal terms are canceled by the rapid oscillations of the exponential factor in theµ→∞limit.This behavior can be referred to as the Fermi velocity super-selection rule.At the leading order in1/µone hasiiV·Dψ+=0,ψ−=−ψ†+(D/⊥)2ψ+ .(28)2µThe previous remarks apply to any theory describing massless fermions at high density.The next step will be to couple this theory in a SU(3)L⊗SU(3)R⊗SU(3)c invariant way to Nambu-Goldstone bosons(NGB)describing the appropriate breaking for the CFL phase(we will not discuss here the determination of the parameters relevant for the U(1)A,Bfields,see[11]).Using a gradient expansion we get an explicit expression for the decay coupling constant of the Nambu-Goldstone boson as well for their velocity. The invariant coupling between fermions and Goldstonefields reproducing the sym-metry breaking pattern of eq.(2)is proportional toγ1Tr[ψT LˆX†]C Tr[ψLˆX†]+γ2Tr[ψT L CˆX†ψLˆX†]+h.c.,(29) with an analogous expression for the right-handedfields.Here the spinors are meant to be Dirac spinors.The trace is operating over the group indices of the spinors and of the Goldstonefields.Since the vacuum expectation value of the Goldstonefields is ˆX = ˆY =1,we see that this coupling induces the right breaking of the symmetry. In the following we will consider only the caseγ2=−γ1∝∆/2,where∆is the gap parameter.Since the transformation properties under the symmetry group of thefields atfixed Fermi velocity do not differ from those of the quarkfields,for both left-handed and right-handedfields we get the effective lagrangian density1L=∑v F√Here λa (a =1,...,8)are the Gell-Mann matrices normalized as follows:Tr (λa λb )=2δab and λ9=8π1V ·p ˜V ·p −∆2A ˜V ·p −∆A −∆A V ·p .(34)We are now in position to evaluate the self-energy of the Goldstone bosons through their coupling to the fermions at the Fermi surface.There are two one-loop contributions[16],one from the coupling Πχχand a tadpole from the coupling ΠΠχχ(see eq.(30).The tadpole diagram contributes only to the mass term and it is essential to cancel the external momentum independent term arising from the other diagram.Therefore,as expected,the mass of the NGB’s is zero.The contribution at the second order in the momentum expansion is given byi 21−8ln24π8∑a =1Πa V ·p ˜V ·p Πa .(35)Integrating over the velocities and going back to the coordinate space we getL kin eff =21−8ln23| ∇Πa |2 .(36)We can now determine the decay coupling constant F T through the requirement of getting the canonical normalization for the kinetic term;this impliesF 2T =µ2(21−8ln2)a result obtained by many authors using different methods(for a complete list of the relevant papers see thefirst reference of[2]).We see also that v2=1/3.The other constants appearing in our effective lagrangian can be obtained via a direct calculation of m D and m M[16].This is done evaluating the one-loop contribution to the gluon self-energy.Also in this case there are two contributions,one coming from the gauge coupling to the fermions,whereas the other arises from the next-to leading(inµ)sea-gull contribution to the fermion effective lagrangian in eq.(28).The results wefind are [16,11]1m2D=g2s F2T,m2M=3is a direct consequence of the integration over the Fermi velocity.Therefore it is completely general and applies to all the NGB’s in the theory, including the ones associated to the breaking of U(1)V and U(1)A(v2φ=v2θ=1/3); needless to say that higher order terms in the expansion1/µcould change this result. The breaking of the Lorentz invariance exhibited by the pion velocity different from one,can be seen also in the matrix element 0|J aµ|Πb .Its evaluation gives[16]0|J aµ|Πb =iF Tδab˜pµ,˜pµ=(p0, p/3).(40) The current is conserved,as a consequence of the dispersion relation satisfied by NGB’s.THE LOFF PHASEWe shall now consider massless quarks of three colors and two differentflavors.Atfinite densities we introduce two chemical potentials,µ1andµ2,for the two species in order to mimic the different mass case.We writeµ1−µ2=δµ≪µ=µ1+µ2The condensation in the LOFF phase gives rise to two breaking terms in the fermion lagrangian characterized by two gap parameters∆(s)and∆(v)−12 ˙Φ2−∞∑n=1c n(| ∇Φ|2)n .(44)HereΦmust be thought as a function of the phononfieldφ.Using| ∇Φ(x)|2=4q2+4q f2| ∇φ(x)|2,(45)at the lowest order in the derivatives of the phononfieldφwe get(neglecting a constant term):L(φ,∂µφ)=1。