Higher derivative corrections to near-extremal black holes in type IIB supergravity

Anomalous Differential Resistance Change at the Oscillation Threshold in Quantum-Well Laser Diodes Peter G.Eliseev,Senior Member,IEEE,Pawel Adamiec,Artem Bercha,Filip Dybała,Roland Bohdan,andWitold A.TrzeciakowskiAbstract—Anomalous behavior is investigated of differential re-sistance of laser diodes at the lasing threshold.The regularcase is the known decrease ofthe associated with a dynamicsaturation of the junction voltage.In contrast to this,the anoma-lous case is an increase of the differential resistance(the“positivekink”of).Some regular samples are found to show theanomaly at lower temperature or at high hydrostatic pressure.Theanomaly is discussed in terms of the injection-induced conductivity.Index Terms—Differential resistance,pressure and temperatureeffects,semiconductor lasers.I.I NTRODUCTIONE LECTRICAL diagnostics of laser diode operation in-cludes determination of the oscillation threshold by kinkof the differential resistance of diode measured as a functionof the pumping current.Corresponding experimental demon-stration and interpretation had been done in early studies ofhomojunction lasers[1],[2]and later in studies of double-het-erostructure lasers[3]–[5],and of quantum-well lasers[6].Sometimes,power kinks and spatial mode switching can be de-tected electrically by observation of variation of the differentialresistance.The junction voltage in the laser diode contains the contri-bution of the quasi-Fermi-levelseparation that shows atendency to saturation above the threshold because of satura-tion of the modal gain under stationary oscillation.When thevoltage is saturated but current continues to increase,it is equiv-alent to the disappearance of the differential resistance.Thisleads to the regular behaviorof in laser diodes(isthe total voltage applied to the diode),according to observa-tions in[1]–[6].Namely this regular behavior is a decrease ofthe above the threshold,sometimes down to the residualManuscript received June7,2004;revised August11,2004.This workwas supported in part by the NATO Science for Peace Program under GrantSfP972443.P.G.Eliseev is with the Center for High Technology Materials,University ofNew Mexico,Albuquerque,NM87106USA(e-mail:eliseev@).P.Adamiec and F.Dybala are with the High Pressure Research Center,PolishAcademy of Sciences,01-142Warsaw,Poland and also with the Institute ofPhysics,Warsaw University of Technology,00-662Warsaw,Poland(e-mail:pa@unipress.waw.pl;fd@unipress.waw.pl).A.Bercha is with the High Pressure Research Center,Polish Academy of Sci-ences,01-142Warsaw,Poland and also with the Uzhgorod National University,88000Uzhgorod,Ukraine(e-mail:artem@unipress.waw.pl).R.Bohdan is with the High Pressure Research Center,Polish Academyof Sciences,Sokolowska29/37,01-142Warsaw,Poland(e-mail:roland@unipress.waw.pl).W.A.Trzeciakowski is with the High Pressure Research Center,PolishAcademy of Sciences,01-142Warsaw,Poland(e-mail:wt@unipress.waw.pl).Digital Object Identifier10.1109/JQE.2004.839237series resistance of the electrodes and bulksemiconductor.However,in addition to this regular behavior,there were obser-vations of anomalous sign of the changeof above thethreshold,particularly in the same device but under differenttemperature(see,for example,[5]and[7]where an explana-tion of this anomaly was also reported).In[8],the analysis ofvoltage across the laser diode has been given for the case of elec-trical response to the external optical feedback.The componentassociated with a leakage into claddings is found to have an op-posite sign to the junction voltage component.In[5]and[7],ithas been shown that the resistance change due to injection-in-duced conductivity can be responsible for anomaly of thresholdkink of the differential resistance.In this work we demonstrate the differential electrical charac-teristics versuscurrent in several types of laser diodesin a range of temperature and of pressure.Anomalous cases arefound.Pressure-related anomaly is reported for thefirst time.We present here also a model explaining qualitatively such pres-sure-and temperature-related anomalies.II.E XPERIMENTALWe studied electrical and optical characteristics of differenttypes of lasers but here we shall present only three examplesof GaAs–AlGaAs quantum-well(QW)lasers that revealedanomalous as a function of temperature and pres-sure.Samples I and II were single-mode SQW diode laserswith emission wavelengths at850and790nm,respectively.Both structures were grown by metal–organic chemical vapordeposition(MOCVD)on an n-type100GaAs:Si substrate.For an850-nm diode laser the active layer contained one9-nm-thick QW made of GaAs.The waveguide layers weremade of100-nm-thickAlGa As.The active layer andthe waveguides were nominally undoped.The thicknesses of n(Si)-and p(Zn)-AlGa As cladding layers were1.5m.Both the p-and n-cladding layers were doped toaboutcm.For the790-nm diode,the active layer was made of9-nm-thickAlGa As QW surrounded by110-nm-thickwaveguides with graded Al concentration varying from45%to63%.The active region was sandwiched between n(Si)-andp(Zn)-AlGa As cladding layers with the dopinglevelcm.In both samples,a300-nm-thick p-contactlayer was made of GaAs:Zn with doping level higherthancm.The threshold currents at ambient conditionswere22mA for Sample I and30mA for sample II.The thirdsample was a commercial single-mode laser made by Sanyo(DL-7140-201)emitting at785nm.The threshold current atambient conditions was about30mA.0018-9197/$20.00©2005IEEEFig.1.Plots of IdU=dI versus I at different temperatures for Sample I.The arrows show the values of threshold currents I .In our pressure/temperature experiments we used a piston-cylinder cell made of maraging steel and operating up to 2GPa.It was described in more detail elsewhere [9].A pure gasoline was used as a pressure medium.The pressure was calibrated with the resistance of InSb sensor that gives about 0.1kbar sen-sitivity in the 20kbar range (the absolute pressure accuracy de-pends on the proper calibration of the sensor and is probably around 1kbar).The light emitted by the laser comes out of the cell through the multimode optical fiber (with 50or 100mi-crometer core)[10].For the temperature control we passed cool nitrogen gas through the copper tube wound around the cell;this method allowed to cool it down to about 120K.The tempera-ture was measured by a thermocouple placed inside the pressure cell and its stability was better than 1K.The cell with cooling elements was placed under a small hydraulic press so that the pressure could be varied at low temperature.In the pressure/tem-perature range where our liquid solidi fies it is necessary to vary the temperature at the fixed position of the piston.In such case the pressure changes when we cool down the cell.Even when the liquid solidi fies the pressure remains hydrostatic (to a good approximation).For each pressure we measured the–and the power-cur-rent characteristics together with the spectra at different cur-rents under cw operation.Thederivative was obtained by adding the ac component to the current and detecting the voltage modulation using the lock-in ampli fier.III.R ESULTSIn Fig.1we showthe as a function of current for Sample I.The kinks at threshold can be clearly seen.At lower temperatures,they change sign.The heights of the kinks (mea-sured vertically between the extrapolated curvesfor )are shown in Fig.2.We also show the kink heights for Sample II where a similar anomalous behavior can be observed.In Fig.3the pressure evolution ofthe dependencies for Sample III is shown.Above 6kbar the thresholds (denoted by arrows in Fig.3)increased substantially for this laser due toincreasedFig.2.Heights of the kinks in IdU=dI versus temperature for two lasers (Samples I and II).In both cases the kink changessign.Fig.3.IdU=dI at room temperature (293K)and at different pressures for Sample III.The arrows show the values of threshold currents at each pressure.leakage tothe minima in the claddings.The anomalous be-havior of the kink height at elevated pressure can be seen in Fig.4,where we display both the kink height (left scale)and the threshold current (right scale).However,for some lasers the effects of pressure were much weaker,as can be seen by the curve for Sample I shown in Fig.4.In this case,the effect of pressure on threshold current was also much weaker.IV .D ISCUSSIONA.Origins of Anomalous VoltageThere are relatively small corrections to the regular–curve of diode associated with voltage drop at series structure components sensitive to the laser state of the active region,more speci fically,sensitive to the carrier density in the active layers.One is based on radiation transport inside the diode chip.Another correction is associated with injection-induced conductivity (IIC)provided by drift-diffusion transport in layers adjacent to the junction.An ideal–curve of the p-n junction can be represented bya simpleexpression,where is junc-tionvoltage,is the so-called saturation current (not related toELISEEV et al.:ANOMALOUS DIFFERENTIAL RESISTANCE CHANGE AT THE OSCILLATION THRESHOLD IN QUANTUM-WELL LASER DIODES11Fig.4.Heights of the kinks in IdU=dI (left scale,full symbols)and threshold currents (right scale,empty symbols)versus pressure for (a)Sample III and (b)Sample I.the laser-induced saturation,but related to the saturation under reversebias),is the factor that,in the nondegenerate caseisor (Shockley [11]and Hall [12]cases)orin more general case(is nonideality factor,that increases to3–4in the case of degeneracy [13])orin the case of tun-neling current(is energy parameter of tunneling mechanism [14]).As the operationcurrent,at forward bias is much largerthan ,we can use a following relationship for the junction–curve:(1)Total voltage of the diode in general can be presentedas(2)where )is junction voltage;small non-linearcontribution is related to thephoto-induced processes in the diodechipand to theIIC.If there are no small correction termsof ,the regular effect at the threshold is the decrease of the differen-tial resistance byquantity(3)That is exactly the junction differential resistance just belowthe threshold.If the measured valueis ,the threshold-related step of this valueis(4)The photo-induced term in (2)is included to account for phe-nomena of photoresponse of different parts of the diode chip to own radiation that is generated in the active region (sponta-neous or both spontaneous and stimulated).Usually the spon-taneous emission is con fined by the semiconductor chip and mainly lost by internal absorption.The photoelectric absorp-tion can produce photo-EMP at carrier-separating barriers and photoconductivity in absorbing layers.Scattered laser emission is subjected also mainly to the internal absorption.In an opti-mized laser structure,these photo-effects are suppressed:poten-tial barriers other than working p-n junction are not desirable in the chip.It could be observed in early-generation laser diodes,but in modern commercial diodes,the resistance of the diode is minimized which means there are no uncontrollable barriers.As to the photoconductivity,it can be,for example,produced in narrow-bandgap cap layer or sometimes in the substrate if the bandgap there is smaller than the radiation photon energy.We believe that this effect is small as the dark conductivity of narrow-bandgap layers is quite high,as a rule.For example,the cap layer is doped usually to10cm or higher.Thus we do not expect any substantial photoconductivity effect.Another important origin of the voltage correction is the injection-induced yers adjacent to p-n junc-tion are subjected to carrier injection,namely,the optical con finement (waveguide)layers,barriers between QWs if any,and those parts of the cladding layers where the leakage of excess carrier occurs from the waveguide region.These layers are transparent to the emission of QWs,therefore,the photoconductivity seems to be not in effect.On the other hand,waveguides and barriers are usually not doped,so that their initial resistance can be substantial.Characteristic thickness of involved layersis 10nm or less for QWs and for barriers between them,100–200nm for each waveguide layer,1–2m for each cladding layer.Thus,most important is the resistance of waveguide layers and,if the carrier leakage occurs,the resistance of claddings may also come into play.There is also the possibility of unintentionally formed depletion layers in the laser heterostructure [15].We call again the layer underconsideration as a sensitive layer.A voltage drop on itis,where is thecurrent,is the resistance of the layer.Actual drift transport through the layer is in fluenced by injection of excess carriers.This voltage drop decreases along with thecurrent riseif decreases more steeplythan.The decreaseof is stopped by pinning of quasi-Fermi levels in the active region above the laser threshold.This contributes a positive change of differential resistance.B.Phenomenological Approach to Injection-Induced ConductivityAssuming that the leakage is negligible at the threshold,we can estimate the contribution of the IIC in the waveguiding re-gions of the laser structure.Assume also that the conductivity of the waveguide regions is controlled by injected carriers with adensityof.It is much smaller than the carrier density in QW because carriers are rapidly captured from the waveguide regions to QWs.The total resistance of the waveguide regionis(5)12IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.41,NO.1,JANUARY 2005where is total thickness of the consideredregion,is elec-tronmobility,is ratio of hole mobility to electron mobility,and is the area of the active region.The IIC voltage dropisand the corresponding contribution to the differ-ential resistanceis(6)Below the threshold,this expression gives for the partial IIC differential resistance a negative value if the dependenceofon is steeper than linear.This is just the case.Remember,that is the carrier density in the waveguiding region,therefore it is much less than the density in QWs,and it is maintained by pumping and by thermal activation from QWs.For simplicity,assumethatis power function of ,namely (7)where is a coef ficientandis a power exponent.In thiscase(8)therefore,this value is negativeif .Consider now what does occur at the laser threshold.Assumethat is strictly saturated above threshold as well as the carrier density in the QWs.In this case,we have at the threshold but belowit(9)where subscript th denotes threshold values of variables.Above the threshold the junction differential resistance andderivativeare assumed both equal to zero,and we have at thethreshold but aboveit(10)The threshold related step of the differential resistanceis(11)This expression contains two terms of opposite signs there-fore,depending on parameters the result can change sign.With assumption of (7)weobtain(12)A criterion of anomalous (positive)sign of the differential resistance stepis(13)andif ,there will be no change of the differ-ential resistance at the threshold.It is also seen from (13)that the anomaly is much more probable atlarger ,but,in prin-ciple,small does not exclude the anomaly.In Fig.5curvesofare shown calculated for a similar model of the laser withdifferentas indicated.Other numerical parametersare meV,Fig.5.Calculated value of IdU=dI versus current I for a simpli fied model of laser diode with different values of resistivity R of sensitive layer at the laser threshold fixed at 20mA.An inversion of the sign of the threshold-related kink isillustrated.mA.We had put an equilibrium carrierdensityin the sensitive layer in order to avoid large resistance of the layer at small current.Note that we have assumed a constantvalueofand we onlyvaried (in case of temperature and pressure variationsbothand are affected).The curveforis a regular case with a maximum negative step at the threshold.With increaseofthe step decreases and then changes sign.In frames of this model,the sign inversion occursatsmallerif the powerexponent is larger.The simple power function is taken here in order to obtain analytic results for illustrative purposes.The detailed modeling of the carrier distribution across the separate-con finement heterostructure is rather complicated and is a subject of a number of papers.There are no analytic expressionsfor.From this modeling,it can be seen that occurrence of theanomalous sign is more probable if the thresholdvalueis larger.On the other hand,to exclude this anomaly one has to provide lower resistance of sensitive layers in the structure.Esti-mation by (5)gives the waveguide layer resistance of1at fol-lowingparameters:cm,cmcm/Vs,cm .The carrier density in waveguide layer can be in the range 10–10cm as it is typically 2–3orders lower than that in active region.Effect of cooling is some freeze-out of the conductivity in undoped and low-doped materials,particularly,in waveguide regions.The anomalous kink is observed at lower temperature whereas at room temperature the same diode behaves in a reg-ular manner.This can be explained by increase of the initial re-sistance of the waveguide region at cooling due to freeze-out of conductivity,according to approach reported earlier in [6],[8].Hardly similar effect in the claddings can in fluence the behavior:the thermal leakage of carriers is reduced at lower temperature.Therefore,the region responsible for the anomaly is the undopedwaveguide.Note that the reductionofat lower temperatures should reduce the anomalous termin.Hydrostatic pressure leads to the blue shift of laser emission.In GaAs –AlGaAs QW lasers the shift is 9.8meV/kbar in the wavelength range around 780nm [16].There is also an increase of the threshold current (see Fig.4)that is explained tentativelyELISEEV et al.:ANOMALOUS DIFFERENTIAL RESISTANCE CHANGE AT THE OSCILLATION THRESHOLD IN QUANTUM-WELL LASER DIODES 13by increase of the carrier leakage from active QWs into wave-guide layers and into cladding layers.The leakage is sensitive to the energy barrier.It is known that under pressure,the energy barrier between direct conduction band -minimum in QW andindirect -minima in barrier layers decreases with arate meV/kbar in GaAs –AlGaAs lasersand meV/kbar in Al-GaInP-based lasers [16].At the Al content exceeding 0.43in Al-GaAs the lowest minimum in the conduction band occurs atthe point.Therefore under pressure the QW becomes effectively more and more shallow and alsothe -barriers in the claddings approach the Fermi level in the well.This leads to an increase of electron leakageinto -minima and thus to the increasein .Therefore there is one factor in the (13),namelyincreasing enhancing the anomalous behavior.Due to increased leakage at high pressure we cannot exclude the contribution of claddings to sensitive layers.Important electrical effect of high hydrostatic pressure is also associated with modi fication of the conductivity of undoped waveguide.Deep levels of defects and residual im-purities can move with pressure with respect to the conduc-tion-band edge,both in direct and indirect AlGaAs.In n-type claddings the deep centers due to Si donors (so-called DX cen-ters)form up to four states that can be metastable at low temper-atures.The effect of pressureonin (12)can therefore be complex,entering not onlythrough but alsothroughand .V .C ONCLUSIONTemperature and hydrostatic pressure effects were investi-gated in AlGaAs laser diodes operating in wavelength ranges between 780and 850nm.Spectral tuning and change of threshold current were measured.Electrical characterization by differential–curves shows that there are cases of anomalous behavior that is a positive step of the differential resistance at the threshold instead of regular negative step.These anomalous kinks of differential–curves are identi fied in two typical cases:in some laser diodes the regular behavior at normal cir-cumstances converts into anomalous one under high hydrostatic pressure or under lowered temperature.We suggest that the voltage drop on the sensitive layer in the laser heterostructure produces a small correction to the diode–curve,so the latter includes additional nonlinear electrical component besides the p-n junction.The differential resistance of the sensitive layer contributes to the threshold-related kink with a sign opposite to the regular effect.Therefore,algebraic summation provides sometime regular (negative)cumulative result,but sometime this summation gives an anomalous (positive result).Occa-sional variations of parameters of the sensitive layers in diodes fabricated by different producers give rise to the observation of the anomaly in some samples.High pressure and low tem-perature are factors favorable for the anomaly as they increase the initial resistance of the sensitive layer (and the threshold currents in case of pressure).The sensitive layers are most probably the undoped optical-con finement (waveguide)layers of the separate-con finement heterostructure.The conductivity of this sensitive layer is modi fied by the injected carriers (effect of the injection-induced conductivity).A similar effect can be expected from photoconductivity effect in other sensitive layers.But in well-designed laser diodes there are no candi-dates for photo-sensitive layers,as other components of the heterostructure are either transparent to the emission of active region or low-resistance ones to give no rise of competitive voltage contribution.The voltage correction caused by IIC produces variations of measured differential resistance step associated with details of the laser structure (thickness of sensitive layer,composition and doping level,carrier depletion,etc.).We demonstrated here that low temperature and high pressure are both favorable for anoma-lous behavior.These two factors,temperature and pressure,pro-duce opposite effects on the threshold current in AlGaAs lasers:lowering temperature gives lower threshold,but higher pressure produces higher threshold.But in both cases we noticed the ap-pearance of an electrical anomaly that we associate with injec-tion-induced conductivity.R EFERENCES[1]P.G.Eliseev,I.Ismailov,A.I.Krasilnikov,M.A.Manko,I.Z.Pinsker,and V .P.Strakhov,“About the threshold phenomena in the injection lasers,”in Proc.IX Int.Conf.on the Physics of Semicond.,vol.1,1968,pp.519–523.[2]P.G.Eliseev,A.I.Krasilnikov,M.A.Manko,and V .P.Strakhov,“Studyof injection lasers at direct current,”in Physics of p -n Junctions and Semiconductor Devices ,S.M.Ryvkin and Y .V .Shmartsev,Eds.New York:Plenum,1971,pp.150–155.[3]T.L.Paoli and P.A.Barnes,“Saturation of the junction voltage in stripe-geometry (AlGa)As double-heterostructure junction lasers,”Appl.Phys.Lett ,vol.28,pp.714–715,1976.[4]W.B.Joyce and R.W.Dixon,“Fundamental and harmonic responsevoltage of a sinusoidally current-modulated ideal semiconductor laser,”J.Appl.Phys.,vol.47,pp.3510–3513,1976.[5]L.Vu Van,P.G.Eliseev,M.A.Manko,G.T.Mikaelian,and O.G.Okhotnikov,“V oltage saturation at the injecting contact in laser diode and phenomenon of negative photovoltage,”in Proc.Lebedev Phys.Inst.,vol.166,N.G.Basov,mack,NY ,1986,pp.236–276.[6]P.G.Eliseev,J.Maege,G.Erbert,and G.Beister,“Threshold drop of thedifferential resistance of stripe quantum-well InGaAs/GaAlAs lasers,”Quantum Electron.,vol.25,no.2,pp.99–101,1995.[7]P.G.Eliseev,O.G.Okhotnikov,and G.T.Pak,“The in fluence of carrierleakage from the active layer on the differential resistance of a laser diode,”in Kratk.Soobsch.po Fizike ,Moscow,Russia:FIAN,1984,vol.3,pp.21–25.(in Russian).[8]Y .Mitsuhashi,J.Shimada,and S.Mitsutsuka,“V oltage change acrossthe self-coupled semiconductor laser,”IEEE J.Quantum Electron.,vol.17,no.7,pp.1216–1225,Jul.1981.[9] F.Dybala,P.Adamiec,A.Bercha,R.Bohdan,and W.Trzeciakowski,“Wavelength tuning of laser diodes using hydrostatic pressure,”in Proc.SPIE Optical Devices for Fiber Communication IV ,M.J.Digonnet,Ed.,2003,vol.4989,pp.181–189.[10]R.Bohdan,A.Bercha,P.Adamiec,F.Dybala,and W.Trzeciakowski,“Afiber feed through for a semiconductor laser located in high hydrostatic pressure cell,”Pribory i Technika Eksperimenta ,vol.3,pp.1–3,2004.(in Russian).[11]W.Shockley,“The theory of p-n junctions in semiconductors and p -njunction transistors,”Bell Syst.Tech.J ,vol.28,no.3,pp.435–489,1949.[12]R.N.Hall,“Power recti fiers and transistors,”Proc.IRE ,vol.40,no.11,pp.1512–1518,Nov.1952.[13]K.A.Shore and M.J.Adams,“The effect of carrier degeneracy on trans-port properties of the double heterostructure injection laser,”Appl.Phys.,vol.9,no.2,pp.161–164,1976.[14] D.J.Dumin and G.L.Pearson,“Properties of gallium arsenide diodesbetween 4.2and 300K,”J.Appl.Phys.,vol.36,no.11,pp.3418–3426,Nov.1965.[15]R.F.Kazarinov and M.R.Pinto,“Carrier transport in laser heterostruc-tures,”IEEE J.Quantum.Electron.,vol.30,no.1,pp.49–53,Jan.1994.[16] A.Bercha,P.Adamiec,F.Dybala,R.Bohdan,and W.Trzeciakowski,“Effect of pressure and temperature on AlGaInP and AlGaAs laser diodes,”in Proc.SPIE,Physics and Simulation of Optoelectronic Devices XI ,vol.4986,M.Osinski,H.Amano,and P.Blood,Eds.,2003,pp.613–620.14IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.41,NO.1,JANUARY2005Peter G.Eliseev(M’87–SM’89)was born in St.Petersburg,U.S.S.R.,in1936. He received the Diploma degree in physics from Moscow State University (MSU),Moscow,Russia,in1959,and the Candidtae and Doctor Sci.degrees from the P.N.Lebedev Physics Institute(FIAN),Russian Academy of Sciences, Moscow,in1965and1974,respectively.He was with the Physics Department of MSU from1959to1963and since 1963he has been with FIAN.As an Invited Research Professor,he was with Research Center for Advanced Science and Technology,University of Tokyo (1991),the Ferdinand-Braun Institute,Berlin,Germany(1993-1994),Univer-sity of Tokushima,Tokushima,Japan(1998-1999),and University of Nagoya, Nagoya,Japan(1999-2000).Since1995,he has been the Research Professor at the Center for High Technology Materials,University of New Mexico,Albu-querque,on leave from FIAN.The mainfield of his activity is physics and tech-nology of semiconductor lasers.He is the author or coauthor of several books and more than500scientific papers.Dr.Eliseev was awarded the State Prize of the U.S.S.R.in science and tech-nology(1984)for pioneering development of quaternary heterostructure mate-rials,and is a recipient of the N.Holonyak OSA Award(2004).Since1992, he has been a Correspondent Member of the Russian Academy of Natural Sci-ences.He is a member of OSA.Pawel Adamiec was born in Nowa Deba,Poland,in1976.He received the M.Sc.degree in physics from Rzeszow University,Rzeszow,Poland,in2000. He is currently working toward the Ph.D.degree at Warsaw University of Tech-nology,Warsaw,Poland.He performs experiments at the High Pressure Research Center of Polish Academy of Sciences,Warsaw,Poland.He investigates the optical and elec-trical properties of laser diodes as a function of pressure and temperature. Artem Bercha was born in Chernovtsy,Ukraine,in1965.He graduated from Uzhgorod National University,Uzhgorod,Ukraine,in1987.He received the Ph.D.degree from the Institute for Physics of Semiconductors,Kiev,Ukraine, in1994.His major studies were on optical properties of GaAs–AlAs short period superlattices.From1995to2000,he worked as a Scientific Researcher at the Institute of Physics and Chemistry of Solids and as an Assistant Professor in the Electronic Systems Faculty of Uzhgorod National University.Since2001,he has had a postdoctoral position at the High Pressure Research Center of Polish Academy of Sciences,Warsaw,Poland,and investigates the properties of semiconductor laser diodes under high hydrostatic pressure.Filip Dybala was born in Poland in1975.He received the M.Sc.degree in elec-tronics from Wroclaw University of Technology,Wroclaw,Poland,in2000.He is currently working toward the Ph.D.degree at the Warsaw University of Tech-nology,Warsaw,Poland,and in the High Pressure Research Center of Polish Academy of Sciences,Warsaw.He investigates the properties of laser diodes under high hydrostatic pressure for wavelength tuning.Roland Bohdan was born in Mukatchevo,Ukraine,in1965.He graduated from Uzhgorod National University,Uzhgorod,Ukraine in1987,where he made the-oretical research on short-period superlattices.He is currently working toward the Ph.D.degree at the High Pressure Research Center of Polish Academy of Sciences,Warsaw,Poland.He investigates the properties of visible and infrared laser diodes,in particular blue InGaN–GaN lasers and InGaAsSb lasers.Witold A.Trzeciakowski was born in1952.He obtained the M.Sc.and Ph.D. degrees in physics from Warsaw University,Warsaw,Poland,in1974and1980, respectively.From1974to1980,he worked as an Assistant Professor in the Department of Physics,Warsaw University.In1980,he joined the High Pressure Research Center of the Polish Academy of Sciences where he has been working untill now. He obtained the degree of Associate Professor(docent)in1989and the title of Professor in2000.Initially,he worked on semiconductor theory(shallow and deep impurity states,magnetic and electricfield effects,heterostructure elec-tronic states),then he turned to experimental physics(deformation effects in QWs,Raman scattering)and to device physics(the effect of pressure on laser diodes,electrical and optical pressure sensors).He worked as a Scientist with NRC,Ottawa,Canada,University of Florence,Italy,Universities of Bordeaux and Grenoble,France,University of Valencia,Spain,SUNY at Buffalo,and with the Technical University of Athens,Greece.。

a rXiv:h ep-ph/2180v116Fe b2UCRHEP–T270A short course in effective Lagrangians.∗Jos´e Wudka †Physics Department,UC Riverside Riverside CA 92521-0413,USA Abstract These lectures provide an introduction to effective theories concentrating on the basic ideas and providing some simple applications I.INTRODUCTION.When studying a physical system it is often the case that there is not enough information to provide a fundamental description of some of its properties.In such cases one must parameterize the corresponding effects by introducing new interactions with coefficients to be determined phenomenologically.Experimental limits or measurement of these parameters then (hopefully)provides the information needed to provide a more satisfactory description.A standard procedure for doing this is to first determine the dynamical degrees of freedom involved and the symmetries obeyed,and then construct the most general Lagrangian,the effective Lagrangian for these degrees of freedom which respects the required symmetries.The method is straightforward,quite general and,most importantly,it works!In following this approach one must be wary of several facts.Fist it is clear that the relevant degrees of freedom can change with scale(e.g.mesons are a good description of low-energy QCD,but at higher energies one should use quarks and gluons);in addition,physics at different scales may respect different symmetries(e.g.mass conservation is violated at sufficiently high energies).It follows that the effective Lagrangian formalism is in general applicable only for a limited range of scales.It is often the case(but no always!)that there is a scaleΛso that the results obtained using an effective Lagrangian are invalid for energies aboveΛ.The formalism has two potentially serious drawbacks.First,effective Lagrangian has an infinite number of terms suggesting a lack of predictability.Second,even though the model has an UV cutoffΛand will not suffer from actual divergences,simple calculations show that is is a possible for this type of theories to generating radiative corrections that grow withΛ,becoming increasingly important for higher and higher order graphs.Either of these problems can render this approach useless.It is also necessary verify that the model is unitary.I will discuss below how these problems are solved,an provide several applications of the formalism.The aim is to give aflair of the versatility of the approach,not to provide an exhaustive review of all known applications.II.F AMILIAR EXAMPLESA.Euler-Heisenberg effective LagrangianThis Lagrangian summarizes QED at low energies(below the electron mass)[1].At these energies only photons appear in real processes and the effective Lagrangian will be then constructed using the photonfield Aµ,and will satisfy a U(1)gauge and Lorenz invariances. Thus it can be constructed in terms of thefield strength Fµνor the loop variables A(Γ)=2FIG.1.Graph generating the leading terms in the Euler-Heisenberg effective Lagrangian ΓA·dx.The latter are non-local,so that a local description would involve only F,namely1 L eff=L eff(F)=aF2+bF4+c(F˜F)2+dF2(F˜F) (1)One can arbitrarily normalize thefields and so choose a=−1/4.The constants b,c and d have units of mass−2.Note that the term∝d violates CP.Though we know QED respects C and P,it is possible for other interactions to violate these symmetries,there is nothing in the discussion above that disallows such terms and,in fact,weak effects will generate them.For this system we are in a privileged position for we know the underlying physics,and so we can calculate b,c,d,....The leading effects come form QED which yields b,c∼1/(4πm e)2at 1loop[1].The parameters b and c summarize all the leading virtual electron effects.(see Fig.1).Forgetting about this underlying structure we could have simply defined a scale M and taken b,c∼1/M2(so that M=4πm e),and while this is perfectly viable,M is not relevant phenomenologically speaking as it does not corresponds of a physical scale.In order to extract information about the physics underlying the effective Lagrangian from a measurement of b and c we must be able to at least estimate the relation between these constants and the underlying scales.In addition we also know that d∼ξ/(4πv)with v∼246GeV andξis a very small constant proportional to the Jarlskog determinant[2].The effective Lagrangian can holdterms with radically different scales and limits on some constants cannot,in general,translate to others.In this case the terms are characterized by different CP transformation properties, and it is often the case that such global symmetries are useful in differentiating terms in the effective Lagrangian.The point being that a term violating a given global symmetry at scaleΛwill generate all terms in the effective Lagrangian with the same symmetry properties through radiative corrections.The caveat in the argument being that the underlying theory might have some additional symmetries not apparent at low energies which might further segregate interactions and so provide different scales for operators with the same properties under all low energy symmetries.When calculating with the effective Lagrangian the effects produced by the new terms proportional to b,c are suppressed by a factor∼(E/4πm e)2,where E is the typical energy on the process and E≪m e.Thus the effects of these terms are tiny,yet they are noticeable because they generate a new effect:γ−γscattering.B.(Standard)SuperconductivityThis is a brief summary of the very nice treatment provided by Polchinski[3].The system under consideration has the electronfieldψas its only dynamical variable(the phonons are assumed to have been integrated out,generating a series of electron self-interactions),it respects U(1)electromagnetic gauge invariance,as well as Galilean invariance and Fermion number conservation.Assuming a local description,thefirst few terms in the effective Lagrangian expansion are(neglecting those containing photons for simplicity)L eff= kψ∗k[i∂t−e k+µ]ψk+ ψ∗kψlψqψ∗pδ(k−l−q+p)V klq+ (2)In this equation the relation e k=µdetermines the Fermi surface,while V∼(electron-photon coupling)2on the Fermi Surface(FS)if e k=µ,if p is near the FS one can write p=k+ℓˆn(with e k=µ).Scaling towards the FS impliesℓ→sℓwith s→0.Then assumingψ→s dψthe quadratic terms in the action will be scale invariant provided d=−1/2.The quartic terms in the action then scales as s and becomes negligible near the FS except when the pairing condition q+l=0is obeyed.In this case the quartic term scales as s0and cannot be ignored.In fact this term determines the most interesting behavior of the system at low temperatures(see[3]for full details).C.Electroweak interactionsAgain I will follow the general recipe.I will concentrate only on the(low energy)inter-actions involving leptonfields,which are then the degrees of freedom.Since I assume the energy to be well below the Fermi scale,the only relevant symmetries are U(1)gauge and Lorenz invariances.In addition there is the question whether the heavy physics will respect the discrete symmetries C,P or CP;using perfect hindsight I will retain terms that violate these symmetriesAssuming a local description I have[1]L eff= ¯ψi(i D−m i)ψi+ f ijkl ¯ψiΓaψj ¯ψkΓaψl + (3)where the ellipsis indicate terms containing operators of higher dimension,or those involving the electromagneticfield.The matricesΓare to be chosen among the16independent basis Γa={1,γµ,σµν,γµγ5,γ5}The coefficients for thefirst two terms are befixed by normalization requirements.While a SM calculation gives f∼g2/m2W=1/v2(v≃246GeV)and is generated by tree-level√graphs(see Fig.II C)because of this the scale1/be observed(or bounded)despite the E≪v condition because they generate new effects: C and P(and some of them chirality)violation.FIG.2.Standard model processes generating four fermion interactions at low energies(e.g.. Bhaba scattering)D.Strong interactions at low energiesIn this case we are interested in the description of the interactions among the lightest hadrons,the meson multiplet.The most convenient parameterization of these degrees of freedom is in terms of a unitaryfield[9]U such that U=exp(λaπa/F)whereπa denote the eight mesonfields,λa the Gell-Mann matrices and F is a constant(related to the pion decay constant).The symmetries obeyed by the system are chiral SU(3)L×SU(3)R,Lorenz invariance,C and P.With these constraints the effective Lagrangian takes the formL eff=a tr∂U†.∂U+ b tr∂µU†∂νU∂µU†∂νU+... + (4)I can set a∼F2by properly normalizing thefields.In this case the leading term in the effective Lagrangian will determine all(leading)low-energy pion interactions in terms of the single constant F.The effects form the higher-order terms have been measured and the data requires b∼1/(4π)2.This result is also predicted by the consistency of this approach which requires that radiative corrections to a,b,etc.should be at most of the same size as their tree-level values.6III.BASIC IDEAS ON THE APPLICABILITY OF THE FORMALISMBeing a model with intrinsic an cutoffthere are no actual ultraviolet divergences in most effective Lagrangian computations.Still there are interesting renormalizability issues that arise when doing effective Lagrangian loop computations.Imagine doing a loop calculation including some vertices terms of(mass)dimension higher than the dimension of space-time.These must have coefficients with dimensions of mass to some negative power.The loop integrations will produce in general terms growing withΛthe UV cutoffwhich are polynomials in the external momenta2and will preserve the symmetries of the model[4].Hence these terms which may grow withΛcorrespond to vertices appearing in the most general effective Lagrangian and can be absorbed in a renormalization of the corresponding coefficients.They have no observable effects(though they can be used in naturality arguments[5].Effective theories will also be unitary provided one stays within the limits of their appli-cability.Should one exceed them new channels will open(corresponding to the production of the heavy excitations)and unitarity violating effects will occur.This is not produced by real unitarity violating interactions,but due to our using the model beyond its range of applicability(e.g.it the typical energy of the process under consideration reaches of exceeds Λ).One can,of course,extend the model,but this necessarily introduces ad-hoc elements and will dilute the generality gained using effective theories.For example consider W W Z interactions with an effective Lagrangian of the formL eff=λ(p,k)Wµν(k)Wνρ(p)Zρµ(−p−k)+···;(5) (where Vαβ=∂αVβ−∂βVα)One can then chooseλto insure unitarity is preserved(at least in some processes),for example[6]λ0λ(p,k)=Another common situation where effective Lagrangians appear occurs when some heavy excitations are integrated out.This can be illustrated by the following toy model3S= d n x ¯ψ(i∂−m)ψ+12Λ2φ21+fφ¯ψψ (9) whereφis heavy.A simple calculation givesS eff= d n x ¯ψ(i∂−m)ψ+1+Λ2¯ψψ (10) andL eff=¯ψ(i∂−m)ψ+f2Λ2 n¯ψψ(11) Note that terms with large number of derivatives will be suppressed by a large power of the small factor(E/Λ),if we are interested in energies E∼Λthe whole infinite set of vertices must be included in order to reproduce theφpole.A.How to parameterize ignoranceIf one knows the theory we can,in principle,calculate L eff(or do a full calculation).Yet there are many cases where the underlying theory is not known.In these cases an effective theory if obtained by writing all possible interactions among the light excitations.The model then has an infinite number of terms each with an unknown parameter,and these constants then parameterize all possible underlying theories.The terms which dominate are those usually called renormalizable(or,equivalently,marginal or relevant).The other terms are called non-renormalizable,or irrelevant,since their effects become smaller as the energy decreasesThis recipe for writing effective theories must be supplemented with some symmetry re-strictions.The most important being that the all the terms in the effective Lagrangian mustrespect the local gauge invariance of the low-energy physics(more technically,the one re-spected by the renormalizable terms in the effective action)[7].The reason is that the presence of a gauge variant term will generate all gauge variant interactions thorough renor-malization group evolution.a.Gauge invariantizing Using a simple argument it is possible to turn any theory into a gauge theory[8]and so it appears that the requirement of gauge invariance is empty.That this is not the case is explained here.Ifirst describe the trick which grafts gauge invariance onto a theory and then discuss the implications.Consider an arbitrary theory with matterfields(spin0and1/2)and vectorfields V nµ, n=1,...N.Then•Choose a(gauge)group G with N generators{T n}.Define a covariant derivative Dµ=∂µ+V nµT n and assume that the V nµare gaugefields.•Invent a unitaryfield U transforming according to the fundamental representation ofG and construct the gauge invariant compositefieldsV nµ=−tr T n U†DµU(12) Taking tr T n T m=−δnm,it is easy to see that in the unitary gauge U=1,V nµ=V nµ.Thus if simply replace V→V in the original theory we get a gauge theory.Does this mean that gauge invariance irrelevant since it can be added at will?In my opinion this is not the case.In the above process all matterfields are assumed gauge singlets(none are minimally coupled to the gaugefields).In the case of the standard model,for example,the universal coupling of fermions to the gauge bosons would be accidental in this approach.In order to recover the full predictive power commonly associated with gauge theories,the matterfields must transform non-trivially under G which can be done only if there are strong correlations among some of the couplings.It is not trivial to say that the standard model group is10SU(3)×SU(2)×U(1)with left-handed quarks transforming as(3,2,1/6),left-handed leptons as(1,2,−1/2),etc.,as opposed to a U(1)12with all fermions transforming as singlets[10].B.How to estimate ignoranceA problem which I have not addressed so far is the fact that effective theories have an infinite number of coefficients,with the(possible)problem or requiring an infinite number of data points in order to make any predictions.On the other hand,for example,if this is the case why is it that the Fermi theory of the weak interactions is so successful?The answer to this question lies in the fact that not all coefficients are created equal,there is a hierarchy[9,10].As a result,given any desired level of accuracy,only afinite number of terms need to be included.Moreover,even though the effective Lagrangian coefficients cannot be calculated without knowing the underlying theory,they can still be bounded using but a minimal set of assumptions about the heavy interactions.It is then also possible to estimate the errors in neglecting all but thefinite number of terms used.As an example consider the standard model at low energies and calculate two processes: Bhaba cross section and the anomalous magnetic moment of the electron.For Bhaba scat-tering there is a contribution due the Z-boson exchange(see Fig.II C)e+e−→Z→e+e−generates O=14In addition the coefficient is suppressed by a factor of m e since it violates chirality.11eeeFIG.3.Weak contributions to the electron anomalous magnetic moment The point of this exercise is to illustrate the fact that,for weakly coupled theories,loop-generated operators have smaller coefficients than operators generated at tree level.Leading effects are produced by operators which are generated at tree level.C.Coefficient estimatesIn this section I will provide arguments which can be used to estimate(or,at least bound) the coefficients in the effective Lagrangian.These are order of magnitude calculations and might be offby a factor of a few;it is worth noting that no single calculation has provided a significant deviation from these results.The estimate calculations should be done separately for weakly and strongly interacting theories.I will characterize thefirst as those where radiative corrections are smaller than the tree-level contributions.Strongly interacting theories will have radiative corrections of the same size at any order51.Weakly interacting theoriesIn this case leading terms in the effective Lagrangian are those which can be generated at tree level by the heavy physics.Thus the dominating effects are produced by operators which have the lowest dimension(leading to the smallest suppression from inverse powers ofΛ)and which are tree-level generated(TLG)operators can be determined[11].When the heavy physics is described by a gauge theory it is possible to obtained all TLG operators[11].The corresponding vertices fall into3categories,symbolically •vertices with4fermions.•vertices with2fermions and k bosons;k=2,3•vertices with n bosons;n=4,6.A particular theory may not generate one or more of these vertices,the only claim is that there is a gauge theory which does.In the case of the standard model with lepton number conservation the leading operators have dimension6[12,11].Subleading operators are either dimension8and their contribu-tions are suppressed by an additional factor(E/Λ)2in processes with typical energy E. Other subleading contributions are suppressed by a loop factor∼1/(4π)2.Note that it is possible to have situations where the only two effects are produced by either dimension8 TLG operators or loop generated dimension6operators.In this case the former dominates only whenΛ>4πE.a.Triple gauge bosons The terms in the electroweak effective Lagrangian which describe the interaction of the W and Z bosons generated by some heavy physics underlying the standard model has received considerable attention recently[13].In terms of the SU(2)and U(1)gaugefields W and B and the scalar doubletφthese interactions areL eff=1information about the heavy physics.2.Strongly interacting theoriesI will imagine a theory containing scalars and fermions which interact strongly.Gauge couplings are assumed to be small and will be ignored.This calculation is useful for low energy chiral theories but not for low energy QCD[14,15,9].A generic effective operator in this type of theories takes the formO abc∼λΛ4 φΛψ3/2 b ∂(4π)2/3Λ,Λφ=116π2(17)In terms of U∼exp(φ/Λφ),the operators take the formO abc=1For the case whereφrepresents the interpolatingfield for the lightest mesons PCAC impliesΛφ=fπ[14,9].Thenψ4∝116π2ψ2∂2U2∝1Λ2 ¯ψγµψ ¯ψγµψ + (20)whereψdenotes the electronfield.The calculation is illustrated in Fig.IV D where the loops involving the4-fermion oper-ator are cut-offat a scaleΛ.The SM and new physics(NP)contributions are,symbolically,_4πg ()2_1v 2_1v2_2f Λ_4π()2Λ_2f Λ_4π()2Λ_2f Λ_4π()2Λ_2f Λ_2f Λ_2fΛ+++. . . FIG.4.Radiative corrections to Bhaba scattering in the presence of a 4-fermion interactionSM:116π2+··· NP:f16π2+ (21)Note that this consistent behavior (that the new physics effects disappear as Λ→∞)results form having the physical scale of new physics Λin the coefficient of the operator.Had we used f ′/v 2instead of f/Λ2the new physics effects would appear to be enormous,and growing with each new loop.It is not that the use of f ′/v 2is wrong,it is only that it is misleading to believe f ′can be of order one;it must be suppressed by the small factor (v/λ)ing these results we see that this reaction is sensitive to Λprovidedf (v/Λ)2>sensitivity.If the sensitivity is,say 1%this corresponds to Λ/√V.APPLICATIONS TO ELECTROWEAK PHYSICSWith the above results one can determine,for any given process,the leading contributions (as parameterized by the various effective operator coefficients).Using then the coefficient estimates one can provide the expected magnitude of the new physics effects with onlyΛas an unknown parameter,and so estimate the sensitivity to the scale of new physics.It is important to note that this is sometimes a rather involved calculation as all con-tributing operators must be included.For example,in order to determine the heavy physics effects on the oblique parameters one must calculate not only these affecting the vector bo-son polarization tensors,but also this which modify the Fermi constant,thefine structure constant,etc.as these quantities are used when extracting S,T and U from the data[18].A.Effective lagrangianIn the following I will assume that the underlying physics is weakly coupled and derive the leadingoperators that can be expected form the existence of heavy excitations at scale Λ.The complete list of dimension6operators was cataloged a long time ago for the case where the low energy spectrum includes a single scalar doublet[12]7.It is then straightfor-ward to determine the subset of operators which can be TLG,they are[11]•Fermions: ¯ψiΓaψj ¯ψkΓaψl•Scalars:|φ|6,(∂|φ|2)2•Scalars and fermions:|φ|2×Yukawa term•Scalars and vectors:|φ|2|Dφ|2,|φ†Dφ|2•Fermions,scalars and vectors: φ†T n Dµφ ¯ψi T nγµψjwhere T denotes a group generator andΓa product of a group generator and a gamma matrix.Observables affected by the operators in this list provide the highest sensitivity to new physics effects provided that the standard model effects are themselves small(or that the experimental sensitivity is large enough to observe small deviations).I will illustrate this with two(incomplete)examplesB.b-parityThis is a proposed method for probing newflavor physics[19].Its virtue lies in the fact that it is very simple and sensitive(though it does not provide the highest sensitivity for all observables).The basic idea is based on the observation that the standard model acquires an additional global U(1)b symmetry in the limit V ub=V cb=V td=V ts=0(given the experimental values0.002<|V ub|<0.005,0.036<|V cb|<0.046,0.004<|V td|<0.014, 0.034<|V ts|<0.046,deviations form exact U(1)b invariance will be small).Then for any standard model interaction a reaction to the typen i b−jet+X→n f b−jet+Y(22) will obey(−1)n i=(−1)n f(23) to very high accuracy.The number(−1)#of b jets defines the b-parity of a state(it being understood that the top quarks have decayed).The standard model is then b-parity even,and the idea is to consider a lepton collider8 and simply count the number of b jets in thefinal state;new physics effects will show up as events with odd number of b jets.The standard model produces no measurable irreducible background,yet there are significant reducible backgrounds which reduced the sensitivity toΛ.To estimate these effects I define•ǫb=b−jet tagging efficiency•t c=c−jet mis tagging efficiency(probability of mistaking a c−jet jet for a b−jet •t j=light-jet mis tagging efficiency(probability of mistaking a light-jet for a b−jet so that the measured cross section with k-b-jets is¯σk= u+v+w=k n u ǫu b(1−ǫb)n−u m v t v c(1−t c)m−v ℓw t w j(1−tj)ℓ−w σnmℓ(24) whereσnmℓdenotes the cross section for thefinal state with n b-jets,m c-jets,andℓlight jets.Note that n u ǫu b(1−ǫb)n−u is the probability of tagging u and missing n−u b-jets out of the n available.As an example considerL eff=L sm+f ijLimits from e+e−→t¯c+¯t c+b¯s+¯bs→1b−jet+Xsǫb=50%ǫb=70%2.5fb−1 1.5TeV500GeV 5.0TeV 5.5TeV200fb−110.0TeVThese results are promising yet they will be degraded in a realistic calculation.First one must include the effects of having t c,j=0.In addition there are complications in using inclusive reactions such as e+e−→b+X since the contributions form events with large number of jets can be very hard to evaluate(aside from the calculational difficulties there19are additional complications when defining what a jet is).A more realistic approach is to restrict the calculation to a sample with afixed number of jets(2and4are the simplest) and determine the sensitivity toΛfor various choices ofǫb and t j using this population only.C.CP violationJust as for b-parity the CP violating effects are small within the standard model and so precise measurements of CP violating observable might be very sensitive to new physics effects.In order to study CP violations it is useful tofirst define what the CP transformation is. In order to do this in general denote the Cartan group generators by H i and the root gener-ators by Eα,then it is possible tofind a basis where all the group generators are real and, in addition,the H i are diagonal[20].Define then CP transformation by Transformationsψ→Cψ∗(fermions)φ→φ∗(scalars)A(i)µ→−A(i)µ,(i:Cartan generator)A(α)µ→−A(−α)µ,(α:root)it is easy to see that thefield strengths and currents transform as Aµ,while Dφ→(Dφ)∗.It then follows that in this basis the whole gauge sector of any gauge theory is CP conserving; CP violation can arise only in the scalar potential and fermion-scalar interactions using this basis.In order to apply this to electroweak physics I will need the list of TLG operators of dimension6which violate CP,they are given by9¯ℓe ¯dq −h.c.(¯q u)ε(¯q d)−h.c. ¯qλA u ε ¯qλA d −h.c.¯ℓe ε(¯q u)−h.c. ¯ℓu ε(¯q e)−h.c.|φ|2 ¯ℓeφ−h.c.|φ|2 ¯q u˜φ−h.c. |φ|2(¯q dφ−h.c.)|φ|2∂µ ¯ℓγµℓ|φ|2∂µ(¯eγµe)|φ|2∂µ(¯qγµq)|φ|2∂µ(¯uγµu)|φ|2∂µ ¯dγµdO1= φ†τIφ D IJµ ¯ℓγµτJℓO2= φ†τIφ D IJµ ¯qγµτJ qO3= φ†εDµφ (¯uγµd)−h.cAll operators except O1,2,3violate chirality and their coefficients are strongly bounded by their contributions to the strong CP parameterθ;in addition some chialiry violating operators contribute to meson decays(which again provide strong bounds for fermions in thefirst generation)and,finally,in natural theories some contribute radiatively to fermion masses and will be then suppressed by the smaller of the corresponding Yukawa couplings. For these reasons I will not consider them further.Moreover,since I will be interested in limits that can be obtained using current data,I will ignore operators whose only observable effects involve Higgs particles.With these restrictions only O1,2,3remain;their terms not involving scalars areO1→−igv22 ¯νL W+e L−h.c.O2→−igv22 ¯u L W+d L−h.c.O3→−igv28 ¯u R W+d R−h.c.The contributions from O1,2can be absorbed in a renormalization of standard model coef-ficients whence only O3produces observable effects,corresponding to a right-handed quark current.Existing data(fromτdecays and m W measurements)impliesΛ∼>500GeV One can also determine the type of new interactions which might be probed using these operators[11].The heavy physics which can generate O3at tree level is described in Fig.10. If the underlying theory is natural we conclude that there will be no super-renormalizableSR coupling(unnatural)FIG.5.Heavy violating operators.Wavy lines denote vectors,solid lines fermions,and dashed ones scalars.Heavy lines denote heavy excitations.couplings;in this case O3will be generated by heavy fermion exchanges only10 Notefinally that these arguments are only valid for weakly coupled heavy physics.For strongly coupled theories other CP violating operators can be important,e.g.f10It is true that vertices involving light fermions,light scalars and heavy fermions produce mixings between the light and heavy scales,but this occurs at the one loop level.In contrast cubic terms of orderΛin the scalar potential would shift v at tree level.。

Omnic软件使用指南1．Omnic与系统Omnic是Nicolet公司的在PC机使用最广泛的窗口软件平台上运行的红外软件，从开始在Windows3.1上运行的版本的 1.0到目前的 6.1a，现行的的操作系统Windows98/Me/NT/2000/XP都支持。

EZ-Omnic是简化的软件，一方面价格比较低，同时更加简明，容易掌握，虽然功能比较简单，仍可以满足先当部分用户的需求。

使用的仪器通讯接口有：LTP(并行口)或PCI卡，部分早一些的仪器使用ISA卡。

2．文件结构Omnic 6.0以上版本的缺省的文件分别存在于三个目录中：C:\My Documents\Omnic,在其子目录中分门别类地存放数据与参数等文件，如Spectra存光谱，Param中存设置参数,Quant 存定量方法； C:\Program Files\Omnic,存有驱动与程序文件等,系统的卸载命令在它的子目录Uninstall中；C:\MyDocument\Omnic\Lib,存放谱库，包括购买和自建的谱库。

软件安装的应用程序除了Omnic外还有Bench Diagnostics,这是一个在系统发生故障时进行判断的重要命令，能够检查从接口卡到仪器的各个重要部件。

它们与PDF文件一起置于Thermo Nicolet程序组中，3．启动Omnic软件使用下列方法之一启动Omnic 红外软件系统：1）．在Windows98等的桌面上双击（或者）2）．从Srart→Program→Thermo Nicolet→Omnic(或者从Srart→Program→Omnic5.0→)3）．其他，如Win98中的快捷方式启动。

4．Omnic显示面板：1）．Omnic是一种与窗口软件充分兼容的软件，可以显示一个或多个显示窗口，当显示多个窗口时可以选择平铺（Tile）或层叠（Cascade）方式，但其中只有一个是活动窗口（被选中的）。

光谱图可以在窗口间拖动、复制与粘贴，而且可以把复制的光谱图直接粘贴到其他应用程序的文本文件中，为发表文章或书写报告带来方便。

a r X i v :0807.2377v 1 [h e p -t h ] 15 J u l 2008Shear Viscosity from AdS Born-Infeld Black HolesRong-Gen Caia,1,Ya-Wen Suna,b,2,aInstitute of Theoretical Physics,Chinese Academy of Sciences P.O.Box 2735,Beijing 100190,ChinabGraduate University of Chinese Academy of Sciences,YuQuan Road 19A,Beijing 100049,ChinaAbstractWe calculate the shear viscosity in the frame of AdS/CFT correspondence for the field theory with a gravity dual of Einstein-Born-Infeld gravity.We find that the ratio of η/s is still the conjectured universal value 1/4πat least up to the first order of the Born-Infeld parameter 1/b 2.1IntroductionThe AdS/CFT correspondence[1,2,3,4]has been a useful way to calculate dynamical quanti-ties of strongly coupled gauge theories.A famous example is the discovery of the universality of the ratio of the shear viscosityηto the entropy density s,which is equal to1/4πin all theories in the regimes described by gravity duals[5,6,7,8].This ratio is also conjectured to be a universal lower bound(the KSS bound)for all materials[6].All known materials in nature by now satisfy this bound.In[9,10,11,12],the authors also calculated the shear viscosity of gauge theories with chemical potentials turned on by studying R-charged black holes.With the presence of nonzero chemical potentials the ratio of shear viscosity to entropy density is still1/4π.Also the stringy corrections to the ratio were calculated in[13,14,15,16,17,18] where the corrections to the value1/4πare found to be positive and satisfy the lower bound. More discussions on this KSS bound can be found in[21,22,23,24,25,26,27,28].On the other hand,in[19,20]the authors considered R2corrections in the gravity side and found that the modification of the ratio of shear viscosity over entropy density to the conjectured bound is negative,which means that the lower bound could be violated in that case.The higher derivative gravity corrections they considered can be seen as generated from stringy corrections given the vastness of the string landscape.They gave a new lower bound, 4/25π,based on the causal condition.However,the physical implication of this violation of the bound is still not very clear yet.This motivates us to consider whether higher derivative corrections to the gauge matterfields on the gravity side also affect the value of shear viscosity when chemical potentials are turned on.In[9,10,11,12],the ratio ofη/s for the case of nonzero chemical potential was calculated through Einstein-Maxwell theory on the gravity side.As an example of nonlinear electrody-namics,we consider Einstein-Born-Infeld theory with a negative cosmological constant.This theory can be viewed as a nonlinear extension on the gaugefields in Einstein-Maxwell theory.In this paper we will calculate the shear viscosity of gauge theories with the gravity dual of Einstein-Born-Infeld theory.The background we need is just the AdS Born-Infeld black hole solution[30,31].Calculate the shear viscosity in this background and we can see if the higher derivative corrections to the matterfields which are coupled to gravity will also affect the ratio. And the answer we get is that the ratio is not affected by this higher derivative correction to the gaugefields.This fact along with the modification of the ratio in higher derivative gravity theories may imply that the ratio may be fully determined by the form of the action of gravity in the dual gravity description.Here the action should be viewed as the effective action whichincludes the contribution from stringy corrections.If we do not consider the stringy corrections, the effective action for gravity is just the Einstein-Hilbert gravity term.The universality then implies that gauge theories having gravity duals with the same effective gravity action terms should have the same ratio of shear viscosity over entropy density.In this paper,wefirst present some basic properties of AdS Born-Infeld black holes in Sec.2. We calculate the shear viscosity of gauge theories with the gravity dual of Einstein-Born-Infeld theory through Kubo-formula in Sec.3.Sec.4is devoted to conclusions and discussions.2AdS Born-Infeld black holesIn this section we give some basic properties of the Born-Infeld black hole solution in the presence of a negative cosmological constant infive dimensions.The action can be written asS=1−g R−2Λ+L(F) ,(1)where L(F)=4b2(1− 2b2).The constant b here is the Born-Infeld parameter and has the dimension of mass.In the limit of b→∞,L(F)reduces to the Maxwell form.Thus if we expand L(F)in a series of1/b2,we willfind that the1/b2corrections to the Maxwell action just correspond to the higher derivative corrections of the gaugefields.We can write out the equations of motion explicitly asRµν−12gµνL(F)−2FµλFλν1+F22b2 =0.(3) Since we want to calculate the shear viscosity of the correspondingfield theory living in R1,3,we need a black hole solution with a Ricci-flat horizon.The AdS Born-Infeld black hole solution with a Ricci-flat horizon can be written as[31]ds2=−V(r)dt2+1r2+(b2l2)r2−b b2r6+3q2+3q23,13,−3q23bqb2r6+3q2.(6)Here q is an integration constant which is related to the electric charge of the black hole and l is the AdS radius throughΛ=−6/l2.When b2approaches to infinity,this solution becomes the AdS Reissner-Nordstr¨o m black hole solution.We can get the position of the outer horizon by solving V(r+)=0.For future convenience we define u=r2+/r2and rescale x to the new coordinate systemds2=−V(u)dt2+r2+ul2d x2.(7)Now V(u)becomesV(u)=−mu3+1u−b b2r6+2r4+2F1[12,4b2r6+],(8)and the horizon corresponds to u=1.Then the mass parameter m can be expressed by r+as m= b2l2 r4+−br+b2r6+2r2+2F1[12,4b2r6+].(9) The thermodynamic properties of this black hole has been discussed in[30,31,32,33,34,35]. Here we only give the entropy density of this black hole solution for future uses=1l3.(10)3Shear viscosity from AdS Born-Infeld black holesIn this section we calculate the shear viscosity of thefield theory dual to the black hole back-ground(4)through the Kubo-formula[21,36]:η=limω→01the scalar,vector and tensor perturbations[29].Here we use the simplest one,the tensor perturbation h12.We useφto denote this perturbationφ=h12and writeφin a basis as φ(t,u,x)=φ(u)e−iωt+ip·x.We can get the equation of motion of thisφ(u)by perturbing both sides of the equation of motion(2)to thefirst order ofφ(u)φ′′(u)+Aφ′(u)+Bφ(u)=0.(13)Here to avoid complication we only calculate the shear viscosity up to thefirst order in the parameter1/b2,and in this approximation we haveA=A0+A1,(14) where A0denotes the part of the coefficient A in the limit b2→∞,A0=l2q2(1−2u)u2+r6+(1+u2)16b2r6+(l2q2u2−r6+(1+u))2.(16) Also B can be written as the sum of two parts,B=B0+B1=r6+(¯p2(u−1)(r6+(1+u)−l2q2u2)+r6+¯w2)16b2(u−1)2(r6+(1+u)−l2q2u2)3.(17) Here¯w=l2ω/2r+and¯p=l2p/2r+.A1and B1manifest the contribution of the higher derivative corrections to the gaugefield on the gravity side.To solve forφ(u),we writeφ(u)=(1−u)−iβ¯w F(u)(18) to decide the boundary condition near the horizon,whereβis a constant to befixed.Substi-tuting(18)into(13)and solving it near the horizon u=1,we getβ=β0+β1=r6+4b2(2r6+−l2q2)2,(19)by pure incoming wave boundary condition near u=1.Hereβ1is the contribution of the higher derivative correction.Next we move on to solveφ(u)in the whole spacetime.Becausewe know from(11)that we only need the low frequency behavior ofφ(u)to calculate the shear viscosity,we can expand F(u)in a power series of¯w and¯pF(u)=1+iβ0¯wF0(u)+iβ1¯wF1(u)+O(¯w2)+O(¯p2).(20)In this expansion,F1represents the contribution from the higher derivative correction of the gaugefield.The equations of motion of F0(u)and F1(u)can be derived at thefirst order of¯wseparately,F′′0(u)+A0F′0(u)+11−u=0,(21)andF′′1(u)+A0F′1(u)+β0β1(1−u)+11−u=0.(22)The solutions of these two linear differential equations can be uniquely decided with the bound-ary condition F0(u)|u=0=F1(u)|u=0=0and the constraint that both F0(u)and F1(u)should be regular at the horizon u=1.The solutions areF0(u)=1r6+−3r3+4l2q2+r6+lnu−u4l2q2+r6++r6+−2l2q2u)((r3+ 4l2q2+r6++r3+) +C2lnr6+(1+u)−l2q2u28l6q6(4l2q2+r6+)(l2q2u2−r6+(1+u))×12r24+(1+u)−6l2q2r18+(−9−10u+u2)+4l8q8u(1+4u+u2)+l4q4r12+(30+51u−29u2−2u3)−l6q6r6+(6+2u+16u2+7u3) ,(25)C1and C2are two constants,C1=3(10l10q10r3++25l8q8r9++12l6q6r15+−45l4q4r21+−28l2q2r27+−4r33+)8l8q8.(27)Now we want to get the on-shell action for the perturbation.The on-shell action forφ(u)is a sum of two parts:one is from the bulk action S bulk and the other from the Gibbons-Hawking boundary term S GH.Wefirst expandφ(x,u)= d4k16πG d4k−gg uu and K2=3−gg uu.The K3and K4terms are not relevant to our aim, so we do not explicitly write them here.The Gibbons-Hawking boundary term isS GH=1−hK.(30)Substituting the solution ofφ(u)into(30)and we reachS GH=1(2π)4f(k)f(−k)(K5φkφ−k+K6φ′kφ−k),(31)where K6=−2√16πG d4k2φkφ−k+K2φ′kφ−k)|01 .(32)The Gibbons-Hawking term is itself a boundary contribution,so the total action can be written asS= d4k32πG √16πGi¯wr4+16πGiωr3+wi=1l3.(36)Comparing this with s=1l3,wefinally reachη/s=1/4π.4Conclusions and DiscussionsIn this paper we calculated the ratio of shear viscosity to entropy density in the background of AdS Born-Infeld black holes through AdS/CFT correspondence.Motivated by[19]wefind that though the higher derivative corrections to the gravity term make the universal bound be modified,the higher derivative corrections to the gaugefields have no change on the value of the η/s ratio.Although we calculated the ratio up to thefirst order of the Born-Infeld parameter, we believe the result is nonperturbative.This result may give us some hint on in what sense the lower bound is universal.We learn that when no gravity corrections are considered,the ratio ofη/s is the same for various gravity backgrounds[7]and for gauge theories with nonzero chemical potentials.When gravity corrections are added,the ratio changes and even the lower bound could be violated,but higher derivative corrections to the gaugefields on the gravity side do not change the value ofη/s. The difference between these two corrections is that with the gravity correction the effective action of gravity part is changed while in the latter case the effective action of gravity part is still the Einstein-Hilbert form though matterfields are coupled to gravity in various ways. This may imply that the universality of the bound is just the universality among gaugefield theories which have gravity duals with the same form of gravity term in the effective action, and as long as the gravity part action is the same,the way matterfields are coupled to gravity does not change the value ofη/s.The result of[16]might be viewed as evidence to support this idea,where the author found some evidence of universality of shear viscosity atfinite t’Hooft coupling.Of course,further evidence is needed to confirm this thought.AcknowledgementsThis work was supported in part by a grant from Chinese Academy of Sciences,grants from NSFC with No.10325525and No.90403029.References[1]J.M.Maldacena,Adv.Theor.Math.Phys.2,231(1998)[Int.J.Theor.Phys.38,1113(1999)][arXiv:hep-th/9711200].[2]S.S.Gubser,I.R.Klebanov and A.M.Polyakov,Phys.Lett.B428,105(1998)[arXiv:hep-th/9802109].[3]E.Witten,Adv.Theor.Math.Phys.2,253(1998)[arXiv:hep-th/9802150].[4]O.Aharony,S.S.Gubser,J.M.Maldacena,H.Ooguri and Y.Oz,Phys.Rept.323,183(2000)[arXiv:hep-th/9905111].[5]G.Policastro, D.T.Son and A.O.Starinets,Phys.Rev.Lett.87,081601(2001)[arXiv:hep-th/0104066].[6]P.Kovtun,D.T.Son and A.O.Starinets,JHEP0310,064(2003)[arXiv:hep-th/0309213].[7]A.Buchel and J.T.Liu,Phys.Rev.Lett.93,090602(2004)[arXiv:hep-th/0311175].[8]P.Kovtun,D.T.Son and A.O.Starinets,Phys.Rev.Lett.94,111601(2005)[arXiv:hep-th/0405231].[9]J.Mas,JHEP0603,016(2006)[arXiv:hep-th/0601144].[10]D.T.Son and A.O.Starinets,JHEP0603,052(2006)[arXiv:hep-th/0601157].[11]O.Saremi,JHEP0610,083(2006)[arXiv:hep-th/0601159].[12]K.Maeda,M.Natsuume and T.Okamura,Phys.Rev.D73,066013(2006)[arXiv:hep-th/0602010].[13]A.Buchel,J.T.Liu and A.O.Starinets,Nucl.Phys.B707,56(2005)[arXiv:hep-th/0406264].[14]P.Benincasa and A.Buchel,JHEP0601,103(2006)[arXiv:hep-th/0510041].[15]A.Buchel,arXiv:0801.4421[hep-th].[16]A.Buchel,arXiv:0804.3161[hep-th].[17]A.Buchel,arXiv:0805.2683[hep-th].[18]R.C.Myers,M.F.Paulos and A.Sinha,arXiv:0806.2156[hep-th].[19]M.Brigante,H.Liu,R.C.Myers,S.Shenker and S.Yaida,Phys.Rev.D77,126006(2008)[arXiv:0712.0805[hep-th]];M.Brigante,H.Liu,R.C.Myers,S.Shenker and S.Yaida, Phys.Rev.Lett.100,191601(2008)[arXiv:0802.3318[hep-th]].[20]Y.Kats and P.Petrov,arXiv:0712.0743[hep-th].[21]G.Policastro, D.T.Son and A.O.Starinets,JHEP0209,043(2002)[arXiv:hep-th/0205052].[22]G.Policastro, D.T.Son and A.O.Starinets,JHEP0212,054(2002)[arXiv:hep-th/0210220].[23]T.D.Cohen,Phys.Rev.Lett.99,021602(2007)[arXiv:hep-th/0702136].[24]A.Cherman,T.D.Cohen and P.M.Hohler,JHEP0802,026(2008)[arXiv:0708.4201[hep-th]].[25]J.W.Chen,M.Huang,Y.H.Li,E.Nakano and D.L.Yang,arXiv:0709.3434[hep-ph].[26]D.T.Son,Phys.Rev.Lett.100,029101(2008)[arXiv:0709.4651[hep-th]].[27]I.Fouxon,G.Betschart and J. D.Bekenstein,Phys.Rev.D77(2008)024016[arXiv:0710.1429[gr-qc]].[28]A.Dobado,F.J.Llanes-Estrada and J.M.T.Rincon,arXiv:0804.2601[hep-ph].[29]P.K.Kovtun and A.O.Starinets,Phys.Rev.D72,086009(2005)[arXiv:hep-th/0506184].[30]T.K.Dey,Phys.Lett.B595,484(2004)[arXiv:hep-th/0406169].[31]R.G.Cai, D.W.Pang and A.Wang,Phys.Rev.D70,124034(2004)[arXiv:hep-th/0410158].[32]S.Fernando,Phys.Rev.D74,104032(2006)[arXiv:hep-th/0608040].[33]A.Sheykhi and N.Riazi,Phys.Rev.D75,024021(2007)[arXiv:hep-th/0610085].[34]O.Miskovic and R.Olea,Phys.Rev.D77,124048(2008)[arXiv:0802.2081[hep-th]].[35]Y.S.Myung,Y.W.Kim and Y.J.Park,arXiv:0805.0187[gr-qc].[36]D.T.Son and A.O.Starinets,Ann.Rev.Nucl.Part.Sci.57,95(2007)[arXiv:0704.0240[hep-th]].。

B o o k I IUnit 1A.1.assess2.alliance3.outcome4.ethical5.identity6.ambiguous7.tolerable8.participates9.pursuit10.constructiveB.1.at stake2.were obliged3.the climate of4.feel well-equipped5.beyond my grasp6.cut back7.other than8.rise above9.care about10.is boundedC.1.incompetent2.indulgence3.migrants4.probesplex6.suspense; engagedpassionate; committed8.tolerant9.tempted10.interconnectedD.1. A. Judging from2. B. in which3. C. and4. D. believe5. A. is one of/ is that of6. B. must get7. C. likely8. D. unemployed9. C. as well as/ and10.B. simplerE. 1.what2.graduation3.intend4.getting5.eventually6.survey7.although8.graduates9.transfer10.rise11.attending12.instead13.cause14.because15.attending16.below17.failure18.expectations19.confidencecationKey to the translation from English to Chinese:1.德.汤说过，一切进步，一切发展均来自挑战及由此引起的反应。

自动化专业常用英语词汇Introduction:In the field of automation, understanding and using the correct terminology is essential for effective communication and collaboration. This document aims to provide a comprehensive list of commonly used English vocabulary in the field of automation. The terms are categorized into different sections for ease of reference.1. General Automation Terms:- Automation: The use of technology and machinery to perform tasks without human intervention.- Control System: A system that manages and regulates the operation of machines or processes.- Programmable Logic Controller (PLC): A digital computer used for automation of electromechanical processes.- Human-Machine Interface (HMI): The interface that allows interaction between humans and machines.- Sensor: A device that detects and measures physical quantities, such as temperature or pressure.- Actuator: A device that converts control signals into physical action, such as moving a valve or motor.- Feedback: Information about the output of a system used to make adjustments or corrections.- Process Variable: A physical quantity that is measured and controlled in a process.- Fault Detection and Diagnosis (FDD): The process of identifying and diagnosing faults in a system or process.- SCADA (Supervisory Control and Data Acquisition): A system used to monitor and control industrial processes.2. Control and Instrumentation Terms:- PID Controller: A control loop feedback mechanism widely used in industrial control systems.- Setpoint: The desired value or target for a process variable.- Proportional Gain: The gain factor that determines the proportional response of a control system.- Integral Time: The time taken for the integral action of a control system to eliminate the steady-state error.- Derivative Time: The time taken for the derivative action of a control system to respond to changes in the process variable.- Transmitter: A device that converts a physical quantity into an electrical signal for transmission.- Flowmeter: A device used to measure the flow rate of a fluid.- Pressure Transducer: A device used to measure pressure and convert it into an electrical signal.- Temperature Controller: A device used to maintain a desired temperature in a system.- Level Sensor: A device used to measure and control the level of a liquid or solid material.3. Robotics and Automation Terms:- Robot: A machine capable of carrying out complex tasks automatically.- End Effector: The tool or device attached to the robot's arm to perform specific tasks.- Gripper: A device used to grasp and hold objects.- Manipulator: The mechanical arm of a robot that performs movements and tasks.- Path Planning: The process of finding an optimal path for a robot to reach its target.- Kinematics: The study of the motion of objects without considering the forces that cause the motion.- Artificial Intelligence (AI): The simulation of human intelligence in machines to perform tasks intelligently.- Machine Learning: A subset of AI that allows machines to learn and improve from experience without explicit programming.- Computer Vision: The field of AI that enables computers to interpret and understand visual information.4. Industrial Automation Terms:- Conveyor: A mechanical system used to transport materials or products from one location to another.- PLC Programming: The process of writing instructions for a programmable logic controller.- SCARA (Selective Compliance Assembly Robot Arm): A type of robot arm used for assembly tasks.- DCS (Distributed Control System): A control system used in large industrial processes.- HMI Design: The process of designing user-friendly interfaces for human-machine interaction.- Safety Interlock: A system that ensures safe operation by preventing access to hazardous areas.- Motion Control: The process of controlling the movement of machines or robotic systems.- Batch Control: The control of a process that involves a sequence of steps or operations.Conclusion:This comprehensive list of commonly used English vocabulary in the field of automation provides a solid foundation for effective communication and understanding. By familiarizing yourself with these terms, you will be better equipped to navigate discussions, collaborate with colleagues, and stay up-to-date with advancements in the field of automation. Remember to use these terms accurately and confidently to enhance your professional communication in the automation industry.。

Derivative derivative analysis, also known as second-order differentiation or double differentiation, involves taking the derivative of a function with respect to one variable and then taking the derivative of the resulting expression with respect to another variable. This analysis is commonly used in calculus to study the rates of change of rates of change, such as acceleration or curvature.Let's consider a function \( f(x) \) and its derivative \( f'(x) \). The derivative of \( f'(x) \) with respect to \( x \) is called the second derivative of \( f(x) \) and is denoted as \( f''(x) \) or \( \frac{d^2f}{dx^2} \).Here's a step-by-step explanation of how to find the second derivative:1. **Find the first derivative**: Differentiate \( f(x) \) with respect to \( x \) to find \( f'(x) \).2. **Find the second derivative**: Differentiate \( f'(x) \) with respect to \( x \) to find \( f''(x) \). For example, let's find the second derivative of the function \( f(x) = x^3 \):1. **First derivative**: \( f'(x) = 3x^2 \)2. **Second derivative**: \( f''(x) = \frac{d}{dx}(3x^2) = 6x \)The second derivative of \( f(x) = x^3 \) is \( f''(x) = 6x \).The second derivative provides information about the concavity of the graph of the function. If \( f''(x) > 0 \), the graph is concave up (like a "U"), and if \( f''(x) < 0 \), the graph is concave down (like an "n"). A second derivative of zero at a specific \( x \) value indicates a possible inflection point, where the concavity of the graph changes.Derivative derivative analysis can be extended to higher orders, where you take the derivative of a function with respect to one variable multiple times. Each subsequent derivative provides increasingly detailed information about the function's behavior.。