Ann.N.Y.Acad.Sci.ISSN0077-8923 ANNALS OF THE NEW YORK ACADEMY OF SCIENCESIssue:The Year in Diabetes and ObesityCardiovascular disease and glycemic control in type2 diabetes:now that the dust is settling from largeclinical trialsFrancesco Giorgino,Anna Leonardini,and Luigi LaviolaDepartment of Emergency and Organ Transplantation,Section of Internal Medicine,Endocrinology,Andrology and Metabolic Diseases,University of Bari Aldo Moro,Bari,ItalyAddress for correspondence:Francesco Giorgino,M.D.,Ph.D.,Department of Emergency and Organ Transplantation, Section of Internal Medicine,Endocrinology,Andrology and Metabolic Diseases,University of Bari Aldo Moro,Piazza Giulio Cesare,11,I-70124Bari,Italy.francesco.giorgino@uniba.itThe relationship between glucose control and cardiovascular outcomes in type2diabetes has been a matter of controversy over the years.Although epidemiological evidence exists in favor of an adverse role of poor glucose control on cardiovascular events,intervention trials have been less conclusive.The Action to Control Cardiovascular Risk in Diabetes(ACCORD)study,the Action in Diabetes and Vascular Disease(ADV ANCE)study,and the Veterans Affairs Diabetes Trial(V ADT)have shown no beneficial effect of intensive glucose control on primary cardiovascular endpoints in type2diabetes.However,subgroup analysis has provided evidence suggesting that the potential beneficial effect largely depends on patients’characteristics,including age,diabetes duration,previous glucose control,presence of cardiovascular disease,and risk of hypoglycemia.The benefit of strict glucose control on cardiovascular outcomes and mortality may be indeed hampered by the extent and frequency of hypoglycemic events and could be enhanced if glucose-lowering medications,capable of exerting favorable effects on the cardiovascular system,were used.This review examines the relationship between intensive glucose control and cardiovascular outcomes in type2diabetes,addressing the need for individualization of glucose targets and careful consideration of the benefit/risk profile of antidiabetes medications.Keywords:type2diabetes;HbA1c;macrovascular disease;blood pressure;lipids;hypoglycemia;glucagon-like peptide-1;ADV ANCE;ACCORD;V ADTIntroductionCardiovascular disease(CVD)is the major cause of death in patients with type2diabetes(T2D), as more than60%of T2D patients die of my-ocardial infarction(MI)or stroke,and an even greater proportion of patients have serious burden-some complications.1The impact of glucose low-ering on cardiovascular complications is a hotly debated issue.The United Kingdom Prospective Di-abetes Study(UKPDS)was thefirst clinical trial to provide key evidence of the importance of using in-tensive therapy for diabetes control in individuals with newly diagnosed T2D.However,although the insulin or sulphonylurea-based intensified glucose-control treatment was effective in reducing the risk of major microvascular endpoints,the effects on CVD risk were modest and did not reach statisti-cal significance.2Recent large clinical trials(often referred to as“megatrials”),the Action in Diabetes and Vascular Disease(ADV ANCE),3Action to Con-trol Cardiovascular Risk in Diabetes(ACCORD),4 and the Veterans Affairs Diabetes Trial(V ADT),5 reported no significant decrease in primary cardio-vascular endpoints with intensive glucose control. In the ADV ANCE study,11,140type2diabet-ics were randomly assigned to receive either stan-dard or intensive glucose control,defined as the use of gliclazide plus any other drug required to achieve a glycosylated hemoglobin(HbA1c)level of 6.5%or less(Table1).3After a median follow-up of 5.0years,the mean HbA1c was lower in thedoi:10.1111/nyas.12044Giorgino et al.Intensive glucose control and cardiovascular risk Table1.Age,diabetes duration,median follow-up,HbA1c values,and outcomes in the ACCORD and ADV ANCE studies and the V ADTDiabetes HbA1c(%):duration(year):Median intensive Primary All-cause Age intensive follow-up History versus endpoint HR mortality HR Study(year)versus standard(year)of CVD standard Primary endpoint(95%CI)(95%CI)ACCORD(n=10,251)62.2±6.810vs.10 3.435% 6.4vs.7.5Nonfatal MI,nonfatalstroke,or death fromCVD0.90(0.78–1.04) 1.22(1.01–1.46)ADV ANCE (n=11,140)66±6.08.0±6.4vs.7.9±6.35.032%6.53±0.91vs.7.30±1.26Death fromcardiovascular causes,nonfatal MI,ornonfatal stroke0.94(0.84–1.06)0.93(0.83–1.06)V ADT(n=1,791)60±9.011.5±8vs.11.5±75.640%6.9vs.8.4MI,stroke,death fromCVD,CHF,surgery forvascular disease,inoperable CAD,oramputation forischemic gangrene0.88(0.74–1.05) 1.07(0.81–1.42)intensive control group(6.5%vs.7.3%),with a reduction in the incidence of combined major macrovascular and microvascular events primarily because of a reduction in the incidence of nephropa-thy.There were no significant effects of the inten-sive glucose control on major macrovascular events, death from cardiovascular causes,or death from any cause.Similarly,in the V ADT,1,791suboptimally controlled type2diabetics,40%with established CVD,were randomized to receive either intensive glucose control,targeting an absolute reduction of 1.5%in HbA1c levels,or standard glucose control (Table1).5After a median follow-up of5.6years, HbA1c was lower in the intensive-therapy group (6.9%vs.8.4%).Nevertheless,there was no sig-nificant difference between the two groups in the incidence of major cardiovascular events,or in the rate of death from any cause.ACCORD was another study designed to determine whether intensive glu-cose control would reduce the rate of cardiovas-cular events(Table1).4In this study,10,251type 2diabetics with median baseline HbA1c of8.1% were randomly assigned to receive either intensive therapy targeting an HbA1c level within the normal range,that is,below6.0%,or standard therapy tar-geting HbA1c between7.0%and7.9%.The primary outcome was a composite of nonfatal MI,nonfatal stroke,or death from cardiovascular causes.Even though the rate of nonfatal MI was significantly lower in the intensive therapy arm,thefinding of higher all-cause and cardiovascular cause mortal-ity in this group led to discontinuation of the in-tensive therapy after a mean follow-up of3.5years (Fig.1).Notably,hypoglycemia requiring assistance and weight gain of more than10kg were more fre-quent in the intensive therapy group.The results of the ACCORD study raised concern about not only the effectiveness but also the safety of inten-sive glycemic control in type2diabetics.Prespec-ified subgroup analysis of the participants in this trial suggested that patients in the intensive group without history of cardiovascular event before ran-domization or whose baseline HbA1c level was8.0% or less may have had fewer fatal or nonfatal car-diovascular events than did patients in the standard therapy group.Several recent meta-analyses of randomized con-trolled trials have also investigated the effects of intensive glucose lowering on all-cause mortal-ity,cardiovascular death,and vascular events in T2D.6–10In the largest and most recent meta-analysis by Boussageon et al.,13studies were in-cluded.6Of the34,533patients evaluated,18,315 received intensive glucose-lowering treatment and 16,218standard treatment.Intensive treatment did not significantly affect all-cause mortality or cardiovascular death.The results of this meta-analysis showed limited benefit of intensive glucose-lowering therapy on all-cause mortality and deaths from cardiovascular causes,and a10%reduction in the risk of microalbuminuria.6Results from other meta-analyses have also shown no effects ofIntensive glucose control and cardiovascular risk Giorgino etal.Figure 1.Effects of intensive glucose control on all-cause and cardiovascular mortality and myocardial infarction in the ACCORD study.CV,cardiovascular.∗P<0.05.Adapted from Ref.4.intensive glucose control on all-cause or cardiovas-cular mortality,while indicating a modest15–17% reduction in the incidence of nonfatal MI in these cohorts.7–10Several potential factors could have contributed to limit the potential benefit of intensive glucose-lowering therapies on CVD prevention in T2D in-dividuals in studies such as the ACCORD and AD-V ANCE and the V ADT:(1)Concomitant targeting of other potentiallymore potent cardiovascular risk factors,suchas blood pressure and lipids,might havedampened the favorable effects of controllinghyperglycemia.(2)Intensive control of hyperglycemia could havebeen directed to patients unable to exhibit theexpected benefit due to their specific clinicalcharacteristics(“wrong”patients).(3)Limited benefit might have derived from usingglucose-lowering drugs with no favorable im-pact on the global cardiovascular risk profile.(4)Glucose-lowering drugs might have producedadverse effects on the cardiovascular systemby inducing weight gain and hypoglycemicevents,resulting in somewhat increased riskfor CVD and mortality(“imperfect”drugs).(5)Excess mortality might have potentially re-sulted from using too many drugs and/or toocomplex drug regimens,leading to undesir-able drug–drug interactions with a potentiallyharmful impact on patients’health.These diverse factors and their potential role in the relationship between intensive glucose control and CVD/mortality are outlined in Figure2,and will be discussed individually below.Limited benefit due to other therapies Although several studies have focused on intensive glycemic control to decrease the risks of macrovas-cular and microvascular diseases in T2D,glucose control is only one of the factors to be considered. Comprehensive risk factor management,including blood pressure control,lipid management,weight reduction in overweight or obese individuals,and smoking cessation,are also needed.The results of the ACCORD and ADV ANCE studies and the V ADT should be interpreted in the context of comprehen-sive care of patients with diabetes.Interventions for simultaneous optimal control of comorbidities of-ten present in type2diabetics,such as hyperten-sion and hyperlipidemia,have been shown to be a more effective strategy in reducing cardiovascular risk than targeting only blood glucose levels per se.11 Evidence for an aggressive approach to lipid and blood pressure control was supported by the re-sults from the Steno-2study.11,12In Steno-2,inves-tigators used intensified multifactorial intervention with improved glycemia,renin–angiotensin system blockers,aspirin,and lipid-lowering agents and evaluated whether this approach would have an ef-fect on the rates of death from cardiovascular causes and from any cause.11,12The primary endpoint at 13.3years of follow-up was the time to death from any cause.Intensive therapy was also associated with a significantly lower risk of death from cardiovascu-lar causes and of cardiovascular events.The Steno-2study did demonstrate a difference in levels of glycemia achieved when compared with the AC-CORD study:4,11HbA1c was a mean of8.4%at study entry and7.9%at end of study intervention for the intensively treated group,whereas it was8.8%at baseline and9.0%at study end for conventional treatment.In addition,only a limited proportion of subjects in the intensively treated group reached an HbA1c level of less than6.5%(i.e.,∼15%),and this proportion was not statistically different than in the conventionally treated group,indicating poor success in achieving the prespecified glucose target and somewhat reducing the relevance of the spe-cific intervention on the hyperglycemia component for CVD and microvascular disease prevention.The observational study has continued,and the differ-ences observed in glycemia between intensive andGiorgino et al.Intensive glucose control and cardiovascularriskFigure2.Relationship between intensive glucose control and cardiovascular outcomes and mortality in the ACCORD study and other megatrials.The potential mechanisms affecting this relationship and limiting the clinical benefit are outlined in the box on the left.CV,cardiovascular;CVD,cardiovascular disease.conventional treatments are much less than at endof intervention.Nevertheless,over the long-termperiod of follow-up,intensive intervention with avaried drug regimen and lifestyle modification hadsustained beneficial effects with respect to vascularcomplications and rates of death from any cause andfrom cardiovascular causes.12Blood pressureThe ACCORD study also had an embedded bloodpressure trial that examined whether blood pressurelowering to systolic blood pressure(SBP)less than120mm Hg provided greater cardiovascular protec-tion than a SBP of130–140mm Hg in T2D patientsat high risk for CVD.13A total of4,733participantswere randomly assigned to intensive therapy(SBP <120mm Hg)or standard therapy(SBP<140mm Hg),with the mean follow-up being4.7years.Theblood pressure levels achieved in the intensive andstandard groups were119/64mm Hg and133/70mm Hg,respectively;this difference was attainedwith an average of3.4medications per participantin the intensive group and2.1in the standard ther-apy group.The intensive antihypertensive therapyin the ACCORD blood pressure trial did not consid-erably reduce the primary cardiovascular outcomeor the rate of death from any cause.However,theintensive arm of blood pressure control reduced therate of total stroke and nonfatal stroke,with the estimated number needed to treat with intensive blood pressure therapy to prevent one stroke over five years being89.There were indicators of possi-ble harm associated with intensive blood pressure lowering(SBP<120mm Hg),including a rate of serious adverse events in the intensive arm.It should be noted that of the subjects investigated in the ACCORD glucose control trial∼85%were on antihypertensive medications,and their blood pres-sure levels were126.4/66.9and127.4/67.7in the intensive and standard groups,respectively,a differ-ence that was statistically significant(P<0.001) (Table2).4Thus,the effects of intensive glucose lowering on the CVD and other outcomes were ex-amined in a context in which blood pressure was being actively targeted,with possible differences between the intensively and conventionally treated cohorts.The ADV ANCE study also included a blood pres-sure intervention trial.In this study,treatment with an angiotensin converting enzyme(ACE)inhibitor and a thiazide-type diuretic reduced the rate of death but not the composite macrovascular out-come.However,this trial had no specified targets for the randomized comparison,and the mean SBP in the intensive group(135mm Hg)was not as low as the mean SBP in the ACCORD standard-therapy group.14However,a post hoc analysis of blood pres-sure control in6,400patients with diabetes andIntensive glucose control and cardiovascular risk Giorgino et al. Table2.Blood pressure and lipid levels and use of statin,antihypertensive medications,and aspirin in the ACCORD and ADV ANCE studies and the V ADT participants at study end(adapted from Refs.3–5)ACCORD(n=10,251)aADV ANCE(n=11,140)b V ADT(n=1,791)cStandard Intensive Standard Intensive Standard Intensive Blood pressure(mm Hg)Systolic128±16126±17137±18135±17125±15127±16 Diastolic68±1067±1074±1073±1069±1068±10 Cholesterol(mg/dL)LDL87±3387±33103±41102±3880±3180±33 HDL49±13(♂)49±13(♂)48±1448±1441±1240±1140±11(♀)40±10(♀)Total164±42163±42153±40150±40 Triglycerides(mg/dL)166±114160±125161±102151±94159±104151±173 On statin(%)888848468385 On antihypertensivemedications(%)858389887576 On aspirin(%)767655578586a Intensive(target HbA1c<6%)vs.standard(HbA1c7–7.9%).b Intensive(target HbA1c<6.5%)vs.standard(HbA1c>6.5%).c Intensive(target HbA1c4.8–6.0%)vs.standard(HbA1c8–9.0%).Abbreviations:HDL,high-density lipoprotein;LDL,low-density lipoprotein;♂,men;♀,women.coronary artery disease enrolled in the International Verapamil/Trandolapril Study(INVEST)demon-strated that tight control(<130mm Hg)was not associated with improved cardiovascular outcomes compared with usual care(130–140mm Hg).15In the V ADT,blood pressure,lipids,diet,and lifestyle were treated identically in both arms.By improving blood pressure control in an identical manner in both glucose arms,the V ADT excluded the effect of blood pressure differences on cardiovas-cular events between treatment arms and reduced the overall risk of macrovascular complications dur-ing the trial.5Participants in the V ADT(n=1,791) with hypertension(72.1%of total)received stepped treatment to maintain blood pressure below the tar-get of130/80mm Hg in standard and intensive glycemic treatment groups.Blood pressure levels of all subjects at baseline and on-study were analyzed to detect associations with cardiovascular risk.The primary outcome was the time from randomiza-tion to thefirst occurrence of MI,stroke,conges-tive heart failure,surgery for vascular disease,in-operable coronary disease,amputation for ischemic gangrene,or cardiovascular death.From data analy-sis,increased risk of cardiovascular events with SBP ≥140mm Hg emphasizes the need for treatment of systolic hypertension.Also,this study for the first time demonstrated that diastolic blood pressure (DBP)<70mm Hg in T2D patients was indepen-dently associated with elevated cardiovascular risk, even when SBP was on target.16Lipid profileIt has been widely demonstrated that intensive targeting of low-density lipoprotein(LDL)choles-terol contributes to CVD prevention in T2D.As a consequence,most guidelines suggest a target LDL-cholesterol level below100mg/dL as the primary goal in T2D individuals,with the option of achieving an LDL-cholesterol level below70mg/dL in those with overt CVD.Regarding the overall lipid-lowering approach in the ACCORD glucose control study,it should be observed that mean LDL cholesterol was below90mg/dL in both the intensive and standard glucose control arms,and that88%of subjects were on statin therapy.4Thus, the results from this study do not clarify whether lipid control and glycemic control,respectively,areGiorgino et al.Intensive glucose control and cardiovascularriskFigure3.All-cause mortality in intensive versus standard glycemia groups according to use of antihypertensive medications, statins,and aspirin in the ACCORD and ADV ANCE studies.∗P=0.0305for subjects on aspirin versus subjects not on aspirin. Adapted from Refs.3and18.related or synergistic,since the large majority of enrolled subjects were already receiving a lipid control regimen.Data from the Steno-2trial on combined control of glucose,lipids,and blood pressure levels demonstrated significant short-and long-term benefits from this multifactorial approach.11In the study,the effect seemed to be cumulative rather than synergistic.Recent results from the ACCORD-LIPID study indicate that intensive lipid control(i.e.,addition of afibrate to statin therapy)does not reduce car-diovascular events.17Specifically,the lipid-lowering arm of ACCORD failed to demonstrate any ben-efit of add-on therapy with fenofibrate to LDL-lowering treatment with HMG-CoA reductase in-hibitors(statins)on vascular outcomes in patients with diabetes.However,data from earlier studies and from a subgroup analysis of ACCORD indi-cate a probable benefit of adding treatment with fibric acid derivatives to individuals with persis-tently elevated triglyceride levels and low high-density lipoprotein(HDL)cholesterol despite statin therapy.17In the ACCORD and ADV ANCE studies and the V ADT,a large proportion of subjects,ranging from55%to85%,were also treated with aspirin (Table2).Thus,results from ADV ANCE,ACCORD, and V ADT suggest that a large proportion of par-ticipants in these trials,which were being treated intensively or less intensively for glucose targets, received extensive antihypertensive,lipid-lowering, and antiplatelet medications.Median levels of SBP, SDP,and LDL cholesterol in these cohorts were also indicative of a significant proportion of them be-ing adequately controlled for blood pressure and lipid targets(Table2).Therefore,the possibility ex-ists that active interventions for simultaneous con-trol of hypertension and hyperlipidemia and use of aspirin may have affected the impact of intensive glucose control on cardiovascular outcomes in the megatrials.Subgroups analyses,however,do not ap-parently support this conclusion(Fig.3).Whether patients were on antihypertensive or lipid-lowering medications was not associated with a different out-come of the intensive glucose control on mortality in ACCORD and ADV ANCE patients,even thoughIntensive glucose control and cardiovascular risk Giorgino et al.the different groups were largely unbalanced in size. Only aspirin use in the ACCORD study seemed to modulate the effects of intensive versus standard glucose control on mortality.18The“wrong patient”concept and the need for individualization of glucose targets The ADV ANCE and ACCORD studies and the V ADT provide conflicting evidence of mortality risk with intensive glycemic control.These trials showed an approximate15%reduction in nonfatal MI with no benefit or harm in all-cause or cardiovascular mortality.Potential explanations for the lack of im-pact of intensive glycemic control on CVD can be found in the patients’characteristics.Indeed,these studies were of shorter duration and enrolled gener-ally older patients than previous studies,including the DCCT and UKPDS in which the intensive con-trol had shown better outcomes.In addition,mean diabetes duration was longer and a greater portion of patients had established CV disease in the mega-trials(approximately32–40%)than in earlier trials (Table1).It is also possible that the follow-up of these studies was too short to detect a clinical ben-efit.Consistent with this hypothesis,in the UKPDS no macrovascular benefit was noted in the inten-sive control arm in thefirst10years of follow-up. Nevertheless,posttrial monitoring for an additional 10years(UKPDS80)revealed a15%risk reduction in MI and13%reduction in all-cause mortality in the intensive treatment group.19A possible explanation is that the wrong patients were investigated in the megatrials(Fig.2).Indeed, the population of the ACCORD study may not rep-resent the average patient with T2D in clinical prac-tice.Participants in this trial had T2D on average for10years at the time of enrollment,had higher HbA1c levels than most type2diabetic patients in the United States and most Western countries to-day(average of8.2%at baseline),and had known heart disease or at least two risk factors in addi-tion to diabetes,such as high blood pressure,high cholesterol levels,obesity,and smoking.4In the UKPDS,the benefits of intensive glycemic control on CVD were observed only after a long duration of intervention in newly diagnosed younger patients.2 In older patients with T2D with longer disease dura-tion,atherosclerotic disease may already have been established and thus intensive glucose control may have had little benefit.Conversely,patients with shorter disease duration,lower HbA1c,and/or lack of established CVD might have benefited signifi-cantly from more intensive glycemic control.20,21 Relevant to this concept,the V ADT showed that ad-vanced CVD,as demonstrated by computed tomog-raphy(CT)-detectable coronary artery calcium,was associated with negative outcomes.In a substudy co-hort of301T2D participants in V ADT,the ability of intensive glucose therapy compared with standard therapy to reduce cardiovascular events was exam-ined based on the extent of coronary atherosclerosis as measured by a CT-detectable coronary artery cal-cium score(CAC).The data showed that there was a progressive diminution of the benefit of intensive glucose control with increasing CAC.In patients with CAC≤100,1of52individuals experienced an event(HR for intensive therapy=0.08;range, 0.008–0.770;P=0.03),whereas11of62patients with a CAC>100had an event(HR=0.74;range, 0.46–1.20;P=0.21).Thus,this subgroup analy-sis indicates that intensive glycemic therapy may be most effective in those with less extensive coronary atherosclerosis.22Why does intensive treatment of hyperglycemia appear to be ineffective in reducing cardiovascular events in T2D with advanced atherosclerosis?Two potential mechanisms may be involved.First,once the atherosclerotic plaque has developed,modified lipoproteins,activated vascular cells,and altered im-mune cell signaling may generate a self-propagating process that maintains atherogenesis,even in the face of improved glucose control.Second,advanced glycosylation end-product formation,which may be involved in CVD,is not readily reversible and may require more than three tofive years of in-tensive glucose control to be reverted.Thus,the presence of multiple cardiovascular risk factors or established CVD may have reduced the benefits of intensive glycemic control in the high-risk cohort of ACCORD,ADV ANCE,and V ADT compared with the low-risk population of the UKPDS cohort,of whom only a minority had prior CVD.23The pres-ence of long-standing disease and prolonged prior poor glycemic control may be additional factors ac-celerating the progression of atherosclerotic lesions in T2D.The goal of individualizing HbA1c targets has gained more attraction after these recent clinical trials in older patients with established T2D failed to show a benefit from intensive glucose-loweringGiorgino et al.Intensive glucose control and cardiovascular riskTable3.Potential criteria for individualization of glucose targets in type2diabetes2–5,25,29HbA1c HbA1c Criterion<6.5–7.0%7.0–8.0% Age(years)<55>60 Diabetes duration(years)<10>10Life expectancy(years)>5<5 Possibility to performIGC for>5yearsYes No Usual HbA1c level(%)<8.0>8.0 CVD No Yes Prone to hypoglycemia No No Reduction of HbA1clevel upon IGCYes NoAbbreviations:CVD,cardiovascular disease;IGC,in-tensive glucose control.therapies on CVD outcomes.Recommendations suggest that the goals should be individualized,such that(1)certain populations(children,pregnant women,and elderly patients)require special con-siderations and(2)more stringent glycemic goals (i.e.,a normal HbA1c<6.0%)may further re-duce complications at the cost of increased risk of hypoglycemia.24For the latter,the recommen-dations also suggest that less intensive glycemic goals may be indicated in patients with severe or frequent hypoglycemia.With regard to the less intensive glycemic goals,perhaps consideration should be given to the high-risk patient with mul-tiple risk factors and CVD,as evaluated in the ACCORD study.4Thus,aiming for a HbA1c of 7.0–8.0%may be a reasonable goal in patients with very long duration of diabetes,history of se-vere hypoglycemia,advanced atherosclerosis,sig-nificant comorbidities,and advanced age/frailty (Table3),even though with what priority each one of these criteria should be considered is not clear at present(current recommendations from scientific societies also do not provide this spe-cific information).In younger patients without documented macrovascular disease or the above-mentioned conditions,the goal of attaining an HbA1c<6.5–7.0%may provide long-lasting bene-fits.In patients with limited life expectancy,more liberal HbA1c values may be pursued.In deter-mining the HbA1c target for CVD prevention,one should consider that at least three tofive years are usually required before possible differences in the incidence of nonfatal MI in T2D are observed.2 Thus,a clinical setting that allows tight glucose control to be implemented for this period of time should be available.Finally,an excess mortality was observed in the ACCORD study in those T2D in-dividuals who showed an unexpected increase in HbA1c levels upon institution of intensive glucose control.25Accordingly,patients exhibiting the pat-tern of worsening glycemic control when exposed to intensive treatment should be set at higher glucose targets(Table3).The above guidelines are apparently incorpo-rated into the updated version of the American Diabetes Association and the European Associa-tion for the Study of Diabetes recommendations on the management of hyperglycemia in nonpreg-nant adults with T2D.The new recommendations are less prescriptive and more patient centered.In-dividualized treatment is explicitly defined as the cornerstone of success.Treatment strategies should be tailored to individual patient needs,preferences, and tolerances and based on differences in age and disease course.Other factors affecting individual-ized treatment plans include specific symptoms,co-morbid conditions,weight,race/ethnicity,sex,and lifestyle.26Current“imperfect”glucose-lowering drugsThe inability of ACCORD,ADV ANCE,and V ADT to demonstrate significant reductions in CVD out-comes with intensive glycemic control also suggests that current pharmacological tools for treating hy-perglycemia in patients with more advanced T2D may have counterbalancing consequences for the cardiovascular system.The available agents used to treat diabetes have not been conclusively shown to reduce macrovascular disease and,in some in-stances,their chronic use may promote negative cardiovascular effects in diabetic subjects despite improvement of hyperglycemia.Importantly,these adverse cardiovascular side effects appear in several instances to be directly due to the mode of drug ac-tion.Selection of a treatment regimen for patients with T2D includes evaluation of the effects of med-ications on overall cardiovascular risk.27 Sulfonylureas have the benefit of acting rapidly to lower glucose levels but,unfortunately,on a。
a rXiv:h ep-th/04968v16Se p24An Introduction to Free Higher-Spin Fields N.Bouatta a ,p`e re a ∗and A.Sagnotti b †a Physique th´e orique et math´e matique and International Solvay Institutes Universit´e libre de Bruxelles Campus de la Plaine CP 231,B-1050Bruxelles,Belgium b Dipartimento di Fisica Universit`a di Roma ”Tor Vergata”INFN,Sezione di Roma ”Tor Vergata”Via della Ricerca Scientifica 1,00133Roma,Italy February 1,2008Abstract In this article we begin by reviewing the (Fang-)Fronsdal construction and the non-local geometric equations with unconstrained gauge fields and parameters built by Fran-cia and the senior author from the higher-spin curvatures of de Wit and Freedman.Wethen turn to the triplet structure of totally symmetric tensors that emerges from free String Field Theory in the α′→0limit and to its generalization to (A)dS backgrounds,and conclude with a discussion of a simple local compensator form of the field equations that displays the unconstrained gauge symmetry of the non-local equations.Based on the lectures presented by A.Sagnotti at the First Solvay Workshop on Higher-Spin Gauge Theories held in Brussels on May 12-14,2004PACS numbers:11.15.-q,11.10.-z,11.10.Kk,11.10.Lm,11.25.-w,11.30.Ly,02.40.Ky ULB-TH/04-22ROM2F-04/231IntroductionThis article reviews some of the developments that led to the free higher-spin equations introduced by Fang and Fronsdal[1,2]and the recent constructions of free non-local geo-metric equations and local compensator forms of[3–5].It is based on the lectures delivered by A.Sagnotti at the First Solvay Workshop,held in Brussels on May2004,carefully edited by the other authors for the online Proceedings.The theory of particles of arbitrary spin was initiated by Fierz and Pauli in1939[6], that followed afield theoretical approach,requiring Lorentz invariance and positivity of the energy.After the works of Wigner[7]on representations of the Poincar´e group,and of Bargmann and Wigner[8]on relativisticfield equations,it became clear that the positivity of energy could be replaced by the requirement that the one-particle states carry a unitary representation of the Poincar´e group.For massivefields of integer and half-integer spinrepresented by totally symmetric tensorsΦµ1...µs andΨµ1...µs,the former requirements areencoded in the Fierz-Pauli conditions( −M2)Φµ1...µs=0,∂µ1Φµ1...µs =0,(i∂/−M)Ψµ1...µs=0,(1)∂µ1Ψµ1...µs=0.(2)The massivefield representations are also irreducible when a(γ-)trace conditionηµ1µ2Φµ1µ2...µs =0,γµ1Ψµ1...µs=0.(3)is imposed on thefields.A Lagrangian formulation for these massive spin s-fields wasfirst obtained in1974by Singh and Hagen[9],introducing a sequence of auxiliary tracelessfields of ranks s−2,s−3, ...0or1,all forced to vanish when thefield equations are satisfied.Studying the corresponding massless limit,in1978Fronsdal obtained[1]four-dimensional covariant Lagrangians for masslessfields of any integer spin.In this limit,all the auxiliary fields decouple and may be ignored,with the only exception of thefield of rank s−2,while the two remaining traceless tensors of rank s and s−2can be combined into a single tensor ϕµ1···µssubject to the unusual“double trace”conditionηµ1µ2ηµ3µ4ϕµ1···µs=0.(4) Fang and Fronsdal[2]then extended the result to half-integer spins subject to the peculiar “tripleγtrace”conditionγµ1γµ2γµ3ψµ1···µs=0.(5) It should be noted that the description of masslessfields in four dimensions is partic-ularly simple,since the massless irreducible representations of the Lorentz group SO(3,1)↑are exhausted by totally symmetric tensors.On the other hand,it is quite familiar from supergravity[10]that in dimensions d>4the totally symmetric tensor representations do not exhaust all possibilities,and must be supplemented by mixed ones.For the sake of sim-plicity,in this paper we shall confine our attention to the totally symmetric case,focussing on the results of[3–5].The extension to the mixed-symmetry case was originally obtained in[11],and will be reviewed by C.Hull in his contribution to these Proceedings[12].12From Fierz-Pauli to FronsdalThis section is devoted to some comments on the conceptual steps that led to the Fronsdal [1]and Fang-Fronsdal[2]formulations of the free high-spin equations.As afirst step, we describe the salient features of the Singh-Hagen construction of the massive freefield Lagrangians[9].For simplicity,we shall actually refer to spin1and2fields that are to satisfy the Fierz-Pauli conditions(1)-(2),whose equations are of course known since Maxwell and Einstein.The spin-1case is very simple,but the spin-2case already presents the key subtlety.The massless limit will then illustrate the simplest instances of Fronsdal gauge symmetries.The Kaluza-Klein mechanism will be also briefly discussed,since it exhibits rather neatly the rationale behind the Fang-Fronsdal auxiliaryfields for the general case. We then turn to the novel features encountered with spin3fields,before describing the general Fronsdal equations for massless spin-s bosonicfields.The Section ends with the extension to half-integer spins.2.1Fierz-Pauli conditionsLet usfirst introduce a convenient compact notation.Given a totally symmetric tensorϕ, we shall denote by∂ϕ,∂·ϕandϕ′(or,more generally,ϕ[p])its gradient,its divergence and its trace(or its p-th trace),with the understanding that in all cases the implicit indices are totally symmetrized.Singh and Hagen[9]constructed explicitly Lagrangians for spin-sfields that give the correct Fierz-Pauli conditions.For spin1fields,their prescription reduces to the Lagrangian L spin1=−12(∂·Φ)2−M22(∂µΦνρ)2+α2(Φµν)2,(8)where thefieldΦµνis traceless.The corresponding equation of motion readsΦµν−αDηµν∂·∂·Φ−M2Φµν=0,(9)whose divergence implies1−α2+1Notice that,in deriving these equations,we have made an essential use of the condition thatΦbe traceless.In sharp contrast with the spin1case,however,notice that now the transversality condition is not recovered.Choosingα2would eliminate some terms,but one would still need the additional constraint∂·∂·Φ=0.Since this is not a consequence of thefield equations,the naive system described byΦand equipped with the Lagrangian L spin2is unable to describe the free spin2field.One can cure the problem introducing an auxiliary scalarfieldπin such a way that the condition∂·∂·Φ=0be a consequence of the Lagrangian.Let us see how this is the case, and add to(8)the termL add=π∂·∂·Φ+c1(∂µπ)2+c2π2,(11) where c1,2are a pair of constants.Taking twice the divergence of the resulting equation for Φµνgives (2−D) −DM2 ∂·∂·Φ+(D−1) 2π=0,(12) while the equation for the auxiliary scalarfield reduces to∂·∂·Φ+2(c2−c1 )π=0.(13) Eqs.(12)and(13)can be regarded as a linear homogeneous system in the variables∂·∂·Φandπ.If the associated determinant never vanishes,the only solution will be precisely the missing condition∂·∂·Φ=0,together with the condition that the auxiliaryfield vanish as well,π=0,and as a result the transversality condition will be recovered.The coefficients c1and c2are thus determined by the condition that the determinant of the system∆=−2DM2c2+2((2−D)c2+DM2c1) (14)−(2(2−D)c1−(D−1)) (15) be algebraic,i.e.proportional to the mass M but without any occurrence of the D’Alembert operator .Hence,for D>2,c1=(D−1)2(D−2)2.(16)The end conclusion is that the complete equations implyπ=0,∂·∂·Φ=0,(17)∂·Φν=0, Φµν−M2Φµν=0,(18) the Fierz-Pauli conditions(1)and(2),so that the inclusion of a single auxiliary scalarfield leads to an off-shell formulation of the free massive spin-2field.32.2“Fronsdal”equation for spin2We can now take the M→0limit,following in spirit the original work of Fronsdal[1].The total Lagrangian L spin2+L add then becomesL=−12(D−2)(∂µπ)2,(19) whose equations of motion areΦµν−∂µ∂·Φν−∂ν∂·Φµ+2D−2π−∂·∂·Φ=0.(21)The representation of the massless spin2-gaugefield via a traceless two-tensorΦµνand a scalarπmay seem a bit unusual.In fact,they are just an unfamiliar basis offields, and the linearized Einstein gravity in its standard form is simply recovered once they are combined in the unconstrained two-tensorϕµν=Φµν+12(∂µϕνρ)2+(∂·ϕµ)2+12ηµνF′=0,that only when combined with theirtrace imply the previous equation,Fµν=0.The massless case is very interesting by itself,since it exhibits a relatively simple instance of gauge symmetry,but also for deducing the corresponding massivefield equations via a proper Kaluza-Klein reduction.This construction,first discussed in[13],is actually far simpler than the original one of[9]and gives a rationale to their choice of auxiliaryfields.Let us content ourselves with illustrating the Kaluza-Klein mechanism for spin1fields. To this end,let us introduce afield1A M living in D+1dimensions,that decomposes as A M=(Aµ(x,y),π(x,y)),where y denotes the coordinate along the extra dimension.One can expand these functions in Fourier modes in y and a single massive mode corresponding to the D-dimensional mass m,letting for instance A M=(Aµ(x),−iπ(x))exp(imy)wherethe judicious insertion of the factor−i will ensure that thefieldπbe real.The D+1-dimensional equation of motion and gauge transformationA M−∂M∂·A=0,(26)δA M=∂MΛ(27) then determine the D-dimensional equations−m2 Aµ−∂µ(∂·A+mπ)=0,−m2 π+m(∂·A+mπ)=0,(28)δAµ=∂µΛ,δπ=−mΛ,where the leftover massive gauge symmetry,known as a Stueckelberg symmetry,is inherited from the higher dimensional gauge symmetry.Fixing the gauge so thatπ=0,one canfinally recover the Proca equation(7)for Aµ.The spin-2case is similar,and the proper choice is ϕMN(ϕµν,−iϕµ,−ϕ)exp(imy),so that the resulting gauge transformations readδϕMN=∂MΛN+∂NΛM,δϕµν=∂µΛν+∂νΛµ,δϕµ=∂µΛ−mΛµ,(29)δϕ=−2mΛ.In conclusion,everything works as expected when the spin is lower than or equal to two, and the Fierz-Pauli conditions can be easily recovered.However,some novelties do indeed arise when then spin becomes higher than two.Let us try to generalize the theory to spin-3fields by insisting on the equationsFµνρ≡ ϕµνρ−(∂µ∂·ϕνρ+perm)+(∂µ∂νϕ′ρ+perm)=0,(30)δϕµνρ=∂µΛνρ+∂νΛρµ+∂ρΛµν,(31) that follow the same pattern,where perm denotes cyclic permutations ofµνρ.In this formulation,there are no auxiliaryfields and the traceϕ′of the gaugefield does not vanish.Let usfirst remark that,under a gauge transformation,F transforms according toδFµ1µ2µ3=3∂µ1∂µ2∂µ3Λ′.(32)Therefore,F is gauge invariant if and only if the gauge parameter is traceless,Λ′=0.This rather unnatural condition will recur systematically for all higher spins,and will constitute a drawback of the Fronsdal formulation.We can also see rather neatly the obstruction to a geometric gauge symmetry of the spin-3Fronsdal Lagrangian along the way inspired by General Relativity.Indeed,the spin-3Fronsdal equation differs in a simple but profound way from the two previous cases,since both for spin1and for spin2all lower-spin constructs built out of the gaugefields are present,while for spin3only constructs of spin3(ϕµνρ),spin2(∂·ϕµν)and spin1(ϕ′µ)are5present.Actually,de Wit and Freedman[14]classified long ago the higher-spin analogs ofthe spin-2Christoffel connectionΓmu;ν1ν2:they are a hierarchy of connectionsΓµ1...µk;ν1ν2ν3,with k=1,...(s−1),that contain k derivatives of the gaugefield.They also noticed thatthe analog of the Riemann tensor,that for spin3would beΓµ1µ2µ3;ν1ν2ν3,would in generalcontain s derivatives of the gaugefield,and related the Fronsdal operator F to the trace of the second connection.One way to bypass the problem is to construct afield equation with two derivatives depending on the true Einstein tensor for higher spins,that however,as we have anticipated, contains s derivatives in the general case.This can be achieved,but requires that non-local terms be included both in thefield equations and in the Lagrangian.This approach will be further developed in section3.Another option is to compensate the non-vanishing term in the right-hand side of(32)by introducing a newfield,a compensator.As we shall see,this possibility is actually suggested by String Theory,and will be explained in section4.3.But before describing these new methods,let usfirst describe the general Fronsdal formulation for arbitrary spin.In this way we shall clearly identify the key role of the trace condition on the gauge parameter that we already encountered for spin3and of the double trace condition on thefield,that willfirst present itself for spin4.2.3Fronsdal equations for arbitrary integer spinWe can now generalize the reasoning of the previous paragraph to arbitrary integer spins. Since we shall use extensively the compact notation of[3],omitting all indices,is it useful to recall the following rules:(∂pϕ)′= ∂p−2ϕ+2∂p−1∂·ϕ+∂pϕ′,∂p∂q= p+q p ∂p+q,∂·(∂pϕ)= ∂p−1ϕ+∂p∂·ϕ,(33)∂·ηk=∂ηk−1,ηk T(s) ′=k[D+2(s+k−1)]ηk−1T(s)+ηk T′(s).In this compact form,the generic Fronsdal equation and its gauge transformations read simplyF≡ ϕ−∂∂·ϕ+∂2ϕ′=0,(34)δϕ=∂Λ.(35) In order tofind the effect of the gauge transformations on the Fronsdal operators F,one must compute the termsδ(∂·ϕ)= Λ+∂∂·Λ,δϕ′=2∂·Λ+∂Λ′,and the result is,for arbitrary spin,δF= (∂Λ)−∂( Λ+∂∂·Λ)+∂2(2∂·Λ+∂Λ′)=3∂3Λ′,(36)6where we used−∂(∂∂·Λ)=−2∂2(∂·Λ),and∂2∂Λ′=3∂3Λ′.Therefore,in all cases where Λ′does not vanish identically,i.e.for spin s≥3,the gauge invariance of the equations requires that the gauge parameter be tracelessΛ′=0.(37) As second step,one can derive the Bianchi identities for all spins computing the terms∂·F= ∂ϕ′−∂∂·∂·ϕ+∂2∂·ϕ′,F′=2 ϕ′−2∂·∂·ϕ+∂2ϕ′′+∂∂·ϕ′,∂F′=2 ∂ϕ′−2∂∂·∂·ϕ+3∂3ϕ′′+2∂2∂·ϕ′.Therefore,the Fronsdal operator F satisfies in general the”anomalous”Bianchi identities∂·F−12∂3ϕ′′,(38)where the additional term on the rightfirst shows up for spin s=4.In the Fronsdal construction,one is thus led to impose the constraintϕ′′=0for spins s≥4,since the Lagrangians would vary according toδL=δϕ F−12F′=−32η∂·F′,(40)where the last term gives a vanishing contribution toδL if the parameterΛis traceless. To reiterate,this relation is at the heart of the usual restrictions present in the Fronsdal formulation to traceless gauge parameters and doubly tracelessfields,needed to ensure that the variation of the lagrangianδL=δϕG,(41) where G=F−12.4Masslessfields of half-integer spinLet us now turn to the fermionicfields,and for simplicity let us begin with the Rarita-Schwinger equation[15],familiar from supergravity[16]γµνρ∂νψρ=0,(42)that is invariant under the gauge transformationδψµ=∂µǫ,(43)whereγµνρdenotes the fully antisymmetric product of threeγmatrices,normalized to their product when they are all different:γµνρ=γµγνρ−ηµνγρ+ηµργν.(44) Contracting the Rarita-Schwinger equation withγµyieldsγνρ∂νψρ=0,(45)and therefore thefield equation for spin3/2can be written in the alternative form∂/ψµ−∂µψ/=0.(46) Let us try to obtain a similar equation for a spin-5/2field,defining the Fang-Fronsdal operator Sµν,in analogy with the spin-3/2case,asSµν≡i(∂/ψµν−∂µψ/ν−∂νψ/µ)=0,(47) and generalizing naively the gauge transformation toδψµν=∂µǫν+∂νǫµ.(48)Difficulties similar to those met in the bosonic case for spin3readily emerge:this equation is not gauge invariant,but transforms asδSµν=−2i∂ν∂µǫ/,(49)and a similar problem will soon arise with the Bianchi identity.However,as in the bosonic case,following Fang and Fronsdal[2],one can constrain both the fermionicfield and the gauge parameterǫso that the gauge symmetry hold and the Bianchi identity take a non-anomalous form.We can now consider the generic case of half-integer spin s+1/2[2].In the compact notation(33)the equation of motion and the gauge transformation read simplyS≡i(∂/ψ−∂ψ/)=0,(50)δψ=∂ǫ.(51)8SinceδS=−2i∂2ǫ/,(52) to ensure the gauge invariance of thefield equation,one must demand that the gauge parameter beγ-traceless,ǫ/=0.(53) Let us now turn to the Bianchi identity,computing∂·S−12∂/S/,(54)where the last term contains theγ-trace of the operator.It is instructive to consider in detail the individual terms.The trace of S isS′=i(∂/ψ′−2∂·ψ/−∂ψ/′),(55) and therefore,using the rules in(33)−12(∂/∂ψ′−2∂∂·ψ/−2∂2ψ/′).(56) Moreover,the divergence of S is∂·S=i(∂/∂·ψ− ψ/−∂∂·ψ/).(57)Finally,from theγtrace of SS/=i(−2∂/ψ/+2∂·ψ−∂ψ′),(58)one can obtain−12(−2 ψ/+2∂/∂·ψ−∂∂/ψ′),(59) and putting all these terms together yields the Bianchi identity∂·S−12∂/S/=i∂2ψ/′.(60)As in the bosonic case,this identity contains an“anomalous”term thatfirst manifests itself for spin s=7/2,and therefore one is lead in general to impose the“tripleγ-trace”conditionψ/′=0.(61) One can also extend the Kaluza-Klein construction to the spin s+1/2in order to recover the massive Singh-Hagen formulation.The reduction from D+1dimensions to D dimensions will turn the masslessfieldψ(s)D+1into massivefields of the typeψ(s)D,ψ(s−1)Dandψ(s−2)D,while no lower-rankfields can appear because of the tripleγ-trace condition(61).Ina similar fashion,the gauge parameterǫ(s−1)D+1reduces only to a singlefieldǫ(s−1)D,as a resultof theγ-trace condition(53).Gaugefixing the Stueckelberg symmetries one isfinally leftwith only twofieldsψ(s)D andψ(s−2)D that contain precisely theγ-traceless tensors introducedby Singh and Hagen in[9].93Free non-local geometric equationsIn the previous section we have seen that is possible to construct a Lagrangian for higher-spin bosons imposing the unusual Fronsdal constraintsΛ′=0,ϕ′′=0(62) on thefields and on the gauge parameters.Following[3,4],we can now construct higher-spin gauge theories with unconstrained gaugefields and parameters.3.1Non-local Fronsdal-like operatorsWe can motivate the procedure discussingfirst in some detail the relatively simple example of a spin-3field,whereδFµνρ=3∂µ∂ν∂ρΛ′.(63) Our purpose is to build a non-local operator F NL that transforms exactly like the Fronsdal operator F,since the operator F−F NL will then be gauge invariant without any additional constraint on the gauge parameter.One canfind rather simply the non-local constructs13 [∂µ∂·F′νρ+∂ν∂·F′ρµ+∂ρ∂·F′µν],(65)13 [∂µ1∂µ2F′µ3+∂µ2∂µ3F′µ1+∂µ3∂µ1F′µ2]=0,(67)F newµ1µ2µ3≡Fµ1µ2µ3−12(ηµ1µ2F newµ3′+ηµ2µ3F newµ1′+ηµ3µ1F newµ2′),(69)and one can easily verify that∂·Gµ1µ2=0.(70) This identity suffices to ensure the gauge invariance of the Lagrangian.Moreover,we shall see that in all higher-spin cases,the non-local construction will lead to a similar,if10more complicated,identity,underlying a Lagrangian formulation that does not need any double trace condition on the gaugefield.It is worth stressing this point:we shall see that,modifying the Fronsdal operator in order to achieve gauge invariance without any trace condition on the parameter,the Bianchi identities will change accordingly and the “anomalous”terms will disappear,leading to corresponding gauge invariant Lagrangians.Returning to the spin-3case,one can verify that the Einstein tensor Gµ1µ2µ3followsfrom the LagrangianL=−12(∂·ϕµ1µ2)2−32(∂µϕ′µ1)2+3ϕ′µ1∂·∂·ϕµ1+3∂·∂·∂·ϕ12∂·∂·∂·ϕ,(71)that is fully gauge invariant underδϕµ1µ2µ3=∂µ1Λµ2µ3+∂µ2Λµ3µ1+∂µ3Λµ1µ2.(72)For all higher spins,one can arrive at the proper analogue of(67)via a sequence of pseudo-differential operators,defined recursively asF(n+1)=F(n)+1 F(n)′−1 ∂·F(n),(73) where the initial operator F(1)=F is the classical Fronsdal operator.The gauge transfor-mations of the F(n),δF(n)=(2n+1)∂2n+12n∂F(n)′=− 1+1 n−1ϕ[n+1],(75)where the anomalous contribution depends on the(n+1)-th traceϕ[n+1]of the gaugefield, and thus vanishes for n>(s/2−1).The Einstein-like tensor corresponding to F(n)G(n)=n−1p=0(−1)p(n−p)!so that G(n)follows indeed from a Lagrangian of the typeL∼ϕG(n).(78) We shall soon see that G(n)has a very neat geometrical meaning.Hence,thefield equations, not directly the Lagrangians,are fully geometrical in this formulation.3.2Geometric equationsInspired by General Relativity,we can reformulate the non-local objects like the Ricci tensor F(n)introduced in the previous section in geometrical terms.We have already seen that, following de Wit and Freedman[14],one can define generalized connections and Riemann tensors of various orders in the derivatives for all spin-s gaugefields as extensions of the spin-2objects asΓµ;ν1ν2⇒Γµ1...µs−1;ν1...νs,Rµ1µ2;ν1ν2⇒Rµ1...µs;ν1...νs,and actually a whole hierarchy of connectionsΓµ1...µk;ν1...νs whose last two members arethe connection and the curvature above.In order to appreciate better the meaning of this generalization,it is convenient to recall some basic facts about linearized Einstein gravity. If the metric is split according to gµν=ηµν+hµν,the condition that g be covariantly constant leads to the following relation between its deviation h with respect toflat space and the linearized Christoffel symbols:∂ρhµν=Γν;ρµ+Γµ;ρν.(79) In strict analogy,the corresponding relation for spin3is∂σ∂τϕµνρ=Γνρ;στµ+Γρµ;στν+Γµν;στρ.(80) It is possible to give a compact expression for the connections of[14]for arbitrary spin,Γ(s−1)=1s−1k ∂s−k−1∇kϕ,(81)where the derivatives∇carry indices originating from the gaugefield.This tensor is actually the proper analogue of the Christoffel connection for a spin-s gaugefield,and transforms asδΓα1···αs−1;β1···βs∼∂β1···∂βsΛα1···αs−1.(82) That is a direct link between these expressions and the traces of non-local operators of the previous section.From this connection,one can then construct a gauge invariant tensor Rα1···αs;β1···βs that is the proper analogue of the Riemann tensor of a spin-2field.12We can thus write in a more compact geometrical form the results of the iterative procedure.The non-localfield equations for odd spin s=2n+1generalizing the Maxwell equations∂µFµ;ν=0are simply1n−1R[n];µ1···µ2n=0,(84) that reduce to R;µν=0for spin2.The non-local geometric equations for higher-spin bosons can be brought to the Fronsdal form using the tracesΛ′of the gauge parameters,and propagate the proper number of degrees of freedom.Atfirst sight,however,the resulting Fronsdal equations present a subtlety[4].The analysis of their physical degrees of freedom normally rests on the choice of de Donder gauge,D≡∂·ϕ−12∂∂/F s(ψ),(86)that generalize the obvious link between the Dirac and Klein-Gordon operators.For in-stance,the Rarita-Schwinger equationγµνρ∂νψρ=0implies thatS≡i(∂/ψµ−∂µψ/)=0,(87) while(86)implies thatSµ−1 S/=i∂/Non-local fermionic kinetic operators S(n)can be defined recursively asS(n+1)=S(n)+1 S(n)′−2 ∂·S(n),(89) with the understanding that,as in the bosonic case,the iteration procedure stops when the gauge variation2nδS(n)=−2i n∂S(n)′−1 n−1ψ/[n],(91)2nand lack the anomalous terms when n is large enough to ensure that thefield equations are fully gauge invariant.Einstein-like operators andfield equations can then be defined following steps similar to those illustrated for the bosonic case.4Triplets and local compensator form4.1Stringfield theory and BRSTString Theory includes infinitely many higher-spin massivefields with consistent mutual interactions,and it tensionless limitα′→∞lends itself naturally to provide a closer view of higher-spinfields.Conversely,a better grasp of higher-spin dynamics is likely to help forward our current understanding of String Theory.Let us recall some standard properties of the open bosonic string oscillators.In the mostly plus convention for the metric,their commutations relations read[αµk,ανl]=kδk+l,0ηµν,(92) and the corresponding Virasoro operators1L k=2α′pµand pµ−i∂µsatisfy the Virasoro algebra[L k,L l]=(k−l)L k+l+D√α′L0.(95)14Taking the limitα′→∞,one can then define the reduced generatorsl0=p2,l m=p·αm(m=0),(96) that satisfy the simpler algebra[l k,l l]=kδk+l,0l0.(97) Since this contracted algebra does not contain a central charge,the resulting massless models are consistent in any space-time dimension,in sharp contrast with what happens in String Theory whenα′isfinite.It is instructive to compare the mechanism of mass generation at work in String Theory with the Kaluza-Klein reduction,that as we have seen in previous sections works for arbitrary dimensions.A closer look at thefirst few mass levels shows that,as compared to the Kaluza-Klein setting,the string spectrum lacks some auxiliary fields,and this feature may be held responsible for the emergence of the critical dimension!Following the general BRST method,let us introduce the ghost modes C k of ghost number one and the corresponding antighosts B k of ghost number minus one,with the usual anti-commutation relations.The BRST operator[17–19]Q=+∞−∞[C−k L k−12α′C k,b k1α′B0(102)for k=0allows a non-singularα′→∞limit that defines the identically nilpotent BRSTchargeQ=+∞−∞[c−k l k−kwhere˜Q= k=0c−k l k and M=1s!ϕµ1···µs(x)αµ1−1···αµs−1|0 ,+∞s=21(s−1)!Cµ1···µs−1(x)αµ1−1···αµs−1−1b−1|0 ,(108)|Λ =∞ s=1i2(∂µϕ)2+s∂·ϕC+s(s−1)∂·C D+s(s−1)2C2,(113)16where the D field,whose modes disappear on the mass shell,has a peculiar negative kinetic term.Note that one can also eliminate the auxiliary field C ,thus arriving at the equivalent formulationL =−+∞ s =012(∂µϕ)2+s2(∂·D )2.(114)in terms of pairs (ϕ,D )of symmetric tensors,more in the spirit of [20].For a given value of s ,this system propagates modes of spin s ,s −2,...,down to 0or 1according to whether s is even or odd.This can be simply foreseen from the light-cone description of the string spectrum,since the corresponding physical states are built from arbitrary powers of a single light-cone oscillator αi 1,that taking out traces produces precisely a nested chain of states with spins separated by two units.From this reducible representation,as we shall see,it is possible to deduce a set of equations for an irreducible multiplet demanding that the trace of ϕbe related to D .If the auxiliary C field is eliminated,the equations of motion for the triplet take the formF =∂2(ϕ′−2D ),(115) D =12∂∂·D .(116)4.2(A)dS extensions of the bosonic tripletsThe interaction between a spin 3/2field and the gravitation field is described essentially by a Rarita-Schwinger equation where ordinary derivatives are replaced by Lorentz-covariant derivatives.The gauge transformation of the Rarita-Schwinger Lagrangian is then surpris-ingly proportional not to the Riemann tensor,but to the Einstein tensor,and this variation is precisely compensated in supergravity [10]by the supersymmetry variation of the Ein-stein Lagrangian.However,if one tries to generalize this result to spin s ≥5/2,the miracle does not repeat and the gauge transformations of the field equations of motion generate terms proportional to the Riemann tensor,and similar problems are also met in the bosonic case.This is the Aragone-Deser problem for higher spins [22].As was first noticed by Fradkin and Vasiliev [23],with a non-vanishing cosmological constant Λit is actually possible to modify the spin s ≥5/2field equations introducing additional terms that depend on negative powers of Λand cancel the dangerous Riemann curvature terms.This observation plays a crucial role in the Vasiliev equations [24],dis-cussed in the lectures by Vasiliev [25]and Sundell [26]at this Workshop.For these reasons it is interesting to describe the (A)dS extensions of the massless triplets that emerge from the bosonic string in the tensionless limit and the corresponding deformations of the compensator equations.Higher-spin gauge fields propagate consistently and independently of one another in conformally flat space-times,bypassing the Aragone-Deser inconsistencies that would be introduced by a background Weyl tensor,and this17。
⽣活⼤爆炸第⼆季笔记分享TBBT205学习笔记（欧⼏⾥德备选项）1.俚语give a damn:give a shit,give a fuck care2.普通级词汇the crow flies：两点间的直线距离conn：交通⼯具的控制部分speed bump：减速带tricky：形容⼈狡猾，形容问题难brain-teaser：⼜有趣⼜难的问题，脑筋急转弯speak up：If you speak up, you say something, especially to defend a person or protest about something, rather than just saying nothing.畅所欲⾔state of the art：最先进的remaster：重新录制增强⾳效的电影或磁带，碟⽚等rear-end：追尾homo novus（novushomo）：新新⼈类plebeian：平民，普通⼈3.爆炸级词汇helium 氦mercury 汞ytterbium 镱molybdenum 钼magnesium 镁manganese 锰europium 铕mendelevium 钔meitnerium 麦4.Sheldon语录Sheldon：Then I hereby invoke what I'm given to understand is an integral part of the implied covenant of friendship....The favor.Penny：Oh, dear God.Sheldon：I'm sorry, I didn't realize I was interrupting your morning prayers.When you're done, we'll go.可以把⼈情说的这么复杂，真Sheldon也。
a r X i v :h e p -e x /0105063v 1 22 M a y 2001QUARTIC GAUGE BOSON COUPLINGS RESULTS AT LEPM.MUSY CERNThe study of charged and neutral boson vertices has been performed in different production channels at the LEP experiments.Decay rates and kinematic properties of these events are exploited to set constraints on the corresponding gauge couplings.1IntroductionThe study of quartic gauge boson couplings (QGCs)has become possible due to the recent theoretical developments on this topic.The presence of QGCs affects the data collected for different final states at LEP.The continous increase of the centre-of-mass energy in e +e −collisions allowed for the precise measurement of the W-pair cross section and couplings.Now,also the study of radiative W-pair events,e +e −→W +W −γ,has become possible.The Standard Model (SM)predicts the existence of quartic gauge boson couplings leading to W +W −γproduction via s -channel exchange of a γor a Z boson as shown in Fig.1a.The contribution of these two quartic Feynman diagrams with respect to the other competing diagrams,mainly initial state radiation,is negligible at the LEP centre of mass energies.Nonetheless,the process leading to the W +W −γfinal state can be sensitive to anomalous contributions to the SM quartic vertices W +W −γγand W +W −Z γ.The theoretical framework of Ref.[1]is used for the parametrisation of such anomalous couplings.The existence of Anomalous QGCs would also affect the e +e −→νe ¯νe γγprocess via the W +W −fusion Feynman diagram containing the W +W −γγvertex 2(see Figure 1b).In the SM the reaction e +e −→ν¯νγγproceeds predominantly through s -channel Z exchange and t -channel W exchange,with the two photons coming from initial state radiation,whereas the SM contribution from the W +W −fusion is again negligible at LEP.AQGCs would enhance the ν¯νγγproduction rate,especially for the hard tail of the photon energy distribution and for photons produced at large angles with respect to the beam direction.eeeeγγFigure 1:Feynman diagrams containing a four boson vertex leading to the (a)W +W −γ,(b)ν¯νγγ,and (c)Z γγfinal states.Finally,the Z γγproduction (see Fig.1c)can also be exploited to derive limits on AQGCs,as it will be discussed in the next paragraph.2QGCs Analyses at LEPTable 1shows the different data sets used for the AQGCs analyses at LEP.L3W +W −γ189-202GeV189-202GeV189GeVZ γγ130-202GeV Opal E γ>10GeV|cos θγ|<0.94cos θfγ<0.9–√s (GeV)σW W γ (p b )00.10.20.30.40.5Figure 2:Measured cross section for the process e +e −→W +W −γ.The resulting cross sections for the L3experiment are:σW W γ(188.6GeV)=0.29±0.08±0.02pb σW W γ(194.4GeV)=0.23±0.10±0.02pb σW W γ(200.2GeV)=0.39±0.12±0.02pbwhile for Opal,the measurement at 189GeV gives σW W γ=0.136±0.037±0.008pb,where the first error is statistic and the second systematic.All the obtained results are compatible with the SM expectations.Figure 2shows the L3measured cross section as a fuction of√The information from theν¯νγγtotal cross section,together with the information derived from the shape and normalisation of the photon spectra in W+W−γevents,produces the fol-lowing ADL combined limits on charged AQGCs using the data set of Table1(first two lines):−0.022GeV−2<a0/Λ2<0.021GeV−2−0.043GeV−2<a c/Λ2<0.058GeV−2−0.022GeV−2<a n/Λ2<0.020GeV−2,in good agreement with the SM expectation of zero for each coupling.The Feynman diagram of Fig.1c involves only neutral bosons,and it is not present in the SM.Besides,as it has been recently indicated3,under more general assumptions the quartic couplings in the neutral sector,still named a0/Λ2,and a c/Λ2in Ref.[1],can be regarded as independent from those in the charged sector.For this reason,on the experiimental side,the convention was used to keep apart the measurements in these channels not performing any combinations,in the wait for a unified theoretical approach.To obtain limits on AQGCs in the neutral sector,the hadronic Zγγ→qqγγevents are used. The signal consists in an enhancement in the rate of high energy photons,while the background mainly comes from doubly radiative returns to the Z.Figure3shows the Zγγcross section as predicted by the EEZGG2program a.The same program is used to model AQGCs and tofit these couplings to the observed data distributions of P Tγ1(for L3),Eγand max|cosθγi|(for Opal). The following LEP combined bounds are derived4:−0.0048GeV−2<a0/Λ2<0.0056GeV−2−0.0052GeV−2<a c/Λ2<0.0099GeV−2All experimental observations are compatible with the SM predictions.3ConclusionsThe measurements of rare processes as Zγγ,W+W−γandν¯νγγconstitute an important test of the SM.Preliminary results on AQGCs were presented here,including new results in the W+W−γchannel up to the centre-of-mass energy of202GeV with a significant increase in the final precision.References1.J.W.Stirling and A.Werthenbach,Phys.Lett.B446,369.2.J.W.Stirling and A.Werthenbach,Eur.Phys.J.C14(2000)103.3.G.Belanger and F.Boudjema,Nucl.Phys.B288(1992)201.4.The Opal Collaboration,ICHEP abs572(2000),Opal PN452;The L3Collaboration,ICHEP abs.505(2000),CERN-EP/2000-006.Recoil Mass (GeV)E v e n t s /5 G e V246810121400.511.5√s/GeVOPAL Preliminaryσ(e +e -→q q γγ) /p bFigure 3:Cross section for the process e +e −→Z γγas a function of√。
