JHEP10(2006)014 July 7, 2006 hep102006014.pdf and a October 4, 2006 August 24, 2006 ctive field theories Received: Accepted: Published: ous quartic gauge boson IR manifestation of this , edding this model into con- eading irrelevant operators, ies can not arise in quantum Alberto Nicolis uctuations around non-trivial , ia world model modifying gravity ymmetry breaking that predict and perturbative string theory. olution, and implying a surpris- ences, abc evidence for the existence of su- ng previously seen in physics, and -matrices satisfy the usual analyt- tly exhibit macroscopic non-locality S . Conversely, any experimental sup- -matrix satisfies canonical analyticity Z S and http://jhep.sissa.it/archive/papers/jhep102006014 /j W Sergei Dubovsky, [email protected] a , [email protected] , Published by Institute of Physics Publishing for SISSA ∗ b Nima Arkani-Hamed, a Beyond Standard Model, Renormalization Group. We argue that certain apparently consistent low-energy effe [email protected] On leave from INFN, Pisa, Italy. ∗ 60th October Anniversary Prospect, 7a,E-mail: 117312 Moscow, Russ Jefferson Physical Laboratory, Harvard University, Cambridge, MACERN 02138, Theory U.S.A. Division, CH-1211Institute Geneva for 23, Nuclear Switzerland Research of the Russian Academy of Sci [email protected] [email protected] SISSA 2006 b c a c ° Causality, analyticity and an IRcompletion obstruction to UV Allan Adams, Riccardo Rattazzi described by local, Lorentz-invariant Lagrangians, secre and cannot be embedded in any UV theory whose Abstract: constraints. The obstructionwhich involves must the be signsrestriction strictly of is positive a that to set thebackgrounds, ensure “wrong” of making UV signs it l impossible lead analyticity. to to defineing superluminal local, An IR causal fl breakdown ev of thefield effective theories theory. or Such weakly effective coupled theor string theories, whose icity properties. This conclusionin applies to the the IR, DGP giving brane- atrolled simple stringy backgrounds, explanation and for to thenegative models difficulty anomalous of of quartic electroweak couplings emb s for the port for the DGPcouplings model, at or future measured accelerators,perluminality negative would and signs constitute macroscopic for non-locality direct unlike anomal almost anythi incidentally falsify both local quantum field theory Keywords: JHEP10(2006)014 6 9 1 3 5 13 28 29 14 22 27 28 30 33 ull theory? To a string led to GR as the unique hese constraints appear to is view, the early study of persymmetric vacua [1]. If a resounding “no.” The hope ther possible effective theories um gravity would be so strong but merely one point in a vast sistent supersymmetric theories e, the rank of the gauge group n the properties of the low-energy – 1 – With the discovery of D-branes and the duality revolution, t 3.1 The trouble3.2 with Lorentz invariance Global problems3.3 with causality The fate of fate 7.1 Little string7.2 theory Non-commutative theories 1. Introduction Can every low-energy effective theory be UV completed into a f theorist in 1985, the answerwas to that this the question consistency would conditions haveas on been a to full more theory of orlow-energy quant effective less theory, uniquely and that single thesimply out infinite couldn’t number the of be standard o extended modelperturbative to heterotic coup strings a yielded full manytheory theory. constraints o invisible In to support the ofwas effective restricted th field to theorist. be smaller For than instanc 22. have evaporated, leaving us with a continuouscoupled infinity of to con gravity andthe very low-energy likely theory a describing our huge universe discretum is of not non-su unique Contents 1. Introduction 2. Examples 3. Signs and superluminality 4. Analyticity and positivity constraints 5. The DGP model 6. Positivity in the chiral lagrangian 7. Examples from string theory 8. Gravity 9. Discussion JHEP10(2006)014 at UV scattering (four-dimensional) ar in the “anything cua in string theory e operators, small fluc- y effective field theory ity in model-building in microscopic theory. This fy gravity in the infrared, erhaps even the hierarchy questions to environmental scopic scales to a non-local surely any consistent model ency conditions on quantum analyticity to causality is an [5] and more recent ideas on del — are unlikely to be tied cal Lagrangian, and Lorentz- ll show that some apparently ties of our vacuum — such as reducing the detailed study of -invariant notion of causality, nstructing interesting theories ering amplitude must be non- s, while these theories are local emingly intractable fine-tuning f events. Moreover, in general icity and unitarity imply posi- to the field equations are again her Lorentz-invariant nor local. ns of certain higher-dimensional ictly positive. The inconsistency opagate superluminally, making her-derivative interactions as the h UV and IR avatars. hial interest. This situation is not ersion relations and unitarity im- amplitudes. One particular such ned by local, Lorentz-invariant La- low energy -matrix satisfying the usual analyticity condi- – 2 – S Given these developments, it is worth asking again: can ever In this note, we wish to argue that the pendulum has swung too f The UV face of the problem is also easy to discern: assuming th The IR face of the problem is that, for the “wrong” sign of thes We will focus on models in which the UV cutoff is far beneath the landscape of vacua of thethe underlying values theory, of then the the dimensionlessto proper the couplings structure of of the the standard fundamentalits theory mo particle-physical in properties any to direct a way, problemwithout of its only consolations. paroc With apuzzles vast landscape such of as vacua, se theproblem [3], cosmological can be constant solved problemones, by suggesting being [2], demoted new and from models fundamental p for particle physics [4]. be UV completed?certainly encourages The this evidence point forgravity of leave an view room — enormous for if landscapecan huge even be of the numbers embedded va consist of somewherethe consistent in last the theories, five landscape. years has Muchpurely implicitly of from taken the the this activ bottom-up pointhas with of no been view, obvious co particularly embedding true intoincluding any in most notably the the contextHiggs Dvali-Gabadadze-Porrati of phases model attempts of gravity to [6 – modi 8]. goes” direction. Using simpleperfectly and sensible familiar low-enegy arguments, effective field wegrangians, theories wi gover are secretly non-local,and do are not incompatible admit with any a Lorentz microscopic tuations around translationally invariantit backgrounds impossible pr to definebackgrounds, a the Lorentz-invariant equation time-ordering of o constraint motion equation can whose solutions degenerate are on UV-dominated.in macro Thu the sense thatinvariant the in field the equations sense derive thatsolutions, from Lorentz the a transforms macroscopic strictly of IR lo solutions physics of this theory isamplitudes neit satisfy the usualmediately analyticity conditions, imply disp a host of constraints on operators in any non-trivial effective theoryof must theories all which be violate str this positivity condition has bot constraint is that thatnegative, the yielding leading the low same energy positivitysuperluminality condition forward on scatt constraint. the hig Oftivity course constraints the is fact very thatancient well analyt one. known, and the connection of tions. The consistency condition we identify is that the sig JHEP10(2006)014 (2.1) ur positivity dence for either of ... , e mean by a consistent + ce of a violation of these 2 in the moon’s orbit that ntly, since weakly coupled ) colliders indicate evidence -matrix theory— the same iant notion of causality and ective theory are gathered, e UV completed unless the omalies and so on, but most ing theory is certainly non- low-energy effective theories make some comments about µν S s can and cannot arise in the hese deviations are seen and tum field theory, and satisfy insically gravitational limita- -matrix, unlike anything pre- minal signal propagation and ˜ xplanation for why this model ale Λ. F t produces an exactly unitary S d string backgrounds. Similarly, fundamental assumptions about as amplitudes in local quantum ]. ng from string theory are in this strained to be positive, implying . For example, it is easy to check µν c F ules of macroscopic locality, causal- turbative string theory. For exam- ( 4 2 c Λ + 2 ) µν F – 3 – µν F ( 4 1 (1) gauge field. The leading interactions in this theory c Λ U + µν F µν F 4 1 − = -matrix axioms, so any effective field theory which violates o L S anomalous quartic gauge boson couplings. Experimental evi Of course, local quantum field theories have a Lorentz-invar Positivity thus provides a tool for identifying what physic The flipside of this argument is that any experimental eviden Consider the theory of a single negative -matrix for particle scattering at energies beneath some sc Planck scale, so gravity ingravitational unimportant, theories. though Our wetions work will on thus also effective complements field the theories intr recently discussed in [9,satisfy 10 the usual conditions cannot be UVstring completed amplitudes into a satisfy local thefield QFT. same theories— Significa analyticity indeed, properties the Veneziano amplitude arose from argument applies to weaklylocal coupled in many strings. crucial ways, Thus,precise the while sense effective field str just theories as arisi the local same positivity as constraints. those deriving from local quan landscape. Perhaps surprisingly, the toolthat is the a DGP powerful model one violateshas positivity, so providing far a resisted an simple embedding e incertain controlled 4-derivative weakly terms couple in thefor chiral Lagrangian example are that con theanomalous electroweak quartic chiral gauge boson Lagrangian couplings cannot are b positive. positivity constraints would signal a crisisity for the and usual analyticity, r and, almostple, incidentally, falsify the per DGPwill model be makes checked precise byother laser predictions pieces lunar for of ranging deviations experimental experimentswe evidence [11]. would supporting also If the have t macroscopic DGP evidence violation eff for of parametrically locality, fast as superlu well as a non-analyti viously seen in physics.for The samethese conclusion theories would holds therefore if clearly disprovephysics. future some of our 2. Examples Let’s begin with somewe examples will of constrain. the apparentlyeffective consistent Of theory course — loosely we it shouldprecisely, should a be have stable consistent precise vacuum, effective no aboutS an field what theory w is just one tha are irrelevant operators, JHEP10(2006)014 are (2.4) (2.3) (2.7) (2.8) (2.2) (2.5) (2.6) h fermions (1) gauge U N 1 into a Dirac are completely ± 3 , . 2 , non-renormalizable i 1 c . . . ra dimension, given by and a Higgs field 0. Analogously, we can in all familiar UV com- ine coupling nts + π terms. respects the usual axioms > 4 . . . ld action for a tance, consider integrating 4 ) 2 8 , etheless there is a systematic F . − at tree-level is 1 with charges ∂y l 2 c . . . ( Ψ c rg Lagrangian for QED, arising h i 2 + h , 1-loop effects can dominate over + π, h Ψ Λ) which is unitary to all orders in 2 2 N M ) ψ, ψ ) M π , . . . E/ 2 − µ − ∂y +const. Again the leading interactions 2 + ( ) π π∂ ¶ 4 iπ/v ) 5 µ . e ∂h γ ) ∂ → 1+ 0 ( h ∂π π 4 ( π 3 − + ( v µ > 4 c + h Λ positive ∂ 2 h – 4 – i λ ) v i 4 c M + f . The action for + ∂π π = 2 ( = µ µ ) 2 2 ∂ are, they can give the leading amplitudes in an exactly 2 Φ = ( ´ eff v ) π∂ i µ v h L µ c µ − ∂y ∂ dimensionless coefficients. As another example, consider a ( 2 iγ | 2 = , h 1+ − 1 Φ ³ | ¯ Ψ c 1 L ( are forced to be λ = p i 4 L c f )= with a shift symmetry | − π Φ | = ( at tree-level yields the quartic term L V h as a Goldstone boson in a linear sigma model, where ) with the effective lagrangian -matrix at energies far beneath Λ. Of course the theories are x π ( S y There are also other simple 1-loop checks. For example, imag It is easy to check that indeed these coefficients are positive As far as an effective field theorist is concerned, the coefficie Another example involves the fluctuations of a brane in an ext -matrix theory, the S spinor Ψ, the effective Lagrangian is identify to Φ in our UV linear sigma model; for sufficietly large the tree terms comingout from a integrating higgsed out fermion. the Higgs. Grouping For two ins Weyl fermions with a potential unitary united into a complex scalar field Φ, so an infinite tower ofexpansion higher for operators the must scattering be amplitudes included, in non powers of ( arbitrary numbers. Whatever the this ratio. However, we claimof that in any UV completion which pletions of these models.from For integrating instance, out the Euler-Heisenbe electrons at 1-loop, indeed generates field localized to a D-brane also gives the correct sign for al massless scalar field Again we find the correct sign. Related to this, the Born-Infe with Λ some mass scale and Integrating out a field which has the claimed positive sign. are irrelevant, JHEP10(2006)014 4 ) ∂π (2.9) couplings must 4 λφ Goldstone model ex- and e not directly related to he trivial vacua of such 2 he positivity of energy (at eld or perturbative string φ iant vacuua with perfectly eractions (such as the ( -trivial backgrounds depend 2 ity-violating irrelevant lead- fluctuations travel slower or hologies. Of course this non- nes to which our constraints superluminal propagation in s with causality and locality uminating question. m e effective field theory remains ient energies are positive, with eas of effective field theory, but he euclidean path integrals for sociated with a clear instability ities in the low-energy vacuum: non guarantees a unitary low-energy tions satisfy a simple dispersion uum stability. We know for in- rounds. Suppose we expand the finite euclidean actions. nvariant solution of the field equa- ... , is a constant vector. The linearized + µ 4 ) C of the leading irrelevant interactions. ∂π ) determined by the higher-dimension oper- ( ˆ k 4 . Related to this, the euclidean path integrals ( 2 v v – 5 – ) 2 1 signs π , where ∂π µ 48 C = -matrices, why don’t they arise as the low-energy limit = S eff giving only small corrections within the effective theory. 0 π L i c µ ∂ , with the velocity 2 | k ~ | ) ˆ k ( 2 v = 2 ω Let’s begin by establishing the connection between positiv Let’s see how this works in an explicit example. Consider our By contrast, the “wrong” sign for the leading derivative int Note that the positivity constraints we are talking about ar -matrix when we continue back to Minkowski space. in theories with “wrong” signs do not exhibit any obvious pat renormalizable theory must be treated usingthe the healthy standard euclidean id formulation atS least perturbatively 3. Signs and superluminality If models withsensible the and “wrong” unitary perturbative signs have stable, Lorentz-invar ing interactions and superluminalityeffective in theory non-trivial around some backg non-trivialtions. translationally i As long as thevalid. background field Translational is invariance sufficiently ensures small,relation, th that small fluctua critically on these signs,generic with backgrounds. the This “wrong”which in signs are turn leading not leads to presenttheory. to in Exactly familiar how any conflict this microscopically conflict local arises turns quantum out fi to be an ill of any familiartheories UV-complete are theories? well-behaved, the speed of As fluctuations around we will see, while t ators in the lagrangian. The crucial insight is that whether faster than light depends entirely on the panded around the solution apply — have the “wrong”least sign, classically), higher e.g. order higher terms powers can of ensure ( t Indeed, even if the leading irrelevant operators — the only o terms above) are not associatedthe with correct any sign energetic of instabil thethe kinetic terms terms proportional guarantee to that the all grad also be positive. In allalready these visible cases, in the “wrong” thesuch signs low-energy are theories theory. as are Related not to well-defined, having this, non-positive-de t At 1-loop, we generate an effective quartic interaction other familiar positivity constraintsstance that that kinetic follow terms from are vac forced to be positive, and that resulting again in a positive leading irrelevant operator. JHEP10(2006)014 is 3 c (3.2) (3.1) (3.3) ons prop- polarization direction, the oefficient . Expanding in x ing metric” and and 3 c , appearance of the BI , all higher dimension 4 ackground fieldstrength = 0 Λ ed — the speed of fluctu- 2 hen the shift in the speed luminality — the ability to ) omentum verned by a BI Lagrangian luminality and understand ld equations. It does not, . m. It is thus instructive to ν ¿ ² Lorentz-invariant lagrangian, distances. It is interesting to s that Lorentz transforms of ng. Consider for example the µ interaction, µ 0. k = 0 C r their heads. On the other hand, µν . > µ ϕ ˜ ν 2 F C , ( ∂ 1 , µ 4 c = 0 2 leading ∂ c Λ 2 i ) = 0 k · around this background is ϕ . . . 2 i 0 + 32 C ∂ + there exist no superluminal excitations iff the π ( 1, it can easily be measured in the low-energy 2 ). Upon boosting in, say, the ˆ 2 ν 4 3 v – 6 – ) − c Λ C ν vt ¿ ² − µ π µ 2 4 ± C ϕ k around translationally-invariant backgrounds of our + 4 Λ ≡ C 2 t x 4 3 ( µ ∂ µν ϕ ϕ c Λ ∼ k f F µ 1 ( . Note that these conclusions hold independently of the k = 4 1 + 4 − c Λ ϕ v µν η h + 32 µ is the velocity of propagation. This has oscillatory soluti k of the leading interactions in the Lagrangian, 2 µ k C the absence of superluminal excitations requires that the c 3 4 c Λ 4 0, both are both positive 2 − ≥ , 1 1 2 c ) k ' 6= 0 can be computed exactly in terms of the so-called ”open str · 2 ) C v B The case of the electromagnetic field is slightly more involv At this point all the problems usually associated with super + , and thus on µ F plane waves, this reads ations around non-trivial backgrounds now² depends on both m positive. interactions are negligible - all that matters is the send signals back in time, closedsuch timelike effects curves, are etc.—rea appearing within aa theory hyperbolic governed by equation a local ofwork motion through and the a physicalexactly perfectly consequences when stable and of vacuu why this these kind theories of run into super 3.1 trouble. The trouble with LorentzThat invariance the effectivesolutions Lagrangian to is the Lorentz-invariance ensurehowever, field ensure equations that are all inertialequation again frames of solutions motion are to for onGoldstone the fluctuations an model, even fie footi where equation of motion for fluctuations agating in all directions, e.g. effective theory by allowing signals to propagate over large note that in the caseof of the open form strings (1), on( D-branes, the which speed are go of propagation in the presence of a b Since ( Within the regime of validity of the effective theory, is always slower thanLagrangian in the string speed theory satisfies of positivity, light with — which is to say, this but the conclusion iscoefficients completely analogous: particular background field one turns on. Note too that even w of propagation is very small, JHEP10(2006)014 ective Fixing mes a non- . sense; they In order for = 0 , the equation of ϕ 2 ⊥ /v ∂ 1 2 define retarded Green v n another frame. − nditions for a superlumi- ) in which the coefficient β > s in another. Indeed, the the unstable excitations. A able unboosted frame. The owing. Crucially, this never /v ly unstable in another? rray of fluctuations incident ϕ ider a solution in the highly here bounded and oscillatory 2 x e what the original observer ons is not Lorentz-invariant. ime which vary exponentially s going on? How is it possible y rule out as unphysical. By tarded Green function in one equations of motion — initial ∂ egrated — however, oscillatory atically about what constitutes cy in the initial conditions that = 1 to be of order Λ; however, since the orentz-invariance. direction, while modes in other ) to set up an initial value problem zed source at some unremarkable ding to the equation of motion — s a particularly natural choice of When 2 π β he dispersion relation, so we need huge x β look like a small fluctuation sourced 1 ∇ − 2 not and v ( effects obtain well within the domain of validity π − positive appear. ϕ x – 7 – ∂ not t ∂ ) 2 ), are the Lorentz boost of the original solutions. So t v β vβ − − remains tiny, the description of the system in terms of the eff ∓ v 1 (1 π 1, there exists a frame ( µ β ± > π∂ x µ ( 2 + 2 ∂ v f ϕ = 2 t ∂ ϕ ) 2 propagates instantaneously and the equation of motion beco β ϕ 2 the slice. These do not represent instabilities in any usual v − (1 along , vanishes, ϕ The point is that what look like perfectly natural initial co Returning to our question of stability vs instability, cons 2 t Notice that we are dealing with tiny superluminal shifts in t space 1 ∂ dynamical constraint. In this frame itto is evolve simply impossible the field from Cauchy slice to Cauchy slice. of whose solutions, e.g. motion is again perfectly dynamicalsolutions and can to certainly these be int equationsdirections move may be only exponentially in growingthat the or what decaying. looks What’ like a stable system in onenal frame mode looks horrib in onetime-ordering frame of look events likeObservers connected in horribly by relative fine-tuned motion propagating will condition a fluctuati thus disagree sensible rather set dram ofconditions initial that to conditions one observer to lookpoint propagate like turning in with on spacetime a their will locali appearfrom to past the infinity other which aswanted a to conspire bewildering call miraculously a the toframe localized annihilat is source. a mixture Said differently, of the advanced re and retarded Green functions i boost velocities to observe these effects,theory requiring remains both valid for ˙ allof observers, effective ensuring field that theory. these equation of motion becomes far so good. However, if Lorentz invariant combination boosted frame in which weLorentz turn boost on a unambiguously localized maps source thiscrucial for point to one is a that of the solution resulting in configurationby a does the local st source — indeed,but these are rather explicitly stable involves accor turningin on initial conditions at a fixed t initial conditions on pastthe infinity theory thus to be explicitlyfunctions. predictive, breaks In L sufficiently we well-behaved must backgrouds,frame, there choose that i a in which frame such in conspiracies which do to simply represent initial conditionsthe which same we token, a would localizedin normall fluctuation the which original remains frame everyw transformsprevents into the a apparently miraculous unstable conspira mode from turning on and gr JHEP10(2006)014 ng 0. (a) 0, this satisfy itive — C < π < 3 3 c condensate c x' µ C B 0, and imagine = , deep inside the < π A . When 3 µ term in the equation π c ∂ ν ϕ int, 2 t π∂ he condensate, in which ). Inside the bubble the ∂ gn, µ rizontal, the bubble has a int. Thus, in any frame in side, however, fluctuations ionally non-invariant back- ∂ no problem. 1. An observer in the rest ent must pass through zero, A 4 ich Lorentz transformations e ever so slightly superlumi- 3 direction. For sufficiently large nomena seen by this boosted c Λ 0 cannot be prescribed by local t' pagate forward in time, setting x e + 4 he same system in a boosted frame d bubble of ted frame. tuations are only seen to propagate to µν itions. η = 0 vacuum, fluctuations of π = , in our Goldstone model with µ µν x G µ C 0). The solid lines denote the causal cone inside of – 8 – = , 0 π , 0 x C, C and thus propagate not along the light cone but along 2 = ( C µ 3 4 C c Λ 4 B − 0 it is possible for the coefficient of the 1 < ' A , on the boundary at which the wave exits the bubble, proceedi 3 2 c t B v in the equation of motion is negative, while outside it is pos ϕ 2 t ∂ 0). Outside the bubble, in the trivial , Bubbles of non-trivial vacua, 0 , 0 C, It is enlightening to run through the above logic in translat As above, when = ( µ the massless wave equationmove and with propagate velocity along null rays. In localized in spaceC and time. Let’s begin in the rest frame of t building, by suitable arrangement of sources, a finite-size grounds. Consider again the Goldstone model with “wrong” si boosts, the causal conethe dips left below with horizontal, and a small different fluc temporal ordering than in the unboos happens in theories withcarry null sensible initial or conditions timelike to propagation, sensible in initial wh cond in which the bubble moves with a large velocity in the positiv which small fluctuations are constrained to propagate. (b) T somewhere along the boundary ofat the which bubble, point then, the the coeffici equationwhich the of motion causal becomes coneclosed deep again a inside shell constra the on bubble whichhamiltonian dips evolution flow. below from This the timeslice in ho toobserver. fact timeslice helps Consider explain the theframe peculiar sequence phe of of the events depicted bubble in sends figure a superluminal fluctuation from a po of motion of acoefficient rapidly of moving observer to vanish (see figure 1b In the rest frame of the bubble, bubble to a point, a “causal” cone defined by the effective metric Figure 1: cone is broader thannally the (see light figure cone 1a). and However, fluctuationsup since propagat and fluctuations solving always pro the Cauchy problem in this background is still JHEP10(2006)014 ϕ A of a pref- direction and sep- seem unlikely any x h systems, however, he ce, there always exists entz-invariance is not a can’t be predicted from degenerate to constraint B invariance and locality. No- dding such effective theories ntinue forwards in time to sense somewhat analogous to sical Lagrangian is physically at ntaneous appearance of two y. se can be made unless Lorentz- ϕ ariance makes the rate of decay ies of small fluctuations along these t no notion of causality or locality survives x . How is this possible? The resolution is that y . In a highly boosted frame, the sequence of C – 9 – C t or A , so the evolution of B direction. The open cones indicate the local causal cones of y vanishes at ϕ -matrix exists within such theories. Indeed, the existence happening before 2 t S ∂ B Two finite bubbles moving with large opposite velocities in t . C This is not something with which we are familiar, and makes it -fluctuations, and the red line the closed trajectory of a ser Figure 2: arated by a finite distance in the π in a theory which violates positivity. at the speed of light to a distant point, cones. Such closed time-like trajectories make it clear tha events will have Lorentz invariant the coefficient of ered class of framesequations — — those suggests in that which theirrelevant, the Lorentz and field invariance equations of raises do thein doubts not clas UV-complete about theories the whichtice possibility respect that of microscopic systems embe with Lorentz Lorentz superluminal invariant propagation field theories are with in ghosts,invariance this of is which explicitly no broken. sen of This the is vacuum because bysymmetry boost-inv ghost of emission the formallyLorentz-invariance theory infinite can can — only arise the only as decay if an rate Lor accidental symmetr be3.2 made Global finite. problems with In causality suc In the simple system of a single bubble in otherwise empty spa local measurements; instead, aexcitation constraint just requires inside the andand spo outside the bubble, which then co JHEP10(2006)014 µ t (3.6) (3.5) (3.4) tor itself varies. 1 and the µν ¿ G 1 and therefore 4 “lightcones” and Λ > ϕ π/ 00 nstrained, non-local e orientable, at least µ G sm. Consider again the methodology of General r from obvious whether π∂ closed (future directed) π . µ ν s in which no global rest ∂ e field theory description. meaning that there should variant, notion of causality ckground in the Goldstone . unds in which the effective f the bubble and translated π∂ ture directed timelike curves . To see that this is the case, Lorentz generators. Forward good’ class, so we can simply gure 1 it is also evident that o a boosted inertial observer, 0 µ µ t ∂ defined over our spacetime [13]. > , 3 4 lly defined. In particular, the co- . . . c 0, we have coordinate is not globally defined Λ 4 µν t + < G + π to the distance over which ν 3 all see that it does, we do not need to worry ν ˙ c x with which to define µν µ η π∂ x µν µ condensate flying past each other at high = ∂ G 3 π 4 c µν Λ 4 – 10 – as an effective metric holds only in the geometric optics − µν 0 ˙ µν η G , G > = µν is indeed what determines the light cone structure within = 0 µν G ϕ ν µν G ν t G µ ∂ ˙ µ x ∂ µν 0) is globally defined, non-degenrate and time-like. The vec , G 0 , 0 , The metric 2 is = (1 good notion of causal ordering in these theories. µν µ t G any ) are those defined by The Goldstone and the Euler-Heisemberg systems are both tim It is useful to treat this problem with the aid of some formali Strictly speaking the interpretation of σ 2 ( µ Anyway, if a pathology arisesabout already the in case this of limit, long and wavelengths. we sh defines at each space-time pointx the direction of time flow. Fu The second conditiontimelike for curves causality (CTCs). to In hold the presence is of that CTCs, there the be no — it isproblem, multiply and valued causality — is lost. and time evolution again becomes a co for backgrounds withinMoreover, the for domain simple of backgroundsthere like validity are the of no single the CTCs, bubbleexists. so of effectiv that fi a However, sensible, in although both not Lorentz systems, in there exist other backgro the vector effective field theory surely makes sense. Then, for limit in which the wavelengths are short enough with respect note that wave equation for smallsystem, fluctuations around a non-trivial ba velocity and finite impactthere parameter, is as in figure 2 — so it is fa where the dots stand for terms that can be neglected when families of inertial frames inmoving which rest causality frame is of meaningfu thedeclare bubble defines that a evolution time is slicinginto in to other this be frames ‘ by prescribed boostingevolution in with in the the time spontaneously rest in broken highly framebut boosted o it frames may is look unambiguous.frame bizarre t exists However, — there for are example, always two background bubbles of the blobs in figure 1.Relativity Now to that determine we have whether a causalityA metric, is we first meaningfully can requirement apply the isexist that a globally the defined spacetime and non-degenerate be timelike time vector orientable, This equation suggests a natural inverse-metric time-evolution. JHEP10(2006)014 | G ~ B | 1 the = − | ~ direction E for which < | y 2 E | ~ 0, photons in E r coaxial with | 4 < 32 Λ 1 1 c c s given by two super- ty-violating field. The 1, the velocity of small to superluminality and rajectories involves the ure 1. Note that a head mains entirely within the ertainly take us out of the working with the metric , and a corresponding local ngent to the constant time that these pathologies only direction — the past of the static observer. include additional fields. In →− agnetic field with be locally defined but must ng”-signed Euler-Heisenberg hs remain large compared to nally, for he non-inertial metric admits s in this background moving 2 pagate strictly forward in the | s needed to probe these CTCs t fixed Lorentz time represents int within the forward effective ~ , move with velocity E ds, this is generically impossible. me in which the condensate is at | not s the critical value of ~ E ry configuration within the regime 4 32 Λ 1 c becoming large in the overlap region. 2 2 | | 2 ~ ~ ) E E | | 4 4 = 1 in the other. If ∂π 32 32 Λ Λ can remain parametrically small everywhere 1 1 v c c 2 ) − – 11 – ∂π 1 1+ = passes through zero and goes negative — as long as , all local Lorentz invariants are small — indeed they v 2 2 , and polarized along ) Λ to define a single-valued time-like coordinate. ~ B π µ ∼ π × ∂ ~ B ~ E ~ E, ∝ k ~ admit CTCs, and time evolution can constraints. does = 0, such as might be found deep inside a cylindrical capacito 0, we can always use µν ~ B G > globally · 2 ~ ) E Another particularly nice example of such closed timelike t Notice that attempting to define a global notion of causality Note that we have been tacitly working with a single positivi In our Goldstone system, a simple such offending background i . As in the Goldstone example, violations of positivity lead π 1 µ − ∂ a current-carrying solenoid, asorthogonal depicted to in the field, figure 3. Photon and ( propagation of light insystem a in non-trivial eq. background (2.1). of our Consider “wro a homogenous, static electrom in the background, andoccur thus within in the backgrounds effective where theory. ( Note this system propagate superluminally. Moreover, as But it is easy to check that a small separation in a transverse in the figure — is enough to ensure that ( situation is just as bad,particular, we and have in relied heavily some on senseof the existence, rather validity of for worse, eve the if effective we theory,rest, of i.e. a a locally comoving framelocal fra in time-like coordinate. which all If we superluminal have fluctuations two superluminal pro fiel luminal bubbles flying rapidlyon collision past between each the other, tworegime bubbles as of in validity shown the of in same the fig plane effective theory, would with c ( macroscopic violations of causality. Hamiltonian flow, by working inin a our non-inertial intrisically frame flat — spacetime — i.e. runs by into problems when t satisfy are fine-tuned to vanish. Furthermore,remain the within small the fluctuation effectiveΛ regime as long as their wavelengt metric fluctuations diverges as their kinetic term vanishes: this i forward light cone forIn the particular effective the cylinder’s metriclight angular overlaps cone, direction with so is the that ata a each circle CTC po between for the theeffective cylindrical effective theory, plates for metric! a while Note that this configuration re the light cone ofslices the of effective an metric observer at at each point rest becomes with respect ta to the solenoid. Fi in the direction parallel to the current and JHEP10(2006)014 . 1. as µν T µν g , and the propagate r dp/dρ < dp/dρ th a current-carrying 0 , a dramatic violation of t E 0 evolution is a globally con- ion is subtracted from B erluminality is the dominant , CTC’s do not arise as long tion, i.e. if the matter action e gravitational metric bsence of superluminality for superluminality for a general luminal propagation generally luminal modes are allowed, the lities within the matter dynam- ergy momentum tensor satisfies ifold planes in string theory. To a fluid is given by at fixed olated by a negative cosmological if the matter dynamics do not fea- s dips below horizontal, allowing for till within the regime of validity of the 0, small fluctuations at fixed < 1 c – 12 – . When ˆ z should a be future directed time-like vector for any B ν t = follows from the absence of superluminality µν ~ B T and p < ρ , i.e., there should be no energy-momentum flow outside the ˆ r µ r A t = ~ E The field between the plates of a charged capacitor coaxial wi Another well-known energy condition closely related to sup In this form the dominanta energy large condition class follows of from systems. the For a instance, the sound velocity in dominant energy condition For a single derivatively coupled scalar field the absence of light-cone for any observer. This condition is trivially vi constant, as well as negativemake tension it objects meaningful such one as must assume orient that the vacuum contribut future directed time-like well. Therefore, whether gravityleads is to dynamical a or global not, breakdown super of causality. energy condition. It states that Figure 3: purely spacelike evolution all the way around the capacitor effective field theory, the “causal cone” of small fluctuation superluminally. For sufficiently large fieldstrengths, but s solenoid is of the form CTCs, since the affine parameter cannot be globally defined, so locality and causality. strained problem. Now, inas GR, the with energy asymptotically momentum flat tensor space satisfies satisfies certain the restrictions. null energy It condi isture a either remarkable instabilities fact or that superluminalthe modes null energy then condition the [12]. en null Conversely, energy as condition soon is as lost,ics super even [12], in and the CTC’s absence can of in instabi principle appear with respect to th JHEP10(2006)014 s- 0. 0. UV > ≥ ) ) X X ( ( 00 L −L , X 2 · ) ) π µ X ( ∂ 0 ( L ≡ acroscopic locality — is tal requirement than the X ories, which are normally or string theory. Any ex- tive theory in pathological f ity are impossible to realize unds, attempting to foliate ory which allows superlumi- t admit backgrounds where = 0, these conditions agree, iolated. Another possibility equires f frame in which evolution is ion of such effective theories. when local Lorentz-invariants d capable of manifesting this sensible backgrounds, e.g. by stance, if the action contains nvariants remain below a y is a stronger condition. Thus X t, there is no Lorentz-invariant the Lagrangian remain negligi- m field theories or perturbative iently simple backgrounds, both ed backgrounds, these phenom- hus provide spectacular evidence IR on, disagreements about time or- es is obstructed by the existence Lorentz invariant. In both cases, an never be locally defined but is ating effective Lagrangians can in o positivity provides an obstruction – 13 – . This means that even the vacuum must spontaneously 2 ) ∂π . An underlying theory could complete the IR physics in two di small Does this mean that effective theories which violate positiv This is not the same as the dominant energy condition, which r 3.3 The fate of fate What have we learned about physics in a Lorentz invariant the For small fluctuations around thebut for trivial a background general with background,the the absence absence of of superluminalit superluminalitydominant energy is condition. a more direct and fundamen nal propagation only around non-trivialnotion backgrounds? of causality. Firs Second, fordering observers can in be traced relative to moti sharpof violations these of complications locality; can in suffic beeverywhere avoided local by and a causal. judiciousspacetime choice into Third, o (perhaps in non-inertial)of more constant-time closed general slic time-like backgro trajectories,always so globally that constrained. time-evolution c in nature? Not necessarily.principle Rather, be since reconstructed positivity-viol frommeasuring experiments low-energy scattering in amplitudes completely ena in can well-behav be interpretedbackgrounds. as signaling This the is breakdownthought of a to the novel be effec constraint self-consistentcutoff, on so as that effective long UV-sensitive field as higher-dimensionble — operators the all instead, in local these Lorentz-i effective theories break down in the get sufficiently background requires the lagrangian to be a convex function o is that the underlyingnon-locality theory at is arbitrarily fundamentally large non-localpositivity scales, provides an while an IR remaining obstructionOf to course, a no purely known UV well-definedstring theories, complet theories, e.g. realize local such quantu macroscopicto non-locality, s embedding these effectiveperimental field observation of theories a into violationthat of quantum one positivity field of would our t mostsimply fundamental wrong. assumptions about Nature — m break Lorentz invariance, though local physics need not be v tinct ways. One possibilitylocal is Lorentz that invariants the canterms theory get with simply inverse arbitrarlily powers does small of no ( — for in JHEP10(2006)014 i vac (4.2) (4.1) | ) y ( O ) udes fail to x ( O | determines the osition at which i vac h . vac | sented as a sequence of ) y )] d of small fluctuations y ( Λ—the exact position is − urable in the low-energy y advance or retardation s illuminating to interpret / calar particles are interpo- ttering process. The effect e-level graphs depicted in x , φ , me super- or sub- luminal ( ound are visible already at ) atement, 0 kground. This is because the x , in a free field theory, while ionally-invariant background, rators precisely because these se small shifts add up to give ( adv < ort scales. φ uperluminal excitations is a UV implies a theoretical uncertainty [ se points, but the closest we can D | 2 aightforwardly from the fact that ) − y ) vac h y − x − ( x be measured in the effective theory. This ( ret if -matrix theory. To see why the UV properties D can S – 14 – = i vac )] = 0 | y ( )] y ( , φ ) x , φ ( ) φ x ) in the full theory. The Fourier transform of [ ( x φ ( [ | ) vanishes outside the lightcone. is the Feynman propagator, O y i vac − h x vac 2 scattering amplitudes: with the wrong signs, these amplit | ( )) ret → y ( D φ Propagation of a small fluctuation around a background repre Λ on the position of the interaction, or equivalently on the p ) / x ( φ 2 scattering are relevant to superluminal propagation, it i ( T → In a local quantum field theory, the subluminality of the spee Exactly the same logic holds in the interacting theory. The s | vac Figure 4: 4. Analyticity and positivity constraints Interestingly, the UV origins of the IR pathologies we have f scattering events. the level of 2 retarded and advanced Green’s functions as satisfy the standard analyticity axioms of of 2 the propagation of a fluctuationwe on have top of described a corresponds backgroundfigure as 4. to a That sca the the leading re-summationof vertex is order of a derivative 1 all interaction tre our fluctuation emerges afterderivative having involves interacted knowing with the the fieldtake bac at two two arbitrarily points clo in the effective theory is a distance of order 1 consideration makes it clear thatquestion: the presence/absence it of depends s oninteractions the cannot signs be of extrapolated non-renormalizable down ope to arbitrarily sh around translationally invariant backgrounds followslocal str operators commute outsideh the lightcone. Recall that fixed by the microscopicdue UV to physics theory. oneffective In scales theory. a smaller However, than typical duringmany the collision, propagation scattering cutoff an in is events ashift. thus take translat unmeas place, Over large eacha distances contributing macroscopic and the time after sa advance many or scatterings, delay the that lated by some operator implies that Therefore, the vanishing of the commutator as an operator st JHEP10(2006)014 - S vity (4.3) ) in the ), so that ω ( y assumption n background − of the theory. -matrix. x i -matrix theory. ( -matrix theory. S ary value of an argument at the S S B | adv )] does. But exactly D , eld theories with the y ( y of an off-shell formu- − ) he interactions in the ) y y isely in the UV singular- , O localized on the lightcone enting causality; instead, erluminal propagation in ) − − x onal to derivatives of delta x bove argument that we are ( the UV theory is not a local m space, and the pole struc- x ( city and dispersion relations amework of ( antum field theories — what O o follow from the ndex of refraction se as a result of a sequence of is was a vexing question to operties of the tly stable particles) as dictated ow we would know whether the B adv ret rom the dispersion relation and ing theory? The only observable ue to the derivative interactions, out the light cone on scales com- to the axioms of D D ty should be encoded in analyticity − ) y − -matrix satisfies all the usual properties -matrix? After all, when we only have S x S ( B ret -matrix. As such, our conclusions apply equally D – 15 – S is interpreted as i≡ i B | translationally invariant background vac | )] y )] ( y any ( , O ) ) vanishes outside the lightcone. , O x ) y ( x -matrix. It is therefore desirable to see whether the positi ( O − S [ | O x [ ( | B h B ret vac D h ) represents the propagator for small fluctuations about the y Λ. Indeed, this is nothing but an operator translation of the − / x -matrix as a function of kinematic invariants is a real bound ( ) vanishes outside the lightcone since the operator [ S B y D − x This argument may appear too quick — after all, our effective fi The commutator argument is convenient when we have the luxur Indeed, how is causality encoded in the As we will show momentarily, the positivity constraints on t ( . Thus again, i ret B the same conclusion follows for Indeed, where | wrong signs for thegoes higher-dimension wrong operators with the are commutatorities local argument? associated qu The with problem their is onlythe prec being operator effective commutators theories. aquire D functions UV localized on singular the terms light-cone.parable proporti to These 1 serve to fuzz- has a delta function singularityture on is the such mass that shell in momentu lation as in local quantumquantum field field theories. theory, for But instance what ifin happens it string if is a theory perturbative str is the end of last section,collisions explaining with how the superluminality background candealing field. with ari a UV So complete itin theory, the with is commutators. no crucial UV in divergent terms the a constraints we are discussing follow more generally from pr In the end, thereall was the no physical physically consequences transparent of way microcausality of were implem seen t matrix theorists, who wanted to build causality directly in interactions giving rise to scattering are causal or not. Th access to the asymptotic states, it is not completely clear h D that the analytic function with cuts (andby poles unitarity. Of associated course with it exac isproperties unsurprising that — microlocali themediums textbook like explanation glass for relies on the the absence analytic of properties sup of the i effective theories we have beenthe assumed discussing analyticity follow properties directly of f the complex frequency plane. well to perturbative string theories, where the — unsurprisingly, as the Veneziano amplitude arose in the fr It is of course elementary and long-understood that analyti JHEP10(2006)014 s γ ) is s (4.6) (4.5) (4.4) ( A as . The poles 2 h channel Higgs M plane is shown r above Λ, and u s 2 h M linear sigma model; and came out positive in s , 4 ¸ ) 2 is equal to the sum of the h ee-level or fermions at 1- ). At leading order and at ∂π 2 ted is that these positivity I our shown in the figure M u th Higgs mass s,t are those depicted in figure 5 ( − − from the 0); note by crossing symmetry ments are a little old-fashioned u g effective field theories. ... , 2 M h → + M 2 h )+ 2 h ± 2 . M 2 s,t u M ) − t ( = s 3 − ( + s − s M 2 t A t + 2 scattering amplitude at tree level, in the theory πi + ds 2 scattering, ) = 2 2 2 h s → ( s – 16 – 2 γ → ( M s I A 4 3 const. Let’s further look at the amplitude in the − c − Λ = 0. On the other hand, = s → I I · ) )= 2 h λ s,t M s,t ( ( M M )= , 0, and define at infinity, s,t ( 2 → | s t | → ∞ M < | s,t . Of course this amplitude violates unitarity at energies fa is ) t s ( ) is bounded by a constant at infinity — more generally, as long − s ). The analytic structure of this amplitude in the complex M s ( s h |A − A − Analytic structure of the forward 2 ( = A u As a warm-up, let us understand why the coefficient of ( In the full theory, this amplitude has poles at Consider first the case where the theory is UV completed into a ) = s ( bounded by two of our explicitloop. examples— At lowest integrating order out inand the the 6. couplings Higgs Let’s the consider at relevant the diagrams tr amplitude for 2 low-energies, often imply positivity constraints (though since such argu arise from tree-level Higgs exchange we will review themconditions here can in serve detail)—what as is a not powerful well constraint on apprecia interestin exchange. in figure 6. Now consider the contour integral around the cont Figure 5: and of course as where of a Goldstone boson UV completed into a linear sigma model wi the theory needs a UV completion. the full amplitude at tree level is instead forward direction, as A JHEP10(2006)014 I s ππ γ (4.8) (4.9) (4.7) , . In the 2 h . By the s 3 M . Thus, . 2 ψ 2 /s h )] )] → corresponding 2 M h s 4 M s 2 ( Ψ ± M A M ssion it is useful to = 4 = s p? In this case, the ± s scattering, and we are ) is even in ( s = ( A ππ s en the analytic structures A les at order approximation. Let’s near the origin, there will be ) res[ 2 h 1 ng at ) since − 2 M h ion for s ion at 1-loop. The cuts starting at ) is the total cross section for 2 h ) s M − 3 4 ( ) is positive. M ) s s ) σ Λ 2 = is manifestly negative from eq. (4.5), ( falls off sufficiently rapidly at infinity, h ( from the pole at the origin, together − s / 3 ( s 2 σ 3 h 2 scattering amplitude at tree level, with M s ( πδ A I c ( = sσ M − A s → = ( ds = res 2 ψ 3 . Now consider again the contour integral ) where )= s s Z s ( /s ( ) 4 M – 17 – 2 + 2 π at s A ±∞ − sσ ( 4 3 A = c Λ A Im 4 3 as above. )= c = Λ 3 s ( c I ⇒ A 0= )] 2 h i² arises more generally from unitarity. Indeed, as M 3 + c 2 = h s M ( at its poles. Since the integral of the discontinuity across the cut disc[ Ψ 3 A − × s /s ) ) res[ s πi ( Analytic structure of the forward 2 (2 → A correspond to Ψ pair production. / ) must be positive. However for the purpose of the future discu 1 s 2 Ψ 3 ( × c M A 4 Of course this is not an accident. In fact the difference betwe What about the case with the fermions integrated out at 1-loo ± vanishes. As before, there is a contribution to = scattering, we have trace how positivity of of these amplitudes is completely an artefact of the lowest- and so optical theorem this is again related to the total cross sect led to the identical expression for to Ψ pair production,around and the curve extending shown to in theI figure. Again, since with 2 Since by the optical theorem Im simple example at hand the residue of Figure 6: which is manifestly positive since the cross section where the factor of 2 accounts for the pole at analytic structure is shown in figure 7. There is a cut beginni residues of the Goldstone couplings arisings from integrating out a Ferm a contribution from a pole at the origin, as well as from the po JHEP10(2006)014 + ππ 2 h plane M . This 2 s h − M s ( s = ∝ s presecription), omplex i² vicinity of , this only involves open Γ ts of section 3. Here, and ve a cut going all the way s h g ce is visible on the physical gauge boson exchnage. The e (tree-level) low-energy act, this is the usual general it-Wigner form heory satisfies the usual ana- s and becomes largest in the lytic structure is the same for tronger than those observable actly stable particles). Narrow erywhere in the complex plane, osons to the Cartan subalgebra, he second sheet. − i M citly that an analogous argument der the amplitude for gauge boson 2 h M 2 scattering amplitude. Poles associated . At 1-loop, we see the non-zero Higgs → 2 h positive for an interacting theory, rather M = s – 18 – is reached by continuing the amplitude under the strictly h is reached from above (as per the Γ Γ h s h to be iM 3 c − + i M 2 h h 2 M = − M s General analytic structure of the forward 2 . There is however no pole on the first or physical sheet in the c 1 − ) h Note that analyticity fixes It is instructive to explicity see how perturbative string t Γ h strings, and furthermore if wethe restrict amplitude the external does gauge not b have any contributions from massless analytic structure is exhibited inthe figure full 8. amplitude at Of allstructure course orders, of the and the ana forward the scattering Ψexcept amplitude theory for — as cuts analytic well. on ev the In realresonances f axis appear (and as poles poles associated on with ex the second sheet. than merely non-negative, asin was general, motivated by the the constraints coming IRin from argumen the UV effective analyticity field are theory s in the IR. lyticity and positivity requirements. Wealso can holds also for see perturbative expli stringscattering amplitudes. in Let’s consi type I string theory in 10D. At lowest order in cut to the second sheet.sheet — Of by course a the big presence bump of in the the resonan discontinuity across the cut in the iM — the expected pole at width Γ. As thethe physical resonance region is for seen since the amplitude takes the usual Bre Figure 7: with narrow resonances are reached by going under the cut to t consider the Higgs theory atto 1 the loop. origin The —scattering amplitude the cross will discontinuity now section. across ha theneighborhood The cut of low reflecting the energy th Higgs cross resonance section near grow JHEP10(2006)014 e or (4.13) (4.14) (4.10) (4.11) (4.12) + const. . In fact, π | symmetry s normal UV | π , → soft particles . ¸ ve. π , bounding the , as follows. As ) ) s s t t in 10 dimensions →− 2 − ln 4 − π ln 1 at large . s ,..., − ff exponentially with )Γ( yticity properties, all 2 1 ´ | s − ity. For simplicity, we e 1 + perm) s s that grow faster than 4 π ∗ s = positive up to power suppressed corrections (4.19) cut at ) = )+ Z 2 2 M M = 2 , we can say more. Such theories include, for instance, weakl = = g 3 s ) s is the perturbative quartic coupling in the UV theory, or per ( ( 2 ) 00 00 g s ( M A A 2 1 − sσ s ( Analytic structure of the forward 2 ds . Also, since at energies beneath Λ, 2 cuts This is all we can say in complete generality. However, in the Λ Z in this region must begin with a term of the form ( / , is an analytic function in the complex plane. Its power expa 2 2 0 coefficient. to the dimensionful scale Λ there is a dimensionless weak cou has an expansion in linear sigma model completionsHiggs of mass non-linear and sigma models, s Λ we have that Figure 8: So, said precisely, the forward amplitude Thus we conclude that Because of the derivativem interactions, the second term abo the discontinuity across the cuts is determined by the total theory thus have JHEP10(2006)014 , 2 2 g Λ (5.1) (5.2) (4.20) (4.23) (4.21) (4.24) ¿ s act a lower . For s g , (5) R , appear at order , g , to the lower of the two ontributions from low- G ¶ − in ¶ which modifies gravity at . For example, the forward √ tring theory is a 4D brane localized at an ...... number of constraints on the + + 6 xdy lized on this boundary, with an ing the contour integral 4 s d 12 5 n h 6 s π ´ 15 M 2 4 2 bulk is the string coupling infinite ) s Λ Z + + s +1 g ³ ( 3 n 5 4 8 h 2 n 4 ) amplitude in the forward direction has s A c s M s 2 4 g 5 3 M ) we conclude 0 (4.22) 3 and π g =1 M πi M ∞ ds + X s n + 2 > 2 + g 4 ∼ h n M γ 2 2 – 22 – (4) c , and at larger distances gravity on the brane s I Λ M R πs )= = µ /m g µ s 1 ( λ n s − I with all positive coefficients g √ ≡ tree term quasi-localizes a 4D graviton to the brane up to 2 x s 3 A 5 2 4 0) = 4 d 0) = M → /M 2 → 4 brane M s,t Z ( s,t 2 4 ( ∼ M M M c r = 2 the leading (in weak coupling . However, as in the case of massive gravity [17], there is in f . The large 5 S 5 M M À 4 M Naively, this model makes sense as an effective field theory up the tree amplitude in this theory is of the form precisely as needed for 1-loop unitarity. Thus, by consider Therefore, in a weakly coupled theory, there are an effective theory: scattering amplitude in the Goldstone model is distances of order with reverts to being 5 dimensional. Planck scales scale Note that low-energy cuts, which are absent at leading order an expansion as a polynomial in while the amplitude for gauge boson scattering in 10D type I s string theories, where Λ is the string scale and running through theenergy same cuts argument which don’t (and exist now at ignoring this the order c in both of which of course have all positive coefficients. 5. The DGP model The DGP model islarge an distances. extremely interesting In brane-worldorbifold addition model fixed to point gravity with in aaction a large of Einstein-Hilbert 5D the term bulk, form loca there is JHEP10(2006)014 ) x ( π (5.5) (5.4) (5.3) (5.6) n the ld fixed point. is transformation km. If, at quantum 3 km [18]. It is therefore 10 ition of the brane along rangian is preserved by 3 lthy classical non-linear ∼ ng the “self-accelerating” nvenience. Note that the ally follows from this non- 1 is mode can be isolated by y a direct 5D calculation in − volve at least two derivatives bending mode, and that the [20], it was shown that at loop , e classical theory, the correct race of the energy momentum ically. eath 10 was initiated in [18], where it was . ed. The same holds for the kinetic π . ¤ )—loosely the “brane-bending” mode 3 2 µ ) x c Λ ( Λ = const . . . so the physics is purely four-dimensional, π + ∂π ( + π µ − 4 N ∂ , 2 − 2 ) → ∞ , ) – 23 – , keeping Λ fixed. In this limit both four and five N are generated, and with additional assumptions π → c 4 r ∂π ∂π Λ N π ( [20]. Indeed, the non-linear properties of this theory ) →∞ µ π ∂ →∞ π 4 2 ) = 3( 5 4 ∂ , M L 5 , M /M 4 µ M µ is the only relevant mode, M ) is (in some gauge) related to the canonically normalized T π x ( brane y . The model enjoys of course full 5D Lorentz invariance, but i π 4 1 mM . 2 —however the variation of the cubic term in eq. (5.3) under th reflection symmetry is broken since the boundary is an orbifo = ) π π ∂π brane This symmetry is nothing but 5D Galilean invariance. The pos These results all follow from the fact that the form of the Lag Now, for realistic parameters, the scale Λ corresponds to Λ y → − Naively this suggest that anyon term every in the Lagragian should in the fifth dimension is a total derivative, and thereforeterm, vanishes ( once integrat by about the structureproperties of of the the UV theory survive theory, quantum-mechanically. [20] argued thata the constant shift hea in the first derivative of interesting to consider loop correctionsshown in that this the theory, tree-level as cubiclevel term is only not operators renormalized. of In the form ( leading to the effective action [18] are what allow it to be experimentally viable, at least class dimensional gravity are decoupled and level, all operators of the form are generated, then,quantum despite theory would the lose interesting all predictivity features at distances of ben th The unusual normalizationLagrangian of is the derivatively kinetic coupledπ term as is expected for forAll later a the co brane- interesting phenomenologysolution of the (which DGP is model actuallyref. — plagued [19]) includi by as ghosts, well aslinear as confirmed classical the b Lagrangian modification with totensor the the for scalar matter lunar coupled fields orbit as to — ( the actu t at which a single 4D scalar degree of freedom decoupling limit in which taking a decoupling limit as — becomes strongly coupled [18]. The classical action for th JHEP10(2006)014 , 2 e ) π ∂φ ( (5.7) 3 1 3Λ ortional − φ = π type terms in the 3 . N , as needed to reproduce π term in the forward 2 2 ) ) 2 φ s (1) Goldstone boson , ν ∂π ∂ U as contributions from the ¢ . c Goldstone theory is the µ scattering has a tree-level n ) (5.8) ∂ T a ) ( 4 rs of Λ and there will be no venient — first, because it makes 2 π e of ( ) goes to zero and a 5D Lorentz for a scalar. The only thing 2 -matrix. As we saw in the last bility. , u ππ ∂ 4 S ∂φ ( ( terms in the effective action that tives on any ,t 6 forward amplitude at even higher m 1 4 Λ brane ∂ s N y ( the coupling to matter is simple: a linear ¡ 2 te the cubic interaction term, but the theory ) µ rm O O ∂ some ∂π + + 2 3 3 ) s u ∂π + ( symmetry), would have the same leading cubic 6 3 π – 24 – t Λ of such ( π ¤ + 3 terms is the only thing making this effective theory 1 3 Λ s term, which gives rise to an →− N . Instead, in the DGP model, this operator is forced 2 4 − ) π ) 2 absence )= ) ∂π ∂π as a Galilean transformation. This symmetry forces the s,t ∂π it is impossible to complete an effective theory for a scalar ( M Lowest order scattering amplitude in the DGP model. brane 2 scattering amplitude is non-zero. The field redefinition = 3( to the trace of the energy momentum tensor y 0, this amplitude vanishes; and in particular the piece prop this property of the theory makes it impossible to UV complet L 4 → → strictly positive t (and hence the πT/M CP Figure 9: precisely 2 amplitude. However, the cubic form of the action is most con vanishes identically. of course there will be → However, Indeed, the absence of the ( 2 Of course it is possible to make field redefinitions to elimina piece. We conclude that s 3 2 amplitude, must be to vanish by the Galilean symmetry. The amplitude for Lagrangian to take the form exchange contribution from the DGPhigher term order (see term, figure but 9) they as all well begin at order which violates transformation acts on gives it a chance for non-linear health and experimentalinto via a UV theory with usual analyticity conditions on the section, the coefficient of the ( the 2 In the forward limit Lagrangian. And again, it is the is not free, the tree-level 2 coupling of the form the Galilean symmetry simply manifest, and second, because eliminates the DGP term but generates quartic terms of the fo that is all further interactions involve at least two deriva orders, but these wills involve even more suppression by powe special in any sense. After all, a generic UV theory yielding to the brane becomes flatter and flatter, the ‘velocity’ that can distinguish thepresence DGP of the scalar Galilean Lagrangian symmetry and from the a associated generi absenc interaction, which is the lowest order derivative coupling JHEP10(2006)014 2 ϕ local cally (5.9) (5.10) , not 2 ) ϕ . This is an ! is sourced by Ω r ∂ and that of ˙ = 0, reaches a ( π 2 backgrounds can r ) → ∞ !# π ϕ 0 0 r V r r is [20] π -like solution in DGP. ∂ R 3 at ϕ space; for instance for  + / 00 0 r mething that is mpact spherical object of π -trivial n therefore have any sign. , à ent of ( propagation. 3 ¸ n of the speed of propagation of a fluctuation moving along 2 2 Λ r al indication for the validity of uctuation 2 rad − 0.6 into a UV theory with the usual 3+ c ible — the leading interaction term r " µ 3 V c − is larger than 1 for any 2 R 0.5 ) + π 2 rad ϕ 1 π c r 18 µ ∂ ). The gradient of this solution is [20] ( ∂ r + 0.4 ( # 4 0 0 0 . The speed r r π → π – 25 – 2 ) q 3 ϕ π 4 · 0.3 Λ µ ~ . Again, this includes any local quantum field theory ∇ 3 ∂ r 4 3+ 3Λ " 0.2 − is the so-called Vainshtein radius of the source. In such a )= -matrix 2 and asymptotes to 1 (from above!) for ˙ r 3 S ϕ ( is given in figure 10: it starts from 4 / 0 0 V 1 0.1 r π ) !# R 4 0 0 cm. Clearly highly boosted observers can observe parametri π r ∼ 2 rad /M 2 20 ∗ r c 1 + versus 10 M 0.8 0.6 0.4 0.2 1.6 1.4 1.2 00 0 π ∼ Λ ( 2 at 2 rad à / / c is the angular part of ( is 3 The speed of radially moving fluctuations in a Schwarzschild 2 2 develops a radial background Λ ) V = 1 π ϕ R , Ω V 3+ ∗ ∂ " R M = A plot of Associated with this, it is easy to see that signals about non ϕ (1) deviation from the speed of light in an enormous region of , the trace of the stress energy tensor. In the presence of a co L Figure 10: maximum of 3 O the Sun, in the equation above; on the solution eq. (5.9) analyticity properties for the or perturbative string theory.the Conversely, DGP any model experiment can then be taken as the direct observation of so travel superluminally. It is trivialis to see cubic, that and this therefore is around possfor a small background, fluctuations the modificatio isAnd indeed linear simple in physical the backgrounds background allow superluminal field and ca QFT or string theory. T mass where the radial direction is given by the ratio between the coeffici with a shift symmetry of the form where ( Schwarzschild-like solution the quadratic action for the fl JHEP10(2006)014 . x 0 π the , so ν field 0 ation ∂ (5.11) (5.12) terms. µ π ∂π ∂ 4 ) ∂π assumed tions. Once this , and after a while 0 replaced by paradoxes as we dis- π 2 and negligible eric Goldstone theory, ν then the field equation ∂ ere the cubic dominates 0 e there we C his is not easy to check: bserve the peculiar time- signals are possible since π ike curve the two masses µ 2 homogeneous and isotropic term saves the day. We can C ∂ ϕ . n regions do not overlap, the taken into account. It would 4 ike solution we just described an observe parametrically fast ) ν r in the background, sition of two Schwarzschild-like e should appear in the doxical field configuration, here ut we also have the ( nd understand whether a closed to the other, and vice versa. In ∂π ϕ∂ ; but this region cannot be larger . µ 4 ) ∂ 2 for constant / 2 ) 1 0 / 0 ∂π x 5 π π 2 Λ ) this interaction gives a contribution to ¤ ∂ field boosted towards each other in the x ( µν 0 π Λ, too small a time interval to be measured ∼ η π / in order for the effective theory to make sense. 1 − 3 0 – 26 – L δc ¿ Λ π ν ∼ ∂ give rise to the same closed timelike curve problems ¿ µ max 0 y ∂ is positive there are no superluminal excitations. ( π max δt 2 3 4 δt 2 ∂ ) Λ grows linearly with ∂π 0 = L ∂π δ is exactly of the form eq. (3.1), with ϕ , since 0 π 2 , so the maximum time advance/delay we can measure for a fluctu 3 /∂ Λ / Λ 0 term starts dominating the kinetic Lagrangian of the fluctua π p 2 4 ∂ ) ∼ has no a priori positivity property. However the ( 0 ∼ L ∂π π The correction to the propagation speed inside the region wh If we turn on a background with constant second derivatives, Having found superluminal propagation, we run into the same It is instructive to contrast this with what happens for a gen ν δc ∂ µ than that the effect of the cubic dominates over that of ( certainly set up in some region a background with constant Exactly as in the∂ DGP analysis, it appears that superluminal the quadratic Lagrangian for the fluctuations which is linea traveling all across the ‘superluminal region’ is is In the presence of a generic background field for the fluctuation In such a case we immediately get Now, we would normally require fast propagation, and indeedreversed if sequence they of boost events. toobackground configurations much It for they which is can evensignal also observers o at propagation. easy rest to c find spatially cussed in section 2.direction For with instance a two small blobs separation of in happens, if the coefficient of ( the ( as in the two boosted blob Goldstone examples. However, whil presence of suitable sources thatwe could expect give something rise to more. ourfeatures Since para the superluminal simple propagation, Schwarzschild-l a closed timelike curv actually sourced by twoa masses quick boosted estimate towards showsmust each pass that other. so in close order T presence to to each of other one close that, mass theother even induces closed if words, sizable the their timel non-linearities full Vainshtei close solutionsolutions is — not new just non-linear thebe anisotropic linear interesting corrections superpo to must further be investigatetimelike such curve a really configuration arises. a where the leading interaction is still the same cubic term, b JHEP10(2006)014 (6.2) (6.1) · · · estigate these . In both cases + term have to be 1 − e can not have a 2 4 a classical theory, ¤ ) ˜ Λ the effective cutoff ) ity properties, there U 3 ∂π ∼ ime of validity of the Gravity on the brane, ν Λ 0 ∂ int on the DGP model π † horter length scales. In 2 À U ct of the cubic dominates re obstacles to making any distinguishes it from more µ he cubic term, a conclusion f the ( volve trying to make some /∂ ory like eq. (5.3) consistent 0 and the cutoff of the theory ∂ milarly, it appears that other Λ ve graviton. However, just as ious sections, which also have π sional theory in more detail. ments of the previous section. , parametrized by the unitary ymmetries in quantum gravity 2 ective field theories in particle Pl theory suffers from a lack of a tr( p ∂ ations of motion; this property ated to a violation of the usual £ M 5 ∼ L fixed. This suggests that there is a L pointing in a specifc isospin direction + 4 2 π µ ¤ ) x /M 2 µ 5 U c µ M ∂ † )= U – 27 – x µ ( ∂ 3 Λ. In this case too the superluminal time advance π / tr( holding 0 £ π 4 2 L ∂ → ∞ p )+ 5 ∼ U , of the form . We have seen that a simple physical principle — requiring µ 3 ˜ /M Λ ∂ σ 4 † U M → ∞ µ ∂ 2 4 tr( /g , while in [9], it is the attempt to keep 5 2 in any sensibly causal theory — it would be interesting to inv , f a 5 M σ a = /M À iπ L 4 e 4 M = M U There is also an interesting connection between our constra Of course, even in brane models respecting the usual UV local We have uncovered a subtle inconsistency of the DGP model. As There is a solution of the equations of motion with which we can take to be fixed, but send 1 subluminal signal propagation — prohibitsgeneral the physical DGP principles limit. — Si such— as block the absence taking of the global weakinteraction s coupling physically weaker limit. than In bulk both gravity. cases, there a 6. Positivity in the chiral lagrangian There are similar positivityphysics. conditions Consider in for more instance familiarfield the eff SU(2) chiral Lagrangian limit of questions from the geometrical perspective of the five dimen and the “weak gravity”interaction conjecture much weaker of than [9]. bulktaking gravity Both — in situations DGP in it is the 4D it has well-defined, two-derivative,underlies Lorentz the invariant healthy equ non-linearbrutal modifications behavior of of gravity, suchin the as the the theory simple theory scalar and of fieldLorentz a theory invariant massi examples two-derivative studied equations in ofLorentz-invariant the notion prev motion, of the causality, whichanalyticity properties is of in scattering turn amplitudes. rel are DGP terms induceddecoupling on limit the with brane. What we have shown is that w positive, it must be setwe by could the have same also reached scale from as the the dispersion coefficient relation of argu t scale is raised from Λ to inside the effective theory.assumptions However about [20] the argued UV thateffective physics in theory can a to the be muchparticular, made larger in to background the fields extend presence and the of to reg much a s strong background field the quartic saves the day. Thus, not only does the coefficient o is unmeasurably small:over the the size quartic of is of the order region of in the which UV the effective cutoff, effe JHEP10(2006)014 , . k πf 4 W/Z nality ∼ 0 (with , and we positive, 2 > 5 , /g 5 4 , 1 4 L at the scale Λ − ). These oper- ity constraints L e 5 , µ 4 D L aint is then actually → µ s a local quantum field ∂ our present example was e, satisfy the usual analyt- is approximately constant around this background. It 3 le dependence. On the other itivity bound is a non-trivial from the high-energy physics cally smaller than Λ onstraint on the UV physics. ± inear sigma model, there may π rgies of order Λ at low-energies are dominated re the leading irrelevant inter- ian is the same 5 The situation is perhaps more erature long ago [24]. µ π , al SU(2) is a good approximate raints following from dispersion precision electroweak physics— 5 lf through the higher-dimension 4 ∂ , 4 L case a low fundamental scale close ng the dynamics of the longitudi- L to propagate subluminally we must ± π as well as 0 (6.3) 3 and (1) gauge symmetry π > 3 U 5 π , 4 × , the matching contribution to – 28 – evaluated near this scale are positive. Of course, g L 5 , 4 L bosons. While it is most likely, given precision electrowea = 1, they represent anomalous quartic couplings for the U W/Z induced off the lowest-order 2-derivative term pushes much larger than the cutoff scale; in order to avoid superlumi 5 , 4 -matrix theory, give rise to effective theories with positiv R L S logarithmically scale dependent; so the positivity constr are 5 , 4 L Of course the pion chiral Lagrangian follows from QCD which i Naturally, the existence of these sorts of positivity const which must be positive. 7. Examples from string theory 7.1 Little string theory As we have seen, UV theoriesicity which properties are of local or, what is the sam the derivatives covariantized for the SU(2) ators are not associated withinstead, the in well-known constraints unitary of gauge theory, so these conditionsinteresting must for neccessarily the be electroweak chiral satisfied. nal Lagrangian components governi of the constraints, that the UV completionalso involves be Higgses more and exotic a possibilities, l includingto in the the electroweak extreme scale. Thisoperators in physics the should effective manifest Lagrangian, itse symmetry, and the assuming constraint custodi on the electroweak chiral Lagrang theory are verydiscussed well (though known, perhaps though not widely not recognized) said in the very lit explicitly; from the low-energy running contribution,constraint. and hence the pos can independently identify the matching contribution to will dominate over the log-running contribution down to ene by the log runningHowever contribution in and theories with there a is weak coupling no interesting c and so in a theory without a weak-coupling expansion, the log running of over a length scale we should demand that that running a constraint on the running couplings at energies parametri have In our previous Abelian examples,actions the in 4-derivative the terms theory, we andhand, so did not have any logarithmic sca Indeed, we can imagine turning on a background where is then easy to check that in order for both We can look at the small fluctuations of both JHEP10(2006)014 r µν η (7.1) 0) su- , , and far N perluminality (1) ion 3.2, so we U that the LST is coefficients in weakly dimensional (2 prove 4 etions of higher-dimen- theory, which is a non- F T is really non-local, the six lly non-local. However, as operators in the low-energy open string modes localized Minkowski metric. To check system in string theory is a er grounds, we can use them 4 modified by higher-dimension on-local theory. But if we also nd find that they are zero or D Yang-Mills coupling but no µν ide the underlying Minkowski on-zero value of antisymmetric F es to ocal CFT. On the other hand, five dimensions, maximally su- r a “non-local” theory. This is- . uminality constraints, since the B is no well-defined way to extract λ ν s are dominated by long-distance erluminality and macroscopic non- B -theory, which does not have a weakly µλ terms. related to the closed string metric B 4 M 2 ) F -channel poles, and the dispersion relation 0 µν t G πα ’s are much heavier than the little string scale, (2 – 29 – W − µν η = µν (1)’s have been computed [21]. They are all positive. This G U terms have in fact been determined by a variety of methods. Fo 4 F ) theory has a Coulomb branch where it is Higgsed to is a direct analogue of the effective metric discussed in sect N along the brane world-volume. It is instructive to see how su ( U µν µν G B coupling between the 4 Some of these The prime candidate for such a test is of course However, we can certainly study non-gravitational UV compl F expect its causal cone to be contained in the light cone of the The metric in the following way [22], on D-branes propagate in the effective metric constraints work in this case. In the presence of non-zero stack of parallel D-branes intensor flat field space-time with constant n 7.2 Non-commutative theories Probably the best studied example of the Lorentz-violating coupled theories where thenon-locality did signs not are put in determined. an appearance Thus, in if the LS SYM theory — ifdramatically any non-local. of them have the “wrong” sign, this would instance, the is perhaps not surprising, since they are related by dualiti the out along the Coulomb branch, where the we saw in the lastinformation section, about in higher-dimension gravitational theories, operators there notion from of superl the correct metric tooperators, use for the while GR lightcone just canlightcone. be gravity already Associated with bends this,graviton all exchange gravitational in signals amplitude the forward ins direction, with coupled description, and is thought by many to be fundamenta on certain leading irrelevant interactions thatlocality. forbid sup If wenegative, experimentally then we measure have such direct evidencehappen interactions for to a a know fundamentally some n ofas these a operators locality theoretically, test on for oth the UV completion. sional gauge theories, especiallypersymmetric supersymmetric Yang-Mills ones. theories In are UV completed into the arguments can’t be used. perconformal theory, which although6D mysterious is super-Yang still Mills a isgravitational l string UV theory completed with intosmall string the dimensionless tension coupling. 6D set This little bysue is the can string another be 6 candidate probed fo if we can determine the coefficient of the JHEP10(2006)014 µν ne η ne”. In suppressed Pl /M strings decouple from non-commutative field heory of gravity in flat of open modes vanishes ational fields we can cer- ric by 1 og of our superluminality vanishing of commutators ski space has a metric ry, it is natural to define 2 ich is broken by the higher- ) on of massive charged string inkowski light-cone”. On the urse require that the electric j ity parameter. Interestingly, t massless particles propagate ) and calculate its norm with l gauge invariant operators in i n the usual light cone. Note that mate Lorentz-invariance (with causality because these solitons faster than the speed of “light” -cone, the cone of null geodesics eory of a spin-2 field interacting ij ,n changes its singature. Physically . In a theory with gravity, there is c which is the true Lorentz metric B − µν , but in the presence of strong field as = (1 j G n µν µ j n B E i n − i – 30 – E [23]. This fact however, neither contradicts to our ( − µν . One finds , and it is natural to ask whether there are constraints G = µν Pl is either space-like or null with respect to the open string ν G M n µ n µν G µ n irradiating from a point; we call this the “gravity light-co . The effect of any higher dimension operator on the GR lightco µν µν g g . Consequently, i -matrix. 0 S B suppressed operators from our considerations. = i Pl E /M However, we might take another track. For weak enough gravit It is easy to see that there can’t be any straightforward anal In the limit of large magnetic field the propagation velocity the weak field approximation we are considering, the fact tha and a well defined light-cone, whichother we hand shall a refer classical to gravitational asof field the defines “M the a full new light metric operators. There is alsoof no local analog operators of outsidegravity our the — arguments light-cone. one using There of the space are the is no reasons the loca the only observable in a quantum t tainly think of General Relativitywith as matter a Lorentz fields invariant in th Minkowski space. The underlying Minkow can then completely be absorbed into a redefinition of the met on 1 constraints in GR.the The light-cone reason by is that in a gravitational theo still propagate inside light coneof of the the closed underlying string theory. metri 8. Gravity So far, we have been discussinga non-gravitational natural theories UV cutoff scale of order superluminality constraints nor leads to any problems with as compared toopen the modes speed and of dynamicstheory. closed of modes. At the the openopen classical In sector string level is this metric this described playing limitdimensional theory the by closed role the operators exhibits of approxi proportional thethese to Lorentz theories metric) the allow wh soliton non-commutativ solutionsas which defined can by the propagate open string metric at this point vacuum becomesmodes. unstable towards pair producti well. For the background offield the is electric smaller type than one the should of critical co value when metric metric, implying that its causalthis cone conclusion is holds indeed not contained only in for small values of where respect to the open string metric this let us pick up an arbitrary light-like vector JHEP10(2006)014 4 (8.2) (8.1) (8.3) rsion nt, the contri- ravitational field. pproximation we are e can further constrain is negative somewhere, ible to constrain higher- ν . he gravitational field has is null with respect to the k s either to instabilities at ) Notice that a violation of µ r µ obeying the above equation we have been discussing so etric, which means that the k at the Minkowski light-cone k rs ago in [25]. µ + , nstance a plane gravitational ce the Einstein action already ull geodesics move outside the ce we want to preserve Lorentz k µν tion of these particles with the x ground field. Indeed we can ask nteracting with the background T tor ) = 0. Einstein’s equations then µν ry one, and negative definite. trajectories inside the Minkowski he source the wave field decays slower ying Minkowski light-cone. r, in the background field itself, and ion in the matter sector [12]. The µν . − t h η πG T trenghts, but the potentials themselves both ( 1 2 = 0 16 µν linear ν − − T k due to the very same sources that emitted µ r 1 µν k r µν )= ) h 3 h ( d µ µν µν – 31 – gravitational field ouside the sources is therefore ∂ h Z h η − G 2 1 4 . µν − − η µν ( h µν retarded h )= ( x can be made positive only if ¤ t, ν ( , k definite sign. In fact in the geometric optics limit the dispe µν µ µν h k h = 0. The + µν h h µν physically means that even if gravitational waves are prese η a priori = µν h µν . g is the sources’ stress-energy tensor. Outside the sources w /r µν is not a negative-definite matrix, then there exists a T However we must wonder how we actually set up the background g We are therefore led to the conclusion that, in the weak field a This is a satisfying result. However, it also makes it imposs At first this seems to be trivially possible, since the effect t µν At first this seems to contradict the usual fact that far from t 4 h For solving Einstein’s equations we need toinvariance, fix we the choose gauge. De Sin Donder gauge, that is time-like withparticle respect can to travel the outside underlyingwave Minkowski has the m a Minkowski non light-cone. negative-definite For i the gravitational waves is always larger than the oscillato in, it is impossible tounderlying set Minkowski light-cone. up a Gravity ‘bends’ gravitationallight-cone. all field null This such that conclusion n was also reached a number ofdimension yea operators using superluminality arguments — sin but this is inthe contradiction Null with Energy the Conditionarbitrarily Null under short time-scales Energy very or broad Condition. tonegativity assumptions superluminal of lead propagat the gauge by setting bution of the static Newtonian potential to than the static one. This is indeed true at the level of field-s decay like 1 where along the gravity light-conebackground takes into gravitational account field. thefar, interac This the is propagation very of similar signalswhether in to it a is what Lorentz-violating possible back lies to inside turn the on gravity agravitational one, gravitational field, so can field that actually such travel massless th outside excitations, the i underl on the dispersion relation ofas a such massless it particle has is no read If full metric Now it is clear that relation is simply the statement that the particle wave-vec JHEP10(2006)014 s (8.7) (8.5) (8.4) (8.8) (8.6) 0. On tells us → t poses it is is also ap- t , Pl 0 the leading M → for fixed t ever, there is some has a good behavior ed at infinity; in fact l, massless 1-graviton ) NS bosons of type II u 4 1 tes. We can isolate the essed by ) → ∞ Regge u ee-level graviton exchange s 1 4 by the lowest order massless M -behaved limit as − oefficients of higher-dimension 0, f the form and consider the amplitude for ee amplitude has a contribution )Γ(1 + der operators can’t change this sart bound. t )Γ( → t 4 1 , t 2 scattering in the forward direc- 1 4 , − → is by definition the contribution to the 2 t K Regge s )Γ( with all positive coefficients s )Γ(1 + s t u N 64 M s 1 4 , and therefore, upon subtracting the tree- 2 1 4 Regge G s s − − + -channel pole, the amplitude is polynomially t ∼ M = Γ( ) – 32 – grav Γ(1 + s,t grav M ( at large K . The amplitude can be expanded as M 4 2 = s M 4 s g /t ) t − ∼ ( M α s K + polynomial in )= 2 scattering amplitude as plane. Thus, we should expect that as 0, 2 t 2 scattering amplitudes. From our previous experience, it i s → s → N s,t,u → t ( G is a factor depending on external polarizations; for our pur M → ) K s,t ( M 0. However, in a gravitational theory, already at tree-leve 0, and it is easy to check that it is indeed polynomially bound → t Let us see how this works explicitly for the scattering of NS- In the context of a UV theory with a weak-coupling factor, how The difficulty of bounding higher-dimension operators suppr → t level graviton contribution that removes the bounded in the complex amplitude is of the form that the amplitude behaves as where as parent in discussing 2 evident that there aretion positivity constraints on 2 exchange dominates the 2 tree-level amplitude from the heavy Regge modes. Of course which again implies an infinite numberoperators of constraints in on the the c theory. strings in 10 dimensions. The full scattering amplitude is o so again, the effect of1-graviton any exchange higher order which operators dramatically are violated swamped the Frois is the tree-level graviton contribution, while pushed signals inside theconclusion Minkowski regardless light-cone, of higher-or their sign. hope. For instance, considermassless weakly closed coupled string string mode theory, scattering.from The graviton lowest order exchange, tr ascontribution well from as Regge fromdiagram. states the The simply tower resulting of by subtracted Regge amplitude subtracting then sta the has tr a well where once again only important that as the other hand, the Regge behavior of the full amplitude as JHEP10(2006)014 s π (8.9) at encode giving rise . . . nce contribution to + dimensional effective 8 s der operators in gravi- be embedded into local (1) es force the leading inter- o full theories that satisfy signals about consistent 6 re the coefficient of some of ψ not ying 5D theory, under which 92160 − ies satisfy these properties, so eld theories described by local, ger constraint that the leading and Lorentz-invariance: in such t. e scattering amplitude, the co- tive. nal excitations around coherent certain gravitational amplitudes troweak chiral Lagrangian, these + spersion-relation arguments force 6 ity exists. The high-energy face of gnal propagation around coherent he low-energy manifestation of this s (1) 4 192 ψ − + 4 s – 33 – that is all that is left in a decoupling limit sending fixed. This scalar theory is controlled by a nice (1) (1) are negative. π 2 n with any negative coefficients, it can’t come from a 4 to vanish, in contradiction to the strict positivity of ψ ψ 2 2 − s s /M with all positive coefficients. 2 5 2 . From our general argument, all the terms beginning with s M s 0) = quartic gauge couplings in unitary gauge. Thus, UV locality term in the forward amplitude must be strictly positive. Thi → Z 2 / s s,t -matrix theory, specifically the analyticity conditions th ( W at large S 4 | s Regge | . This symmetry forces the coefficient of the leading operator keeping Λ = µ M c + π µ → ∞ ∂ come from a weakly coupled string theory — if the short-dista 5 → Similar considerations apply to the chiral Lagrangian, whe Our analysis applies to the DGP model — specifically the four- In effective theories where the particle content or symmetri Thus, while we can’t say anything in general about higher-or not , M are guaranteed to be positive, and indeed π 4 µ 4 this coefficient in any UV local theory. Associated with this, backgrounds can propagate parametrically faster than ligh the 4-derivative terms are forced toturn be into positive. anomalous In the elec implies that such anomalous couplings are necessarily posi theory for the “brane-bending” mode M to an amplitude proportional to symmetry — interpreted as Galilean∂ invariance in the underl requirement precisely guarantees the absencebackgrounds. of In superlumi weakly coupledamplitude theories, is there a is power the series stron in perturbative closed string model. 9. Discussion We have shown that certain apparentlyLorentz-invariant consistent Lagrangians effective are fi secretely non-local. T lurking non-locality is thebackground possibility fields. of This superluminal creates a si theories tension between no causality Lorentz-invariant notion ofthis causality or tension local is thatthe such usual theories axioms can not of be UV completed int locality. Both local quantumwe field have theories provided and a stringQFT’s theor simple and string diagnostic theory. for theories that can actions to be irrelevant operators,positivity completely standard conditions di in theseefficient interactions. of the In leading terms of th it is bounded by tational theories, there is ado simple diagnostic for whether the amplitide is a polynomial in s where all the polygamma functions JHEP10(2006)014 2 → int on the low-energy re of course implicitly ries, and which are in nnections to the physics rious models under con- ysics. uge boson couplings, can or weakly coupled string tz-invariant theories with a t seems possible that there [12], it would be interesting l Lagrangian, precisely the y be consistently embedded e have discussed. ted, and if so, what is the IR lete Quantum Field Theory n afoul of positivity. uminality and the breakdown bles. While our arguments do ically non-local physics unlike e possible to see behind black bles at them, which runsc afoul properties. Another concrete usly breaks Lorentz invariance, erful probe into the validity of eresting to identify constraints instance in [24]. However, the her properties of UV-complete volving fermions, gravitational ive theories that can or can not ymptotic Minkowski space that ance, any experimental support ed in [9]. It would be interesting ous beyond-the-standard-models . interesting varying speed-of-light , a Rindler observer immersed in n see behind the nominal Rindler -matrix — do they lead to interesting S -matrix, namely, positivity of forward 2 S – 34 – Note that our results are quite likely to have interesting co Similarly, it would be interesting to apply positivity to va The UV face of the positivity constraints we have described a All of our considerations in this paper have concerned Loren More generally, we have studied only one very simple constra bosonic scattering amplitudes. For example,deriving it from would higher be int scatteringamplitudes, amplitudes, etc. amplitudes It in wouldmodels also above and be beyond interesting simple to analyticity identify of the ot pathology of a theoryare which many runs more afoul constraints of onin such just UV-complete a which theories, constraint? effective and, I theories in ma particular, string theory constraints on which effective field theories may be UV comple Lorentz-invariant ground state. It isallows the us presence to of get the intonot paradoxes as directly involving apply highly to theories boosted in bub such which the as vacuum Higgs spontaneo phases ofto gravity ask [6 – whether 8] there or are the models any analogous studied in constraints to those w well-known in dispersion theory.positivity constraints In we the describedirectness have context with been of which these discussed the important for arise chira signs from pinpoint local effect UV theories,of and macroscopic the locality, has connection not withfor been superl the appreciated. DGP For model, inst be or taken as negative a signs directanything for indication anomalous ever of quartic the seen existence ga intheory. of Experimental physics, macroscop tests either of insome positivity of then quantum our provide field a most pow firmly theory held assumptions about fundamental ph of horizons. For example,the in translationally theories invariant superluminal violating background positivity horizon ca to see allhole of and Minkowski space. cosmological horizons Similarly, byof it tossing all should the superluminal b usual bub rules aboutlink horizons between and positivity their and thermodynami black-holeto physics was further develop explore such connections. sideration to identifyfact which fundamentally non-local. can Obvious be candidateswith are embedded leading vari derivative in interactions, as local wellmodels as of UV the interest host theo in of cosmology, many of which very likely ru effective field theoryderiving of from a the local, analyticity Lorentz-invariant of UV-comp the UV JHEP10(2006)014 , Phys. Phys. Rev. , ]. ]. , (1987) 2607. 59 ]. , low strong coupling pported by a Junior (2005) 073 . 06 69. Ghost condensation and a hep-th/0409124 hep-th/0312099 . R is partly supported by r for deconfusing him with rsation about it. N. A.-H. achru, Markus Luty, Aaron e, Misha Shifman, Raman , ques Distler, Gia Dvali, Gre- alytic structure of amplitudes Phys. Rev. Lett. f N. A.-H. is supported by the ini, and Arkady Vainshtein for hep-ph/9707380 , JHEP ularly like to thank Aaron Pierce , note. We thank Howard Schnitzer The anthropic principle and the mass Null energy condition and superluminal (2004) 076 [ (2004) 074 [ ]. The string landscape, black holes and hep-th/0509212 de Sitter vacua in string theory Predictive landscapes and new physics at a 10 05 , ]. (1998) 5480 [ . . ]. The accelerated universe and the moon 4d gravity on a brane in 5d minkowski space JHEP JHEP , , – 35 – D 57 Supersymmetric unification without low energy ]. hep-th/0512260 hep-th/0005016 hep-th/0407104 hep-th/0601001 Phys. Rev. , , , hep-ph/0212069 (2006) 025 [ . hep-th/0301240 03 (2000) 208 [ ]; Phases of massive gravity Lorentz-violating graviton masses: getting around ghosts Anthropic bound on the cosmological constant JHEP B 485 , (2003) 024012 [ The string landscape and the swampland (2003) 046005 [ hep-th/0501082 D 68 , hep-th/0405159 D 68 scale of the standard model supersymmetry and signatures for fine-tuning at the LHC [ TeV N. Arkani-Hamed, S. Dimopoulos and S. Kachru, Phys. Lett. consistent infrared modification of gravity scale and vdvz discontinuity gravity as the weakest force Rev. propagation [2] S. Weinberg, [3] V. Agrawal, S.M. Barr, J.F. Donoghue and D. Seckel, [4] N. Arkani-Hamed and S. Dimopoulos, [5] G.R. Dvali, G. Gabadadze and M. Porrati, [6] N. Arkani-Hamed, H.-C. Cheng, M.A. Luty and S. Mukohyama [7] V.A. Rubakov, [8] S.L. Dubovsky, [9] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, [1] S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, [10] C. Vafa, [11] G. Dvali, A. Gruzinov and M. Zaldarriaga, Acknowledgments We would like to thank Paologory Creminelli, Gabadadze, Oliver Leonardo DeWolfe, Giusti, Jac ThomasPierce, Gregoire, Shamit Massimo K Porrati,Sundrum, Valery Sergei Rubakov, Sibiryakov, Igor Leonardo Tkachev,helpful Senator Enrico conversations about Trincher these ideas. A.for A. invaluable would assistance partic during the finalfor revisions pointing of this out elementarilysaid incorrect in things the about first thewould version like an of to thank thisextremely Lubos paper, helpful Motl and and and for illuminating especially aDOE discussions. Michal under helpful Fabinge The contract conve work DE-FG02-91ER40654. o Fellowship from The the work Harvard of Societythe A. of European A. Fellows. Commission is The under su work contract of MRTN-CT-2004-5033 R References [12] S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, JHEP10(2006)014 al , 06 D 73 Phys. , 129 (1999) 032 JHEP , (Proc. Suppl.) 09 ]. 88 Phys. 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