IR Dynamics from UV Divergences: UV/IR Mixing, NCFT, and the Hierarchy Problem
PPrepared for submission to JHEP
IR Dynamics from UV Divergences:UV/IR Mixing, NCFT, and the Hierarchy Problem
Nathaniel Craig & Seth Koren
Department of Physics, University of California, Santa Barbara, CA 93106, USA
Abstract:
The persistence of the hierarchy problem points to a violation of effective fieldtheory expectations. A compelling possibility is that this results from a physical breakdownof EFT, which may arise from correlations between ultraviolet (UV) and infrared (IR)physics. To this end, we study noncommutative field theory (NCFT) as a toy modelof UV/IR mixing which generates an emergent infrared scale from ultraviolet dynamics.We explore the range of such theories where ultraviolet divergences are transmogrifiedinto infrared scales, focusing particularly on the properties of Yukawa theory, where weidentify a new infrared pole accessible in the s -channel of the Lorentzian theory. We furtherinvestigate the interplay between UV-finiteness and UV/IR mixing by studying propertiesof the softly-broken noncommutative Wess-Zumino model as soft terms are varied relativeto the cutoff. While the Lorentz violation inherent to noncommutative theories may limittheir direct application to the hierarchy problem, these toy models provide general lessonsto guide the realization of UV/IR mixing in more realistic theories. a r X i v : . [ h e p - ph ] S e p ontents φ Theory 8
A.1 Scalar Two-Point Function 29A.2 Fermion Two-Point Function 29A.3 Three-Point Function 30
At its heart, the electroweak hierarchy problem is a question of how an infrared (IR) scalecan emerge from an ultraviolet (UV) scale without fine-tuning of UV parameters. Giventhe sensitivity of the Standard Model Higgs mass to UV scales, the expectation of effectivefield theory (EFT) is that the two should coincide. Conventional solutions to the hierarchyproblem introduce both symmetries that control UV contributions to the Higgs potentialand dynamics that generate IR contributions, leading to considerable structure at the weakscale and correspondingly sharp experimental tests. Ongoing exploration of the weak scalehas given no evidence for these solutions, despite their theoretical soundness.In the face of increasingly powerful LHC data in excellent agreement with the StandardModel, it’s worth taking seriously the possibility that Nature may be leading us to theconclusion that there is no new physics at the weak scale . While this is often taken tosuggest the existence of considerable fine-tuning in the Higgs potential, here we pursue an– 1 –lternative idea. Perhaps the apparent violation of EFT expectations at the weak scale isa sign of the breakdown of EFT itself.This statement is not as radical as it may at first seem. That EFT must eventuallybreak down is not a new idea; it has long been known that gravity contains low-energyeffects which cannot be understood in the context of EFT. The fact that black holes radiateat temperatures inversely proportional to their masses [1] necessitates some sort of ‘UV/IRmixing’ in gravity — infrared physics must somehow ‘know about’ heavy mass scales inviolation of a na¨ıve application of decoupling. As a perhaps-more-fundamental raisond’ˆetre for such behavior, the demand that observables in a theory of quantum gravitymust be gauge-(that is, diffeomorphism-)invariant dictates that they must be nonlocal (seee.g. [2–6]), again a feature which standard EFT techniques do not encapsulate. In viewof this, the conventional position is that EFT should remain a valid strategy up to thePlanck scale, at which quantum gravitational effects become important. But once localityand decoupling have been given up, how and why are violations of EFT expectations to besequestered to inaccessible energies? Indeed, the ‘firewall’ argument [7] evinces tension withEFT expectations in semiclassical gravity around black hole backgrounds at arbitrarily lowenergies and curvatures.That quantum gravitational effects will affect infrared particle physics is likewise not anew idea. This has been the core message of the Swampland program [8], which has beencataloging — to varying degrees of concreteness and certainty — ways in which otherwiseallowable EFTs may conjecturally be ruled out by quantum gravitational considerations.These are EFTs which would look perfectly sensible and consistent to an infrared effectivefield theorist, yet the demand that they be UV-completed to theories which include Einsteingravity reveals a secret inconsistency. While this is powerful information, the extent towhich the UV here meddles with the IR is relatively minor — just dictating where onemust live in the space of infrared theories. Even so, they have been found to have possibleapplications to SM puzzles, including the hierarchy problem [9–18].In theory far more flagrant violations of low-energy expectations are permissible —that is, the extent to which quantum gravitational violation of EFT will affect the infraredof our universe is not at all certain. Of course any proposal to see new effects from abreakdown of EFT must contend with the rampant success of the SM EFT in the IR.Certainly a violation of EFT must both come with good reason and be deftly organizedto spoil only those observed EFT puzzles. For the former, the need for quantum gravityis obviously compelling. As to the latter, it is interesting to note that the most pressingmysteries involve the relevant parameters in the SM Lagrangian.Ultimately, our ability to address the hierarchy problem through quantum gravitationalviolations of EFT is limited by our incomplete understanding of quantum gravity. Thismotivates finding non-gravitational toy models that violate EFT expectations on their own,providing a calculable playground in which to better understand the potential consequencesof UV/IR mixing. In this work we pursue the idea that UV/IR mixing may have moredirect effects on the SM by considering noncommutative field theory (NCFT) as such atoy model. These theories model physics on spaces where translations do not commute[19, 20], and have many features amenable to a quantum gravitational interpretation —– 2 –ndeed, noncommutative geometries have been found arising in various limits of stringtheory [21–24]. This noncommutativity bears out the general expectation that the general-relativisticnotion of spacetime should break down in a theory of quantum gravity [36]. Its realizationhere leads directly both to UV/IR mixing in the form of a violation of decoupling and tononlocal effects in interactions. This gives rise to many interesting effects, but particularlyfascinating for our purposes is that UV divergences present in the S-matrix elementsof QFTs on commutative spaces can be transmogrified into new infrared poles in thecorresponding field theory on noncommutative space [37]. An effective field theorist livingin a noncommutative space would have no way to understand the appearance of this infraredscale; its existence is intrinsically linked to the geometry of spacetime and to the far UVof the theory. Such an effective field theorist would see a surprising lack of new physicsaccompanying this pole to explain its presence.It is clear from the outset that the direct application of NCFT to understand thehierarchy problem is immediately hindered by the Lorentz invariance violation which isinherent to these theories. Precisely how fatal this might be is not entirely clear; resultsregarding the extent to which ‘generic’ Lorentz violation is empirically ruled out [38] arepartly circumvented here by the fact that the Lorentz violation is not generic, but comesas part of some larger structure. In this case the novel effects of UV/IR mixing in factonly appear in nonplanar loop diagrams [39] and care is required when interpreting EFTconstraints on Lorentz violation — a point we will emphasize in Section 2. Even so, itis difficult to imagine that observed properties of the weak scale and the wide range ofconstraints on Lorentz violation leave room for NCFT to be directly relevant to puzzles ofthe Standard Model.Thus we make no claim about having solved the hierarchy problem. The value of thiswork is in the exploration of this toy model of UV/IR mixing, which possesses the intriguingfeature that ultraviolet dynamics generate a scale whose lightness would be baffling to aneffective field theorist. As this is the only model (of which we are aware) with this feature— and this feature, at the level of words, increasingly matches the experimental situationwith the Higgs — it’s worth understanding its appearance in as much detail as possible.To make this work self-contained for the contemporary particle theorist, we begin withan extensive introduction. In Section 2, we review quantum field theory on noncommutativespaces with an emphasis on the violation of EFT expectations. In Section 3 we use thistechnology to go over the classic result of [37] which first identified this emergent infraredpole in a Euclidean φ theory. We compute also the effect in dimensional regularization toevince the regularization-independence of the UV/IR mixing effects.In Section 4 we ask how general the effect of UV/IR mixing is within NCFT, which leadsus to study noncommutative Yukawa theory in detail. We find that the scalar propagatoragain develops a new infrared pole at one loop, in contrast with previous work. Intriguingly,the pole in this case is accessible in s -channel scattering in the Lorentzian theory, making Noncommutative branes arising in gauge theory matrix models have also been found to contain emergentgravitational effects, and so have been suggested as novel quantum theories of gravity [25–34]. We do notpursue this perspective here, but refer the reader to [35] for a review of this approach. – 3 –ukawa theory a promising setting for probing phenomenological consequences of UV/IRmixing.In Section 5 we upgrade our model to the softly-broken Wess-Zumino model to studythe interplay between UV-finiteness and UV/IR mixing effects. When the fermion is keptin the spectrum of the theory below the cutoff, the lack of UV sensitivity of the field theoryremoves the light pole. As the fermion is taken above the cutoff, an effective theorist againsees effects past those observed in Wilsonian EFT. These results are expected, but thismodel affords us a concrete demonstration that UV/IR mixing can only have interestinglow-energy effects if the field theory is UV sensitive, and puts this naturalness strategyin stark contrast to conventional approaches. Of course, this also makes addressing thehierarchy problem with UV/IR mixing a potentially Pyrrhic victory: to generate an IRscale, the field theory alone cannot be fully predictive.Finally, in Section 6 we examine the appearance of the emergent light pole in NCFTfrom more general arguments, so as to ascertain the relative importance of nonlocality andLorentz-violation for these effects. The conclusion is inevitably that in this case the twoare inexorably linked, and no strong conclusion about the possibility of finding a light polein a theory with only one or the other is available. However, we provide some directiontoward future explorations into both of these possibilities. We wrap up in Section 7.
In this section we review the salient features of the formulation of noncommutative fieldtheories and the standard formalism for studying their perturbative physics. Useful generalreferences for this background include [40, 41]. Readers familiar with NCFT may wish toskip to Section 3, but we emphasize that our interest is necessarily non-perturbative in theparameter controlling the noncommutativity, unlike much of the earlier phenomenologicalliterature.Physics on noncommutative spaces involves the introduction of a nonzero commutatorbetween position operators [ˆ x µ , ˆ x ν ] = iθ µν , (2.1)where we will refer to θ µν = − θ νµ as the noncommutativity tensor, and we emphasize thatit is covariant under Lorentz transformations. So while it does break Lorentz invariance,it only does so in the way that turning on a magnetic field in your lab chooses a preferredframe. This basic definition is reminiscent of the introduction of a nonzero commutatorin passing from classical mechanics to quantum mechanics. Indeed much of the structureis precisely analogous, including importantly the construction of noncommutative versionsof familiar commutative theories via a quantization map. At an even more basic level, theabove nonzero commutator induces an uncertainty relation∆ˆ x µ ∆ˆ x ν ≥ | θ µν | , (2.2)which immediately makes apparent the presence of UV/IR mixing in this theory. If youattempt to create a wavepacket which is very small in one direction it will necessarily– 4 –e elongated in another, and so we see already the non-trivial mixing of UV and IRmodes. This clearly violates the separation of scales which is baked in to EFT. Thus purelyfrom the defining relation of noncommutative geometry, we see already an indication thatnoncommutative theories should violate EFT expectations.Field theories on this space may be conveniently formulated in terms of fields thatare functions of commuting coordinates imbued with a new field product, known as aGroenewold-Moyal product (or star-product), with position-space representation f ( x ) (cid:63) g ( x ) = exp (cid:18) i θ µν ∂ µy ∂ νz (cid:19) f ( y ) g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) y = z = x = f ( x ) exp (cid:18) i ←− ∂ µ θ µν −→ ∂ ν (cid:19) g ( x ) . (2.3)It is important to observe that this is a nonlocal product, since it contains an infiniteseries of derivative operators. So we see again that one of the tenets of EFT has beenviolated by our basic definition of field theory on noncommutative spaces.With this in hand we may now write down noncommutative versions of familiar theories in terms of commuting coordinates , which will then allow us to use normal QFT methodsto analyze them. First note that this noncommutative quantization will not affect thequadratic part of the tree-level action due to momentum conservation and the antisymmetryof the noncommutativity tensor. For the interacting part of the action the effects ofnoncommutative quantization are not so trivial, but are easy to analyze classically. Asan example, for a simple φ n theory we find L ( NC )int = λn ! n copies (cid:122) (cid:125)(cid:124) (cid:123) φ ( x ) (cid:63) φ ( x ) (cid:63) · · · (cid:63) φ ( x ) . (2.4)Note, importantly, that the star-product has endowed our vertices with a notion ofordering, as it is only cyclically invariant. If we now Fourier transform the action tomomentum space, we find that we can account for the effects of quantization on the tree-level action with a simple modification of the momentum-space vertex factor:˜ V ( k , . . . , k n ) = δ ( k + · · · + k n ) exp i n (cid:88) i 1. Indeed, doing so would give us a series of irrelevant op-erators which would correct the leading interaction. However, once the theory is truncatedat some finite order in θ , we are left with a perfectly local EFT. In other scenarios wherean infinite series of operators appears, this is a valid approximation procedure and allowsone to calculate the leading corrections a theory predicts. But here our definition of NCFT– 5 –ntroduces UV/IR mixing which we expect to violate EFT expectations. Truncating theseries removes these effects entirely, and a theory so defined no longer has anything to dowith NCFT — at least not in the effects we will be interested in, which are nonperturbativein θ as we shall see explicitly in the following sections. There has been much workexpended on these ‘noncommutative-inspired’ theories, but they do not contain UV/IRmixing, and do not capture the most striking and most interesting features of physics on anoncommutative space, from our perspective. With that in mind, we may now proceed to do perturbative quantum field theorycalculations, but we must worry about keeping track of all the phases from each of thevertices. In fact there is another simplification that occurs, as found by Filk [39], whichallows us to simplify the process of finding the phase factor for a diagram to a graph-topological statement. Filk proved two simple rules for the phase factors:1. An internal line which ends on two different vertices can be contracted while keepingthe ordering of the other lines fixed.˜ V ( k , . . . , k n , p ) ˜ V ( − p, k n +1 , . . . , k n ) = ˜ V ( k , . . . , k n ) δ ( k + · · · + k n + p ) (2.7)2. A loop which doesn’t cross any lines can be eliminated. Note that the fixed orderingof the lines at a vertex means that we can now meaningfully speak of lines which door don’t cross each other.˜ V ( k , . . . , k n , p, k n +1 , . . . , k n , − p ) = ˜ V ( k , . . . , k n , k n +1 , . . . , k n ) if n (cid:88) i = n +1 k i = 0(2.8)The proof of these facts relies only on the antisymmetry of θ µν and the fact that eachvertex contains a momentum-conserving delta function. We may make use of this to simplyfind the phase factor of any Feynman diagram. Using the first rule, we can reduce anydiagram to a single vertex, which is a rosette of the external lines and closed loops. Thesecond rule allows us to eliminate loops which don’t cross other lines.If the graph was planar (including, importantly, any tree-level graph), then by def-inition all loops can be eliminated. So all contributions to phase factors from internallines cancel, and we’re only left with an overall phase corresponding to the ordering of theexternal lines, which has remained fixed throughout the reduction process.For a nonplanar graph, in this representation it is easy to see that we only pick upphase factors from lines which cross. The loop gives vertex legs with ± p µ , and for anexternal line which doesn’t cross this loop, both loop legs will be on the same side of it inthe cyclic ordering, and so the two terms will cancel in the sum. Only for an external linewhich crosses it are the ± p on different sides, and so the antisymmetry of θ will make thetwo negative signs cancel to give a coherent phase for this vertex. Thus we define I ij , the We are not the first to issue a warning of this sort — see e.g. [42, 43] in the context of connectingnoncommutativity to the real world, and [44] which discusses the general case of nonlocal interactions. – 6 –ntersection matrix of an oriented graph: I ij = j crosses i from right − j crosses i from left0 line j does not cross i (2.9)Then for any graph G , the contribution Γ( G ) of the phase factors is justΓ( G ) = ˜ V ( { external momenta } ) × exp i (cid:88) ij I ij k i ∧ k j , (2.10)where we’ve defined k i ∧ k j ≡ k µi θ µν k νj .In what follows we will omit the overall external phase when evaluating diagrams, asit will not be important for our purposes. We have now simplified perturbative field theoryon noncommutative spaces down to the simple task of marking line-crossings, at least atthe level of writing down integrands of amplitudes. The triviality of this task for tree-level graphs leads to the interesting feature that tree-level amplitudes on noncommutativespaces are the same as on commutative manifolds, and it is only at loop-level that we finddeviations. We will see in the next section that the loop integration will bring surprisingfeatures.An important issue for the interpretation of NCFTs is that of their unitarity. There isno problem in Euclidean space, but for Lorentzian spacetimes with noncommutativity in thetime directions (‘timelike’ or ‘space-time’ noncommutativity when − k µ θ µρ θ ρν k ν ≡ k ◦ k < This may be interpreted physically as being due to the production of tachyonicstates, which if added to the Fock space of the theory result in a formal restoration of thecutting relations whilst making the nonunitarity explicit [49].This failure of unitarity is well-understood from the stringy perspective. Spatialnoncommutativity appears from a background magnetic field and the field theory limitto a spacelike NCFT is smooth [23]. In the case of timelike noncommutativity, however,approaching the field theory limit forces an electric field to supercritical values whencepair-production of charged strings destabilizes the vacuum [50]. Study of string theorieswith timelike noncommutativity (e.g. ‘noncommutative open string theory’ [50, 51]) isoutside our scope, but there are at least some hints of similar UV/IR mixing effects asthose in the NCFT [52]. We note in passing that there are further interesting connectionsbetween NCFTs and string theories — not only do particles on noncommutative spacesact in many ways like rods of size L ∼ pθ (see e.g. [53–57]), mimicking the behavior ofextended objects, but there have been many hints in the spacelike theories that the curiousIR effects in the NCFT are reproducing effects from closed strings, despite the fact thatthese have been decoupled (e.g. [37, 52, 58–65]). Though it is interesting to note that the special case of ‘lightlike’ noncommutativity is also unitary[47, 48]. – 7 –ithin the realm of field theory, there have long been suggestions that this difficultyis pointing to the need for a modified definition of quantum field theories on timelikenoncommutative spaces (for some early references, see [66–73]). From this perspective, theissue is that such field theories are non-local in time, which renders nonsensical the normaltime-ordering involved in the perturbative Dyson series (at the least). That is, our effectivedefinition of these theories above via the diagrammatic expansion may be too na¨ıve. Aninteresting line of work is to formulate a modification of the standard quantum field theorymachinery to non-local-in-time theories which avoids the unitarity issue by construction.We note that the same UV/IR mixing effects of interest in the two-point function havebeen seen to persist in at least some of these approaches (e.g. [68]). For some recent workon the formulation and properties of nonlocal field theories, see e.g. [44, 74–78].Below we will begin in Euclidean space, where k ◦ k ≥ θ µν ,but will then venture into Lorentzian signature. All of our calculations and the generalfeatures we find, including finding new infrared poles, will hold robustly in spacelike non-commutative theories. However we will comment also on how these features are modifiedwhen timelike noncommutativity is turned on, taking license from the aforementioned hintsthat unitary completions/reformulations of timelike NCFT may retain the UV/IR mixingexhibited in the na¨ıve approach. φ Theory In this section we review the perturbative physics of the noncommutative real scalar φ theory at one loop, which was first studied in detail by Minwalla, Van Raamsdonk, andSeiberg in [37]. In four Euclidean dimensions the action on noncommutative space becomes S = (cid:90) d x (cid:18) ∂ µ φ∂ µ φ + 12 m φ + g φ (cid:63) φ (cid:63) φ (cid:63) φ (cid:19) , (3.1)where we have already used the fact that the quadratic part of the noncommutative actionis the same as the commutative theory to eliminate the star product there. Our objectof interest will be the one-loop correction to the two-point function. In the commutativetheory this is given by a single Feynman diagram, but the noncommutative theory containsboth a planar diagram and a nonplanar diagram. − Γ (2)1 = p k + p k Some early results in this model may also be found in [79, 80]. – 8 –he expressions for these two diagrams now differ — not only in symmetry factor but alsodue to the phase in the integrand. We findΓ (2)1 , planar = g π ) (cid:90) d kk + m Γ (2)1 , nonplanar = g π ) (cid:90) d kk + m e ik µ θ µν p ν . (3.2)We may already see that something interesting should happen, as in the nonplanar diagramthe phase mixes the internal and external momenta. One may intuit that the rapidlyoscillating phase in the UV of the loop integration will dampen the would-be divergence,and indeed we will see that nonplanar diagrams are finite. However, unlike in the casewhere the vertex factor vanishes rapidly for large Euclidean momenta and so ensures UV-finiteness [78], here the damping is in some sense ‘marginal’. This fact will be responsiblefor the interesting feature we will find presently.The simplest method to evaluate noncommutative diagrams is to use Schwinger pa-rameters, recalling the identity k + m = (cid:82) ∞ d α e − α ( k + m ). The presence of the phasein the nonplanar diagram means we must complete the square before going to sphericalcoordinates to get a Gaussian integral. This means that after the momentum integrals weend up with Γ (2)1 , planar = g π (cid:90) d αα e − αm Γ (2)1 , nonplanar = g π (cid:90) d αα e − αm − p ◦ p α (3.3)where again p ◦ q = − p µ θ µν q ν . Moving to Schwinger space trades large- k divergences forsmall- α divergences, which we now smoothly regulate by multiplying the integrands byexp (cid:0) − / (Λ α ) (cid:1) so that the small α region will be driven to zero. Note that a term of thisform already exists in the expression for the nonplanar diagram. After introducing theregulator, we can evaluate the integrals to findΓ (2)1 , planar = g π (cid:18) Λ − m log (cid:18) Λ m (cid:19) + O (1) (cid:19) Γ (2)1 , nonplanar = g π (cid:18) Λ − m log (cid:18) Λ m (cid:19) + O (1) (cid:19) , (3.4)where we’ve defined Λ ≡ / Λ + p ◦ p/ , (3.5)which is the effective cutoff of the nonplanar diagram.The first thing to note is that it seems the UV divergence of the nonplanar diagramhas disappeared — the graph is finite in the limit Λ → ∞ , and so appears to have beenregulated by the noncommutativity of spacetime. In fact the effect is more subtle, asalluded to earlier, and now the UV and IR limits of this amplitude do not commute. If wefirst take an infrared limit p ◦ p → eff → Λ and the ultraviolet divergence– 9 –f the commutative theory reappears. If we take the UV limit Λ → ∞ first we find an IRdivergence p ◦ p , so the noncommutativity has transmogrified the UV divergence into an IRone. Turning to the question of renormalizability, one may na¨ıvely ask if we can absorball UV divergences into a finite number of counterterms. Under this criterion, it is clearthat this procedure works in the noncommutative theory at least when the commutativeversion is renormalizable. In the current case, we may absorb the UV divergences ofthis correction to the two-point function into a redefinition of the physical mass, M = m + g Λ π − g m π log Λ m , and so write down a one-particle irreducible quadratic effectiveaction which has a finite Λ → ∞ limit: S (2)1PI = (cid:90) d p (2 π ) (cid:32) p + M + g π (cid:0) p ◦ p + (cid:1) − g M π log 1 M (cid:0) p ◦ p + (cid:1) + · · · + O ( g ) (cid:33) φ ( p ) φ ( − p ) . (3.6)However, in the Λ → ∞ limit one finds that at one loop the propagator now has two poles. The first is a standard radiative correction to the free pole, but the second hasappeared ex nihilo at one loop: p = − m + O ( g ) p ◦ p = − g π m + O ( g ) , (3.7)where we have assumed that θ µν is full rank. The former is to be interpreted as the on-shell propagation of the particles associated to our fundamental field φ . If θ µν has onlyone eigenvalue 1 / Λ θ — with Λ θ thought of as the scale associated with the breakdownof classical geometry — we have p ◦ p = p Λ θ . We see that the new pole appears at p ∝ g θ m , and so if our field φ lives in the deep UV of the theory, our new pole appears atparametrically low energy scales. To the extent that poles are particles, we appear to havegenerated a new light particle from ultraviolet dynamics. The interpretation of the new pole can be sharpened by considering more carefully thecriteria for renormalizability in Wilsonian EFT. In a Wilsonian picture, we upgrade ourLagrangian parameters to running parameters, and define our theory at the scale Λ as S W ilson (Λ) = (cid:90) d x (cid:18) Z (Λ) ∂ µ φ∂ µ φ + 12 Z (Λ) m (Λ) φ + Z (Λ) g (Λ)4! φ (cid:63) φ (cid:63) φ (cid:63) φ (cid:19) . (3.8) We note here that the failure of a ‘correspondence principle’ between commutative and noncommutativetheories as θ µν → θ -dependence isstarkly different from that of any truncation. Although there is no pole at finite Λ, a scale is still induced in the form of an infrared cutoff ∼ Λ θ / Λ. – 10 –t is immediately apparent from the above calculation that we cannot write the action at alower scale Λ < Λ in this same form by choosing appropriate definitions for Z (Λ) , m (Λ) , g (Λ)— there’s nowhere to put the p ◦ p term! Stated more precisely, for Wilsonian renormalizability we require that we can definethe running couplings such that correlation functions computed from this action convergeuniformly to their Λ → ∞ limits. However, this requirement is flatly violated by thenoncommutation of the UV and IR limits of the diagrams. For any finite value of Λ,the effective action of Equation 3.6 differs significantly from its limiting value for smallmomenta p ◦ p (cid:28) . This is the precise sense in which the violation of Wilsonian EFTappears in this one-loop correction.This brings up the question of how an effective field theorist would describe theuniverse if they unknowingly lived on a noncommutative space. A consistent Wilsonianinterpretation can be regained by including a degree of freedom which can absorb the newinfrared dynamics of the quadratic effective action. Since we need this to involve the φ momentum, this new particle must mix linearly with the φ field. We manufacture its tree-level Lagrangian such that the problematic inverse p ◦ p term in the quadratic effectiveaction of φ is replaced with its Λ → ∞ value for all values of Λ, to satisfy our precisecondition for Wilsonian renormalizability. To see how this works, we add to our tree levelWilsonian action∆ S (Λ) = (cid:90) d x (cid:18) ∂χ ◦ ∂χ + 12 Λ ∂ ◦ ∂χ ) + i √ π gχφ (cid:19) . (3.9)Since χ appears quadratically, we may integrate it out exactly at tree level to find acontribution to the effective action∆ S (Λ) = (cid:90) d p (2 π ) (cid:32) − g π (cid:0) p ◦ p + (cid:1) + g π p ◦ p (cid:33) φ ( p ) φ ( − p ) (3.10)This precisely subtracts off the problematic term in the original 1PI quadratic effectiveaction and adds back its Λ → ∞ limit, as we had wanted. Ignoring the logarithmic term, we are left with an effective action which is manifestly independent of the cutoff Λ, andso satisfies our criterion for Wilsonian renormalizability. We discuss the generalization ofthis procedure in Appendix A. There has been much work on understanding renormalizability of NCFTs, especially with an eye towardfinding a mathematically well-defined four-dimensional quantum field theory with a non-trivial continuumlimit. Renormalizability has been proven for modifications of NCFTs where the free action is supplementedby an additional term which adjusts its long-distance behavior. Such an action is manufactured either byrequiring it manifest ‘Langman-Szabo’ duality [82] p µ ↔ θ − ) µν x ν [83, 84] or by adding a 1 /p ◦ p termto the free Lagrangian [85], the latter of which directly has the interpretation of adding ‘somewhere to putthe 1 /p ◦ p counterterm’. For recent reviews of these and related efforts we refer the reader to [86, 87]. Itwould be interesting to understand fully the extent to which the physics of these schemes agrees with theinterpretation of the IR effects as coming from auxiliary fields [37, 61]. Discussion of the interpretation of logarithmic singularities as being due to auxiliary fields propagatingin extra dimensions may be found in [61]. In Equation 3.9, the four-derivative quadratic action of the auxiliary field can be rewritten as two fieldswith two-derivative actions, one of which is of negative norm and may be thought of as the ‘Lee-Wick – 11 –ow while we have written down an action which identifies the new observed IR polewith a field and in doing so gives our effective action a Wilsonian interpretation, the extentto which χ can be taken seriously as a fundamental degree of freedom is unclear. The newpole is inaccessible in Euclidean space — so one does not immediately conclude there is atachyonic instability — and relatedly, when we na¨ıvely analytically continue this result toLorentzian spacetime this new pole is inaccessible in the s -channel. However, its presenceis still enough to break unitarity for this theory [45], and in fact may still be interpretedas being due to the presence of tachyons [49]. As discussed in Section 2, it is possible thismay be resolved if analytical continuation is adjusted for nonlocal-in-time theories, or itmay be that a UV theory cures this apparent violation.Separately, it is not obvious much has been gained by attributing the new pole to anew, independent field, past acting as a formal tool to regain a notion of renormalizability.Since the only interaction of χ above is linear mixing, its action is not renormalized — anydivergences are instead absorbed into the running of φ parameters — and so no interactionsare generated. Furthermore one is obstructed from integrating out the heavy field φ to comeup with an effective action of χ at low energies by the fact that the kinetic terms of χ arenon-standard, which prevents diagonalization of the quadratic terms in the Lagrangian.Thus it seems it is intrinsically linked with the heavy scalar which begat it.There are further obstructions to asking that this specific mechanism be responsible forthe lightness of an observed particle such as the Higgs. Prime among these is the modified partner’ of the positive norm state [88], viz. L = 12 ∂χ (cid:48) ◦ ∂χ (cid:48) − ∂ ˜ χ ◦ ∂ ˜ χ − 12 4Λ ˜ χ + i √ π g (cid:0) χ (cid:48) − ˜ χ (cid:1) φ, χ (cid:48) ≡ χ + ˜ χ (3.11)One may then wonder if the lightness of the new IR pole may be understood through the regularizationperformed by the Lee-Wick field, as is done for the Higgs in the ‘Lee-Wick standard model’ [89]. However,in that theory the Higgs is kept light because every particle comes with a Lee-Wick partner, and so alldiagrams contributing to corrections to the Higgs mass are made finite. The presence of the Higgs’ Lee-Wickpartner alone is not enough to keep it light. Here, the lightness of χ can be understood diagrammaticallyas being simply due to the fact that its only interaction is linear mixing with φ , and so any correction to itstwo-point function is absorbed into that of the two-point function of φ . A further issue with the Lee-Wickrewriting is that the seeming perturbative unitary of the theory is normally guaranteed by the Lee-Wickpartner being heavy and unstable. But as we take the Λ → ∞ limit in our Wilsonian action, we see that theLee-Wick partner becomes massless as well, in accordance with the result that this theory is non-unitary[45]. We note that in matrix models containing dynamical noncommutative geometries it has been arguedthat emergent infrared singularities should be associated with the dynamics of the geometry (see e.g.[35, 90]). As our field theories are formulated on fixed noncommutative backgrounds, this interpretation isunavailable to us. Note that this peculiar connection regarding (in)accessibility is due to the Lorentz violation. Whilethe normal pole which is inaccessible in Euclidean signature becomes accessible for timelike momentain Lorentzian signature, the Wick rotation affects the noncommutative momentum contraction differently.When taking x → − ix , one also rotates θ ν → − iθ ν such that Equation 2.1 continues to hold for the samenumerical θ µν . For the simplest configuration of full-rank noncommutativity with θ µν block-off-diagonal andonly one eigenvalue 1 / Λ θ , the Euclidean p ◦ p = p / Λ θ becomes a Lorentzian p ◦ p = ( p − p + p + p ) / Λ θ . Soa noncommutative pole which is inaccessible in the Euclidean theory becomes accessible in the Lorentziantheory for spacelike momenta, while a noncommutative pole which can be accessed in the Euclidean theorybecomes accessible in the s -channel in Lorentzian signature. – 12 –ispersion relation of the new field, p ◦ p = O ( g ), which means that the free propagationof this field would be Lorentz violating. We will explore these issues further in the nextsections, as in the Yukawa theory of Section 4 the new pole will appear with the oppositesign and so will offer the prospect of appearing as an s -channel pole.We emphasize that a new infrared scale whose lightness is unexplained in the contextof Wilsonian effective field theory is an exciting feature that makes further exploration ofUV/IR mixing an interesting pursuit. The fact that it here appears as the scale of a pole ina propagator makes the connection to the hierarchy problem captivating, but asking thatthis toy model — where Lorentz violation is at the fore — literally solve the problem forus would be too much. We proceed without further hindrance in exploring NCFT so as tolearn more about the appearance and effects of UV/IR mixing here. A good question to ask is whether, or to what extent, these effects are an artifact of ourchoice of regularization. To demonstrate their physicality, we repeat the calculation of theone-loop correction to the two-point function now in dimensional regularization. We setup our integral in d = 4 − (cid:15) dimensions, having defined g = ˜ g ˜ µ (cid:15) , and we again go toSchwinger space: Γ (2)1 , planar = ˜ g ˜ µ (cid:15) π ) d (cid:90) d d k d α e − α ( k + m ) Γ (2)1 , nonplanar = ˜ g ˜ µ (cid:15) π ) d (cid:90) d d k d α e − α ( k + m )+ ik µ θ µν p ν . (3.12)After completing the square in the nonplanar integral, the momentum integral and theSchwinger integral may then be performed analytically, with the results:Γ (2)1 , planar = ˜ g ˜ µ (cid:15) π ) d/ ( m ) d − Γ(1 − d (2)1 , nonplanar = ˜ g ˜ µ (cid:15) π ) d/ d ( m ) ( d − ( √ p ◦ p ) − d K d − ( m √ p ◦ p ) . (3.13)If we expand the planar graph in the limit (cid:15) → 0, which should be thought of as probingthe ultraviolet, we recover Γ (2)1 , planar = − ˜ g m π ) (cid:20) (cid:15) + ln µ m (cid:21) , (3.14)where in MS we would subtract off the pole and find the renormalization group evolutionof m from the logarithmic term, as usual.The question of dimensional regularization for the nonplanar diagram is a subtle one[91]. If we first take the (cid:15) → This dispersion relation means that χ only propagates in noncommutative directions, and so attemptsto use hidden extra-dimensional noncommutativity to avoid four-dimensional Lorentz violation constraintsseem a phenomenological nonstarter. – 13 –ivergences, and we are simply left with the finite, (cid:15) termΓ (2)1 , nonplanar = g m π ) (cid:20) m p ◦ p − ln 4 m p ◦ p − γ (cid:21) , (3.15)which we have expanded near p ◦ p → p ◦ p → (2)1 , nonplanar = ˜ g m π ) π (cid:15)/ ˜ µ (cid:15) m (cid:15) Γ (cid:16) − (cid:15) (cid:17) + ˜ g π ˜ µ (cid:15) π (cid:15)/ Γ (cid:16) − (cid:15) (cid:17) p ◦ p − (cid:15)/ + O ( p ◦ p ) . (3.16)If we were to now blindly take the (cid:15) → (cid:15) > 2, then wehave incorrectly kept the second term in Equation 3.16, as that term would be at least O ( p ◦ p ). If we were to work in d < 2, expand in p ◦ p → then analytically continue back to d = 4, we would instead find the (cid:15) − poleΓ (2)1 , nonplanar = − ˜ g m π ) (cid:20) (cid:15) + ln µ m (cid:21) , (3.17)and now we recover the UV divergence that was present in the commutative theory, sothat once again we find the UV and IR limits don’t commute.The key to understanding clearly this seemingly ambiguous dimensional regularizationprocedure is that while Γ (2)1 , nonplanar ( p ◦ p ) ∼ (cid:82) d d q d α e − α ( q + m ) − p ◦ p α is convergent in d > p ◦ p > 0, at p ◦ p = 0 it is only convergent for d < 2. Since it is a property of dimensionalregularization that if an integral converges in δ dimensions, it converges to the same valuein d < δ dimensions [92], we may thus perform the integral at d < all p ◦ p andcorrectly find Equation 3.13. It is only when taking the IR limit that we must rememberthe integral was performed in d < We observed in our first example that the UV divergences of the real φ commutativetheory are transmogrified into infrared poles in the noncommutative theory. It is natural While we only presented the calculation of the one-loop correction to the two-point function, [37] goesthrough corrections to the two- and n -point functions for φ n with n = 3 , – 14 –o ask whether this “strong UV/IR duality” [93] is a common feature of all noncommutativetheories.The answer is no, and the simplest counterexample is provided in the case of a complexscalar field with global U (1) symmetry and self-interaction [93]. In the quantization of thescalar potential we have two quartic terms which are noncommutatively-inequivalent dueto the ordering non-invariance, so the general noncommutative potential is V = m | φ | + λ φ ∗ (cid:63) φ (cid:63) φ ∗ (cid:63) φ + λ φ ∗ (cid:63) φ ∗ (cid:63) φ (cid:63) φ, (4.1)where λ and λ are now different couplings. By doodling some directed graphs, one seessimply that the one-loop correction to the scalar two-point function contains planar graphswith each of the λ , λ vertices, but the only nonplanar graph has a λ vertex. There isthus no necessary connection of the ensuing nonplanar IR singularity to the UV divergencein the θ → λ = 0).Another important counterexample is that of charged scalars, the simplest exampleof which is noncommutative scalar QED, which was first constructed in [94]. There isa very rich and interesting structure of gauge theories on noncommutative spaces, a fulldiscussion of which is far beyond the scope of this paper. We refer the reader to [43, 95–99]for discussions of some features relevant to SM model-building. We here satisfy ourselveswith the simplest case, for which we have the noncommutative Lagrangian L = 14 g F µν (cid:63) F µν + ( D µ φ ) ∗ (cid:63) ( D µ φ ) + V ( φ, φ ∗ ) , (4.2)where even though we’re quantizing U (1) we have F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ ∗ , A ν ] dueto the noncommutativity, where [ · ∗ , · ] is the commutator in our noncommutative algebra.The vector fields transform as A µ (cid:55)→ U (cid:63) A µ (cid:63) U † + i∂ µ U (cid:63) U † , where U ( x ) is an elementof the noncommutative U (1) group, which consists of functions U ( x ) = (cid:0) e iθ ( x ) (cid:1) (cid:63) , which isthe exponential constructed via power series with the star-product.The potential and the covariant derivative both depend on the representation we choosefor the scalar. In contrast to commutative U (1) gauge theory, where we merely assign φ acharge, our only choices now are to put φ in either the fundamental or the adjoint of thegauge group. Note that an adjoint field smoothly becomes uncharged in the commutativelimit. Such a field φ transforms as φ (cid:55)→ U (cid:63) φ (cid:63) U † . The covariant derivative is thus D µ φ = ∂ µ φ − ig [ A µ ∗ , φ ]. The gauge-invariant potential then includes both quartic termsin Equation 4.1, in addition to others such as φ ∗ (cid:63) φ (cid:63) φ (cid:63) φ , since the adjoint complex scalaris uncharged at the level of the global part of the gauge symmetry. Strong UV/IR dualitythen should not hold here either.The situation is even worse if φ is in the fundamental, where it transforms as φ (cid:55)→ U (cid:63)φ and φ ∗ (cid:55)→ φ ∗ (cid:63) U − with covariant derivative D µ φ = ∂ µ φ − iA µ (cid:63) φ . It is easy to see in It is important to note that many fundamental concepts which one normally thinks of as depending uponLorentz invariance still hold on noncommutative spaces, due to a ‘twisted Poincar´e symmetry’ [100–103].This includes the unitary irreducible representations, so it is sensible to speak of a vector field. – 15 –his case that the λ interaction term is no longer gauge invariant, and a charged scalarmay only self-interact through V = λ φ ∗ (cid:63) φ (cid:63) φ ∗ (cid:63) φ . Purely from gauge invariance we thussee that a fundamental scalar has no nonplanar self-interaction diagrams in the one-loopcorrection to its two-point function, and so there is no remnant of strong UV/IR dualityto speak of. The question is then whether there are other examples where this strong UV/IRduality does occur, or whether it is perhaps a peculiar feature of real φ n theories onnoncommutative spaces. To answer this, we will study in detail another case of especialphenomenological significance: Yukawa theory. Noncommutative Yukawa theory was firststudied in [106]. Our result on the presence of strong UV/IR mixing differs, for reasonswe will explain henceforth. For reasons that will soon become clear, we will now work directly in Minkowski space, andbegin with a commutative theory of a real scalar ϕ and a Dirac fermion ψ with Yukawainteraction: L (C) = − ∂ µ ϕ∂ µ ϕ − m ϕ + iψ /∂ψ − ψM ψ + gϕψψ. (4.3)When constructing a noncommutative version of this theory, the quadratic part of theaction does not change. However, ordering ambiguities appear for the interaction term,and we in fact find two noncommutatively-inequivalent interaction terms which genericallyappear: L (NC)int = g ϕ (cid:63) ψ (cid:63) ψ + g ψ (cid:63) ϕ (cid:63) ψ. (4.4)These terms are inequivalent because the star product is only cyclically invariant. Inthe analysis of [106], only the g interaction was included. As a result, it was concludedthat this theory contains no nonplanar diagrams at one loop, and the first appear at twoloops as in Figure 1. This immediately tells us that the one-loop quadratic divergence ofthe scalar self-energy will not appear with a one-loop IR singularity, and so rules out theputative strong UV/IR duality of the theory they studied.However, we must ask whether we actually have the freedom to choose g and g inde-pendently. To address that question, we must understand the role of discrete symmetriesin noncommutative theories. For ease of reference we here repeat our definition of thenoncommutativity parameter [ x µ , x ν ] = iθ µν (2.1)It is manifest that the noncommutativity tensor does not transform homogeneously undereither parity or time-reversal, but only under their product: P T : x µ → − x µ ⇒ P T : θ µν → Noncommutative QED also has strange behavior in the gauge sector that runs counter to strong UV/IRduality — the photon self-energy correction gains an infrared singularity from nonplanar one-loop diagrams,even though the commutative quadratic power-counting divergence is forbidden by gauge-invariance. Thetheory is constructed in detail in [104], while more physical interpretation is given in [105], and the possiblerelation to geometric dynamics in the context of matrix models is discussed in [90]. Aspects of noncommutative Yukawa theory have also been studied recently in d=3 in [107], and witha modified form of noncommuativity in [108]. – 16 – ₂ g ₂ g ₂ g ₂ g ₂ g ₂ g ₂ g ₂ Figure 1 : Representative leading nonplanar corrections to the self-energies in the noncommutativeYukawa theory of [106]. Fermion lines have arrows and dashing denotes nonintersection. θ µν . So while any Lagrangian with full-rank noncommutativity unavoidably violates both P and T , it may preserve P T .Since both ϕ and the scalar fermion bilinear are invariant under all discrete symmetries,these symmetries na¨ıvely play no further role in this theory. However, the time-reversaloperator is anti-unitary, and thus negates the phase in the star-product:( P T ) − ( f ( x ) (cid:63) g ( x )) P T = g ( x ) (cid:63) f ( x ) . (4.5)Armed with this, we may now apply CPT to our interaction Lagrangian, to find( CP T ) − L (NC)int CP T = g ψ (cid:63) ϕ (cid:63) ψ + g ϕ (cid:63) ψ (cid:63) ψ. (4.6)Comparing with Equation 4.4, we see that our interactions have been re-cycled! Requiringthat our interactions preserve CPT amounts to imposing( CP T ) − L (NC)int CP T = L (NC)int = ⇒ g = g (4.7)And so the theory of [106] appears to violate CPT. When we instead include bothorderings of interactions the nonplanar diagrams now occur at the first loop order. Further-more, with both couplings set equal the planar and nonplanar diagrams will have the samecoefficients, which reopens the question of strong UV/IR duality for this theory. In thefollowing we will keep g and g distinguished merely to evince how the different verticesappear, but in drawing conclusions about the theory we will set them equal. We note that while the CPT theorem has only been proven in NCFT without space-time noncommuta-tivity [109–112], the difficulty in the general case is related to the issues with unitarity discussed in Section2, and we expect it should hold in a sensible formulation of the space-time case as well. We should note that in the construction of noncommutative QED it has been argued that it is sensibleto assign θ the anomalous charge conjugation transformation C : θ µν → − θ µν ([113] and many others since).The argument is that charged particles in noncommutative space act in some senses like dipoles whose dipolemoment is proportional to θ , and so charge conjugation should naturally reverse these dipole moments. Here,however, our particles are uncharged, and thus we have no basis for arguing in this manner. Furthermore,such an anomalous transformation makes charge conjugation relate theories on different noncommutativespaces M θ → M − θ . The heuristic picture of the CPT theorem (that is, the reason we care about CPT beinga symmetry of our physical theories) is that after Wick rotating to Euclidean space, such a transformationbelongs to the connected component of the Euclidean rotation group [114], and so is effectively a symmetryof spacetime. So it is at the least not clear that defining a CPT transformation that takes one to a differentspace accords with the reason CPT should be satisfied in the first place. – 17 – .3 Scalar Two-Point Function First we consider the planar diagrams, of which there are two: − i Γ ,s,p ( p ) = g ₁ g ₁ p k + p/2k - p/2 + g ₂ g ₂ p k - p/2k + p/2 The ‘symmetrization’ of the momenta of the internal propagators is an importantcalculational simplification. This calculation is textbook save for our Schwinger-spaceregularization, so we will be brief and merely point out the salient features. The sum ofthese diagrams givesΓ (2) ,s,p ( p ) = i ( − (cid:0) ( ig ) + ( ig ) (cid:1) (cid:90) d k (2 π ) ( − i ) Tr (cid:2)(cid:0) M − /k − /p/ (cid:1) (cid:0) M − /k + /p/ (cid:1)(cid:3) (( k + p/ + M ) (( k − p/ + M ) . (4.8)To evaluate this, we must now introduce two Schwinger parameters α , α and thenswitch to ‘lightcone Schwinger coordinates’ which effects the change (cid:82) ∞ d α (cid:82) ∞ d α → (cid:82) ∞ d α + (cid:82) + α + − α + d α − . Regulating the integral by exp (cid:2) − / √ α + Λ (cid:3) , we may then evaluateand isolate the divergences as Λ → ∞ to findΓ (2) ,s,p ( p ) = − ( g + g )2 π (cid:20) Λ − M + p (cid:18) Λ M + p / (cid:19) + . . . (cid:21) (4.9)Turning now to the nonplanar diagrams, there are again two − i Γ (2) ,s,np = g ₂ p g ₁ + g ₂ p g ₁ Each now has one g vertex and one g vertex, which makes it clear why the analysisof [106] found no such diagrams. The two diagrams will come with opposite phase factors, e ip ∧ k and e ik ∧ p , so we can compute one and then find the other by taking p (cid:55)→ − p . In thiscase it’s obvious that after completing the square we will only be left with terms which arequadratic in p , and so the two diagrams give the same contribution. We can thus computeboth terms at the same time.The phase factor in the integrand will modify our change of variables, as it did in the φ case, to give again an effective cutoff for this diagram due to the noncommutativity. WefindΓ (2) ,s,np ( p ) = g g π (cid:90) d q d α d α q (cid:18) M − q + α α ( α + α ) p + p ◦ p α + α ) (cid:19) × e − ( α + α ) ( q + M ) − α α α α p − p ◦ p α α . (4.10)We can now follow the same steps to regulate and integrate this, and again find aclosed-form expression for the pieces which contain divergences. Note that unlike the φ – 18 –alculation, we can already see that the nonplanar expression will not merely be given byΛ → Λ eff , as the change of variables has here modified the numerator of the integrandto give an extra piece to the momentum polynomial multiplying the exponential. And sointegration gives usΓ (2) ,s,np ( p ) = g g π (cid:34) (cid:0) M + p p ◦ p + 40(4 M + p ) p ◦ p Λ (cid:1) K (cid:32) (cid:112) M + p Λ eff (cid:33) + 20 (cid:112) M + p Λ eff (cid:0) − 96 + p p ◦ p + 12 p ◦ p Λ (cid:1) K (cid:32) (cid:112) M + p Λ eff (cid:33)(cid:35) . (4.11)We must now think slightly more carefully about what we want to add to the quadraticeffective action to find a Wilsonian interpretation of this theory. We may isolate the IRdivergence that appears when the cutoff is removed by first taking the limit Λ → ∞ with p ◦ p held fixed, and then expanding around p ◦ p = 0. We may then ask that this samedivergence appears at any value of Λ. To account for this IR divergence, we must add toour effective action∆ S (Λ) = − (cid:90) d p (2 π ) g g π (cid:18) Λ − p ◦ p (cid:19) ϕ ( p ) ϕ ( − p ) , (4.12)which can easily be done through the addition of an auxiliary scalar field as was done inSection 3 and is discussed in more generality in Appendix A. After having added this to ouraction, for small p ◦ p the scalar two-point function now behaves as Γ s ( p ) = − g g π p ◦ p + . . . for any value of Λ. The new pole in this case has the opposite sign as that in 3.9, and sowill be accessible in Euclidean signature, clearly signaling a tachyonic instability. Whilethis puts the violation of unitarity in this theory on prime display, it also means that thispole will be accessible in the s -channel in the Lorentzian theory if we allow for timelikenoncommutativity.We emphasize that any conclusions about the Lorentzian theory with timelike non-commutativity are speculative and dependent upon a solid theoretical understanding ofa unitary formulation of the field theory, and in principle such a formulation could findradically different IR effects than this na¨ıve approach. However, it was found in [68] thata modification of time-ordering to explicitly make the theory unitary (at the expense ofmicrocausality violation) leaves the one-loop correction to the self-energy unchanged in φ theory, and the same might be expected to hold true for Yukawa theory. This makes itworthwhile to at least briefly consider the potential phenomenological consequences of thenew pole.At low energies, the propagator is here modified to m +( p i + p j ) − g g π p i + p j ) ◦ ( p i + p j ) .If we consider scattering of fermions through an s -channel ϕ and take the simple case of anoncommutativity tensor which in the lab frame has one eigenvalue 1 / Λ θ with m (cid:29) Λ θ ,then the emergent pole appears at s = g g π − β β Λ θ m . Here s = − ( p i + p j ) is the invariantmomentum routed through the propagator, and β is the boost of the ( p i + p j ) systemwith respect to the lab frame. The Lorentz-violation here then has the novel effect ofsmearing out the resonance corresponding to the light pole for a particle which is produced– 19 –t a variety of boosts. This is in contrast to the pole at m , which gives a conventionalresonance at leading order. Of course, we have not constructed a fully realistic theoryin any respect, and ultimately it may well be that other Lorentz-violating effects providethe leading constraint. Nonetheless, the lineshape of resonances may be an interestingobservable in this framework.A further feature of this opposite sign of the new pole compared to that in the φ theory is that the unusual momentum-dependence of the two-point function will leadto ordered phases which break translational invariance [37, 115–118]. While a Lorentz-violating background field may possibly be very well constrained, the detailed constraintdepends on its wavelength and the ways in which it interacts with the SM. But this isanother obvious line of exploration for constraining realistic NCFTs. There are again two planar diagrams: − i Γ (2) ,f,p = p g ₁ g ₁ + p g ₂ g ₂ No new features appear in the evaluation of these diagrams, so we merely quote the finalresult:Γ (2) ,f,p = − g + g π (cid:18) M − /p (cid:19) log 4 p Λ m + 2 m ( p − M ) + ( M + p ) + . . . (4.13)We also have two nonplanar diagrams, which again mix the two vertices − i Γ (2) ,f,np = p g ₁ g ₂ + p g ₁ g ₂ Here we find that the different phase factors for each diagram, which we saw wereinconsequential for the nonplanar corrections to the scalar, have an important role. Whenwe complete the square in each of the two cases, we find that one of the diagrams hasan integrand proportional to (cid:16) M − /p α α + α − p µ θ µν γ ν α + α (cid:17) and the other is proportional to (cid:16) M − /p α α + α + p µ θ µν γ ν α + α (cid:17) , so the would-be divergence in pθ will cancel manifestly betweenthe two diagrams. After this everything proceeds as before, and we findΓ (2) ,f,np = − g g π (cid:18) M − /p (cid:19) log 4 p Λ m + 2 m ( p − M ) + ( M + p ) + . . . (4.14)We see that with g = g ≡ g , the fermion quadratic effective action also behaves asexpected from ‘strong UV/IR duality’. The logarithmic divergence of the commutativetheory has been transmogrified in the nonplanar diagrams into IR dynamics via the simplereplacement Λ → Λ eff , and so a p ◦ p → .5 Three-Point Function The correction to the vertex function constitutes further theoretical data toward theWilsonian interpretation of the noncommutative corrections. We calculate the one-loopcorrection in this section and delay the discussion of the use of auxiliary fields to accountfor them until Appendix A.3. We will find that while we can use the same fields to accountfor the modifications to both the propagators and the vertices, the physical interpretationof such fields is unclear.We can compute corrections for each fixed ordering of external lines separately sincethey’re coming from different operators. For simplicity we’ll compute the g ordering,which we will denote Γ ϕψψ ( r, p, (cid:96) ). There are four diagrams in total: one planar diagramwith two insertions of the g vertex, one nonplanar diagram with two insertions of the g vertex, and two nonplanar diagrams with one insertion of each. It is easy to see by lookingat the diagrams that the same expressions with g ↔ g compute the correction to theother ordering, Γ ψϕψ ( r, p, l ).The new feature of this computation is that we now need three Schwinger parameters,and this presents a problem for our previous computational approach. We won’t be able toperform the two finite integrals before expanding in a variable which isolates the divergenceswhen α + α + α → 0, analogously to what we did in 2 d Schwinger space. Instead weslice 3 d Schwinger space such that we can perform the integral which isolates the leadingdivergences first, and then — as long as we’re content only to understand this divergence —we can discard the rest without having to worry about performing the other two integrals.The planar diagram is i Γ ϕψψ ,p ( p, (cid:96) ) = p g ₂ g ₂ g ₁ l , and corresponds to the expressionΓ ϕψψ ,p ( p, (cid:96) ) = − i ( ig )( ig ) (cid:90) d k (2 π ) ( − i ) (cid:16) M − ( /k + /p + /(cid:96) ) (cid:17) (cid:16) M − ( /k − /p − /(cid:96) (cid:17)(cid:0) ( k + p + (cid:96) ) + M (cid:1) (cid:0) ( k − p − (cid:96) ) + M (cid:1) (cid:0) ( k + p − (cid:96) ) + m (cid:1) . (4.15)After moving to Schwinger space, integrating over the loop momentum, and introducing acutoff exp (cid:0) − / (cid:0) Λ ( α + α + α ) (cid:1)(cid:1) , we switch variables to α = ξ η, α = ξ η, α = (1 − ξ − ξ ) η, (4.16)under which (cid:82) ∞ d α (cid:82) ∞ d α (cid:82) ∞ d α → (cid:82) d ξ (cid:82) − ξ d ξ (cid:82) ∞ d η η . Performing the momen-tum integral transfers the divergence for large k to a divergence in small α + α + α = η .This will allow us to find the leading divergent behavior immediately by carrying out the η integral and then expanding in Λ → ∞ . This yieldsΓ ϕψψ ,p ( p, (cid:96) ) = g g π log (cid:0) Λ (cid:1) + finite , (4.17)where we are unable to determine the IR cutoff of the logarithm, but this suffices for ourpurposes. – 21 –he three nonplanar graphs now each receive a different phase corresponding to whichexternal line crosses the internal line i Γ ϕψψ ,np ( p, (cid:96) ) = p g ₁ lg ₁ g ₁ + p lg ₁ g ₂ g ₁ + p lg ₁ g ₂ g ₁ , where the first gets exp [ − i ( k ∧ p + k ∧ (cid:96) + p ∧ (cid:96) )], the second exp [ − i ( k ∧ p + p ∧ (cid:96)/ − i ( k ∧ (cid:96) + p ∧ (cid:96)/ p, (cid:96) → p,(cid:96) → Γ ϕψψ ,np ( p, (cid:96) ) = g ( g + 2 g )16 π log (cid:0) Λ (cid:1) + finite . (4.18)However, if we first take the UV limit Λ → 0, and then expand in small momenta, we findlim Λ →∞ Γ ϕψψ ,np ( p, (cid:96) ) = g π (cid:20) g log (cid:18) p + (cid:96) ) ◦ ( p + (cid:96) ) (cid:19) + g log (cid:18) p ◦ p (cid:19) + g log (cid:18) (cid:96) ◦ (cid:96) (cid:19)(cid:21) + finite , (4.19)where we again see UV/IR mixing, and we note that each nonplanar diagram has beeneffectively cutoff by the momenta which cross the internal line. We discuss the use ofauxiliary fields to restore a Wilsonian interpretation to this vertex correction in AppendixA.3. We now turn our attention to the softly-broken noncommutative Wess-Zumino model asa controllable example of the interplay between UV/IR mixing and the finiteness of thefield theory. We will restrict ourselves to calculating the one-loop correction to the scalartwo-point function. Since the new poles appearing in the quadratic effective action inthe scalar and Yukawa theories are intimately related to the quadratic divergences of thecommutative theories, we will not be surprised to find that this feature will disappear whenboth the scalar and the fermion are present in the EFT below the cutoff. By studying thesoftly-broken theory we can take the fermion above or below the cutoff to smoothly see therelation between the finiteness of the field theory and the effects of UV/IR mixing. Theexactly supersymmetric noncommutative Wess-Zumino model was first discussed in detailin [119], and the absence of an infrared pole in a softly-broken theory was first noted in[105]. The softly-broken Wess-Zumino model was first considered in [42]. The noncommutative Wess-Zumino theory can be suitably formulated in off-shellsuperspace as L = (cid:90) d θ Z Φ † Φ + (cid:90) d θ (cid:18) M Φ + 16 y Φ (cid:63) Φ (cid:63) Φ (cid:19) + h.c. , (5.1) Our one-loop results agree with those of [42] save for their claim that logarithmic IR divergences areabsent in the exactly supersymmetric theory, which contradicts [119]. We will below find a logarithmicIR divergence in the wavefunction renormalization which is independent of the soft-breaking, which isconsistent with the expectations of strong UV/IR duality. – 22 –here Φ is a chiral superfield and we have included a wavefunction renormalization factorin the K¨ahler potential Z = 1 + O ( y ). We can introduce soft supersymmetry breaking bypromoting this factor to a spurion Z = 1 + ( | M | − m ) θ θ † , the only effect of which is tomodify the scalar mass spectrum.Formulating the noncommutative theory including the auxiliary F fields makes itmanifest that we have preserved supersymmetry off-shell. This procedure is in fact preciselythe same as quantizing after integrating out F , and so we end up with a star-product versionof the familiar Lagrangian: −L NCWZ = Z∂ µ φ ∗ ∂ µ φ − iZψ † ¯ σ µ ∂ µ ψ + Z − m φ ∗ φ + 12 M ψψ + 12 M ∗ ψ † ψ † + 12 Z − yφ (cid:63) ψ (cid:63) ψ + 12 Z − y ∗ φ ∗ (cid:63) ψ † (cid:63) ψ † + 12 Z − yM ∗ φ (cid:63) φ (cid:63) φ ∗ + 12 Z − y ∗ M φ ∗ (cid:63) φ ∗ (cid:63) φ + 14 Z − | y | φ (cid:63) φ (cid:63) φ ∗ (cid:63) φ ∗ (5.2)where φ is a complex scalar and ψ is a Weyl fermion. Of course, now that we’ve introducedsupersymmetry breaking we expect to find that there is further renormalization beyondthat associated with Z , but keeping the manifest factors of Z will allow us to easily compareto our expectations for the supersymmetric limit.The calculation of the one-loop correction to the two-point function goes much as thepreviously-demonstrated examples. The presence of the three-scalar interaction gives anew class of diagrams, whose evaluation is routine. The two-component fermions yieldslightly different factors than did the Dirac fermions [120]. Finally, it is important to notethat the results for the diagrams computed in Section 3 cannot be used here, as we musthere regulate uniformly using exp( − / (Λ ( α + α ))) like we did in Section 4. This maybe easily accommodated by writing the integrand in the quartic diagrams as k + m k + m k + m .Adding up all these diagrams and taking the limit where Λ , Λ eff are large, we find thatthe one-loop scalar two-point function may be organized asΓ (2) ,s ≡ Zp + Z − ( m + δm ) (5.3) Z = 1 + y π log (cid:20) ΛΛ eff M (cid:21) + . . . (5.4) δm = y π (cid:0) M − m (cid:1) log (cid:20) ΛΛ eff M (cid:21) + . . . , (5.5)where we make manifest the presence of supersymmetric nonrenormalization in the limit m → M , which acts as a non-trivial check. As expected, the absence of the quadraticUV divergence in the Wess-Zumino model has led to the absence of an infrared pole fromthe noncommutativity, even as the fermion is made arbitrarily heavy relative to the scalar.However, logarithmic UV/IR mixing still occurs.We may repeat this calculation using dimensional regularization and taking note ofthe issues which arose in Section 3.1. Using the same parametrization of the one-loop– 23 –wo-point function as above, the planar diagrams contribute Z planar = 1 + y π (cid:18) (cid:15) + log µ M (cid:19) + . . . (5.6) δm = y π ( M − m ) (cid:18) (cid:15) + log µ M (cid:19) + . . . , (5.7)as expected. The full form of the nonplanar diagrams is unenlightening, but if we take theIR limit p ◦ p → (cid:15) → d < Z nonplanar = 1 + y π log 4 M p ◦ p + . . . (5.8) δm = y π ( M − m ) log 4 M p ◦ p + . . . , (5.9)which has precisely the same correspondence with the Schwinger-space regularization aswe saw for the φ case.We thus see clearly the conflict between supersymmetry and the use of UV/IR mixingto explain low-energy puzzles. UV/IR mixing transmogrified UV momentum dependenceinto IR momentum dependence, and so depended crucially on the sensitivity of our fieldtheory to UV modes. For a theory which is finite as a field theory, the dependence on theUV physics has been removed, and so we see no interesting IR effects.Of course, in the presence of a cutoff Λ it is also possible to study the behavior ofthe scalar two-point function when M (cid:29) Λ (cid:29) | M − m | as the fermion is taken abovethe cutoff while keeping the scalar light. This corresponds to taking M/ Λ , M/ Λ eff > , Λ eff are large. This gets rid of the nonplanarYukawa-type diagrams and, as one might expect, results in a return of UV sensitivity inthe scalar EFT below the cutoff, foreshadowing a return of the UV/IR mixing effects. Thescalar mass-squared in this limit becomes δm = y π (cid:0) M + 16Λ + 8Λ (cid:1) + . . . . (5.10)and UV/IR mixing reappears at the quadratic level. So our EFT intuition isn’t totally outthe window; it’s been broken in a controlled way, and we can smoothly interpolate betweentheories with and without UV/IR mixing by taking the states responsible for finitenessabove the cutoff. This sharpens the sense in which UV/IR mixing can do somethinginteresting in the IR as long as the field-theoretic description of our universe is never finite.Ultimately, this highlights a central challenge for approaching the hierarchy problemvia UV/IR mixing. The hierarchy problem is particularly sharp when the full theory isfinite and scale separation is large, in which case the sensitivity of the Higgs mass tounderlying scales is unambiguous. But UV/IR mixing effects potentially relevant to thehierarchy problem are absent in this case, and emerge only when finiteness is lost. Thistension is not necessarily fatal to UV/IR approaches to the hierarchy problem – ultimately– 24 –he UV sensitive degrees of freedom are not the ones we would wish to identify with theHiggs – but it bears emphasizing.Moreover, there is a possible loophole in the general argument that finiteness must besurrendered in order to generate a scale from UV/IR mixing. The presence of interestingeffects in the IR here depends solely on the UV sensitivity of the nonplanar diagrams. The‘orbifold correspondence’ [121–123] provides non-supersymmetric field theories constructedvia orbifold truncation of N > To attempt to formulate a realistic theory which uses UV/IR mixing to solve extanttheoretical puzzles, it would be useful to have an understanding of which features of NCFTwere responsible for the curious infrared effects discussed above. This would be helpfulwhether one wishes to test out these ideas in any of the many proposed modifications ofNCFT, or to write down other toy models which share some features of NCFT but arebased upon different principles.Qualitatively, the two unusual features involved in the formulation of NCFT areLorentz invariance violation and nonlocality. However, it is obvious that one may havetheories with one or both of these features without the interesting effects we have seen.The answer then is not so simple as pointing to one axiom or another of EFT which has beenbroken, but depends sensitively on the way in which they are broken. We briefly exploretwo ways we may better understand the interplay here between nonlocality and Lorentz-violation and how they come together to cause surprising low-energy effects. We first give ageneral argument based on the way nonlocality appears to postdict the form of the violationof EFT expectations. We then phenomenologically examine the loop integration appearingin our NCFT calculations to diagnose what caused the appearance of the IR pole. Thiswill lead us to discuss an avenue toward investigating (or manufacturing) such effects innonlocal, Lorentz-invariant theories.To see how EFT expectations may be violated, consider the peculiar way in which thenoncommutative effects in the one-loop action (e.g. Equation 3.6) induce nonlocality. InWilsonian EFT, integrating out momentum modes p (cid:38) Λ produces a nonlocal theory atthose scales, or equivalently on distances x (cid:46) / Λ. However, particles on a noncommutativespace can be thought of as rods of size L ∼ pθ [53–57]. This tells us that in a NCFT weshould expect nonlocality to be present for scales x (cid:46) pθ . Comparing the two scales, we seethat we should find nonlocal effects past those expected in Wilsonian EFT for < pθ . Herethis momentum-dependent nonlocality occurs in a Lorentz-violating way. This expectationwas exactly borne out in the examples above, where we saw that the one-loop effectiveaction in momentum space is nonlocal for p ◦ p (cid:29) / Λ [37].– 25 –urely from this analysis of the form of nonlocality, we may conclude there will be abreakdown of Wilsonian renormalization. After we remove the cutoff, the theory shouldbe nonlocal on all scales p ◦ p > 0. But if we compute a correlation function at a large-but-finite Λ, the theory will still be local for momenta p ◦ p < / Λ , and so will greatly differfrom the continuum result. So our surprising discovery of the non-uniform convergence ofcorrelation functions in the examples above is understood easily from this picture.While this sort of momentum-dependent nonlocality may seem ad hoc , it has beensuggested previously for separate purposes. It has been argued [125] that quantum gravityshould obey a ‘Generalized Uncertainty Principle’ ∆ x (cid:38) (cid:126) ∆ p + (cid:96) p ∆ p , with (cid:96) p the Plancklength, based on the use of Hawking radiation to measure the horizon area of a blackhole. This gives precisely the same sort of momentum-dependent nonlocality as we sawabove. We refer the reader to [126] for a review of the Generalized Uncertainty Principle,[127, 128] for similar conclusions within string theory, and [129] for a more general review ofthe appearance of an effective minimal length in quantum gravity. It would be interestingto investigate other field theories which obey such uncertainty principles and determinewhether UV/IR mixing causes similar features as appear in NCFT. For theories whichviolate Lorentz invariance, care must be taken to avoid arguments that even Planck-scaleLorentz violation is empirically ruled out [38, 130].We may also attempt to phenomenologically diagnose what caused the appearanceof the IR pole from the form of the loop integration. The presence of an exponential ofmomenta was clearly crucial, and this implies a necessity of nonlocality. It’s also clear thatthe modification of the cutoff in the nonplanar diagrams Λ (cid:55)→ Λ eff , which rendered thediagrams UV finite in a way that brought UV/IR mixing, was a result of the contractionbetween the loop momentum and the external momentum. Less obviously, one may see thatany quadratic term in loop momentum in the exponential would have erased this feature,as after momentum integration one would find an integrand ∼ α + , and any divergencewill have disappeared. Heuristically, the quadratic suppression in loop momentum is toostrong and regulates the UV divergence entirely independently of the cutoff, so no UV/IRmixing appears. NCFT disallows such terms as a result of momentum contractions beingperformed with an antisymmetric tensor, and this particular mechanism seems to implythe necessity of Lorentz invariance violation. However, this argument only considers smalldeviations from the form of the integral in NCFT. Further discussions of the form of loopintegrals with generalizations of the star-product may be found in [131, 132].Likely a better approach to understand the prospect for finding features similar to thatof NCFT in a Lorentz invariant theory is to back up and study formulations of Lorentzinvariant extensions of NCFT. This is accomplished by upgrading the noncommutativitytensor θ µν from a c -number to an operator. This was proposed already by Snyder in 1947[19], and this approach has been revived a number of times more recently (e.g. [133–137]).Schematically, this results in an action containing an integral over θ µν S = (cid:90) d x d θ W ( θ ) L ( φ, ∂φ ) , (6.1)where W ( θ ) is a ‘weighting function’, and the Lagrangian is still defined using the star-– 26 –roduct. The challenge in this approach for our purposes is in devising a method fornonperturbative calculations in θ , which as we saw above was necessary to preserve thefeatures of UV/IR mixing.Searching more generally for Lorentz invariant theories which contain UV/IR mixingwill likely allow more promising phenomenological applications. That such theories shouldexist can be broadly motivated by quantum gravity, as any gravitational theory is expectedboth to be nonlocal and to have UV/IR mixing. That Lorentz violation should be present isless clear. A particularly interesting line of development is to then understand in detail theclass of nonlocal theories that would have UV/IR mixing of a sort similar to that discussedhere. Recent work toward placing nonlocal quantum field theories on solid theoreticalground [44, 78] is clearly of sharp interest here, though the larger goal is quite distinct.The nonlocality studied in these works is designed to render the field theory UV-finite, andso the nonlocal vertex kernels are chosen precisely to avoid the introduction of new polesby ensuring these are momentum-space entire functions which vanish rapidly in Euclideandirections. The nonlocal vertices of NCFT manage to introduce new poles by oscillatingas p → ∞ , which presumably allows for the appearance of new ‘endpoint singularities’[138, 139], though a full examination of the Landau equations in NCFT has not (to ourknowledge) been performed. Our interest is thus in a disjoint class of nonlocal theories,where new poles can appear in interesting ways. Classifying the space of such theories anddeveloping an approach to systematically understand their unitarity properties seems wellmotivated. The lack of evidence for conventional solutions to the hierarchy problem has placed particlephysics at a crossroads. While it is possible that the answer ultimately lies further down thewell-trodden path of existing paradigms, the appeal of less-travelled paths grows greaterwith every inverse femtobarn of LHC data.In this work we have ventured to take seriously the apparent failure of expectationsfrom Wilsonian effective field theory regarding the hierarchy problem by investigating aconcrete framework — noncommutative field theory — in which Wilsonian EFT itselfbreaks down. Not only does noncommutative field theory violate Wilsonian expectations, itprovides a sharp instance of UV/IR mixing: ultraviolet modes of noncommutative theoriescan generate an infrared scale whose origin is opaque to effective field theory. To theextent that UV/IR mixing has any relevance to the hierarchy problem, the emergence ofan infrared scale seems to be among the most promising effects. Although the real-worldapplicability of these theories is likely limited by their Lorentz violation, they nonethelessprovide valuable toy models for exploring the potential relevance of UV/IR mixing toproblems of the Standard Model.To this end, we have surveyed existing results on noncommutative theories with an eyetowards ‘strong UV/IR duality’ — the transmogrification of UV divergences into infraredpoles at the same order. This led us to a detailed analysis of noncommutative Yukawatheory, perhaps the most useful toy model for thinking about the hierarchy problem (insofar– 27 –s the Yukawa sector of the Standard Model is responsible for the largest UV sensitivityof the Higgs mass, and highlights the relative UV insensitivity of the fermion masses). Inthe noncommutative theory, the presence of both inequivalent Yukawa couplings impliesthe same strong UV/IR duality exhibited by real φ theory: a quadratic divergence in theone-loop correction to the scalar mass from fermion loops gives rise to a simple IR pole,while a logarithmic UV divergence in the one-loop correction to the fermion mass fromscalar loops give rise to only a logarithmic IR divergence. Intriguingly, the infrared pole inthe scalar two-point function appears accessible in the s -channel in the Lorentzian theory,a feature which gives it particular phenomenological relevance.We then introduced softly-broken supersymmetry as a way to explore the interplaybetween (in)finiteness and UV/IR mixing. Choosing soft terms in order to keep the scalarlight as the fermion mass is varied concretely illustrates several expected features. StrongUV/IR duality is preserved in the sense that both UV and IR divergences are absent atquadratic order (and persist at logarithmic order) when both the scalar and the fermionare in the spectrum. However, infrared structure reappears as the fermion mass is raisedabove a fixed cutoff and (quadratic) finiteness is lost. This underlines the sense in whichUV/IR mixing may only ever play an interesting role when the field theory is quadraticallyUV sensitive at all scales, a scenario in which the hierarchy problem is less concrete.Finally, building on the lessons from the toy models considered here, we have high-lighted a variety of interesting lines of exploration in theories featuring nonlocality with orwithout Lorentz violation that may be of relevance to the hierarchy problem.While the prospect that UV/IR mixing will solve outstanding theoretical problemsin the low-energy universe is possibly fanciful, now is the time for such reveries. Theparadigms of the past few decades of particle theory are under considerable empiricalpressure, and innovative approaches are needed. At the very least, by pushing the limitsof EFT we stand to learn more about the broad spectrum of phenomena possible withinquantum field theory. Acknowledgements We thank Nima Arkani-Hamed, Matthew Brown, Andy Cohen, Tim Cohen, BriannaGrado-White, Alex Kinsella, Harold Steinacker, Terry Tomboulis, Timothy Trott, andYue Zhao for valuable discussions. We are grateful to Tim Cohen, Isabel Garcia-Garcia,and Robert McGehee for comments on a draft of this manuscript. This work is supportedin part by the US Department of Energy under the Early Career Award DE-SC0014129 andthe Cottrell Scholar Program through the Research Corporation for Science Advancement. A Wilsonian Interpretations from Auxiliary Fields In this appendix we discuss various generalizations of the procedure introduced in [37, 61]to account for the new structures appearing in the noncommutative quantum effectiveaction via the introduction of additional auxiliary fields.– 28 – .1 Scalar Two-Point Function It is simple to generalize the procedure discussed in Section 3 to add to the quadraticeffective action of φ any function we wish through judicious choice of the two-point functionfor an auxiliary field σ which linearly mixes with it. In position space, if we wish to addto our effective Lagrangian ∆ L eff = 12 c φ ( x ) f ( − i∂ ) φ ( x ) , (A.1)where f ( − i∂ ) is any function of momenta, and c is a coupling we’ve taken out for conve-nience, then we simply add to our tree-level Lagrangian∆ L = 12 σ ( x ) f − ( − i∂ ) σ ( x ) + icσ ( x ) φ ( x ) , (A.2)where f − is the operator inverse of f . It should be obvious that this procedure is entirelygeneral. As applied to the Euclidean φ model, we may use this procedure to add a secondauxiliary field to account for the logarithmic term in the quadratic effective action as∆ L = 12 σ ( x ) 1log (cid:2) − ∂ ◦ ∂ (cid:3) σ ( x ) − gM √ π σ ( x ) φ ( x ) , (A.3)where we point out that the argument of the log is just 4 / (Λ p ◦ p ) in position space. Wemay then try to interpret σ also as a new particle. As discussed in [61], its logarithmicpropagator may be interpreted as propagation in an additional dimension of spacetime.Alternatively, we may simply add a single auxiliary field which accounts for boththe quadratic and logarithmic IR singularities by formally applying the above procedure.But having assigned them an exotic propagator, it then becomes all the more difficult tointerpret such particles as quanta of elementary fields. A.2 Fermion Two-Point Function To account for the IR structure in the fermion two-point function, we must add an auxiliaryfermion ξ . If we wish to find a contribution to our effective Lagrangian of∆ L eff = c ¯ ψ O ψ, (A.4)where O is any operator on Dirac fields, then we should add to our tree-level Lagrangian∆ L = − ¯ ξ O − ξ + c (cid:0) ¯ ξψ + ¯ ψξ (cid:1) , (A.5)with O − the operator inverse of O . In the Lorentzian Yukawa theory of Section 4, if weadd to the Lagrangian∆ L = − ξ M − i /∂/ M − ∂ / (cid:20) log (cid:18) − ∂ ◦ ∂ (cid:19)(cid:21) − ξ + g √ π (cid:0) ξψ + ψξ (cid:1) . (A.6)we again find a one-loop quadratic effective Lagrangian which is equal to the Λ → ∞ valueof the original, but now for any value of Λ.– 29 – .3 Three-Point Function We may further generalize the procedure for introducing auxiliary fields to account for IRpoles to the case of poles in the three-point effective action. It’s clear from the form of theIR divergences in Equation 4.19 that they ‘belong’ to each leg, and so na¨ıvely one mightthink this means that the divergences we’ve already found in the two point functionsalready fix them. However those corrections only appear in the internal lines and werealready proportional to g , and so they will be higher order corrections. Instead we mustgenerate a correction to the vertex function itself which only corrects one of the legs.To do this we must introduce auxiliary fields connecting each possible partition ofthe interaction operator. However, while an auxiliary scalar χ coupled as χϕ + χψψ would generate a contribution to the vertex which includes the χ propagator with the ϕ momentum flowing through it, it would also generate a new ( ψψ ) contact operator,which we don’t want. To avoid this we introduce two auxiliary fields with off-diagonaltwo-point functions, a trick used for similar purposes in [61]. By abandoning minimality,we can essentially use an auxiliary sector to surgically introduce insertions of functions ofmomenta wherever we want them.We can first see how this works on the scalar leg. We add to our tree-level Lagrangian∆ L = − χ ( x ) f − ( − i∂ ) χ ( x ) + κ χ ( x ) ϕ ( x ) + κ χ ( x ) (cid:63) ψ ( x ) (cid:63) ψ ( x ) . (A.7)Now to integrate out the auxiliary fields we note that for a three point vertex, one may usemomentum conservation to put all the noncommutativity between two of the fields. Thatis, χ ( x ) (cid:63) ψ ( x ) (cid:63) ψ ( x ) = χ ( x )( ψ ( x ) (cid:63) ψ ( x )) = ( ψ ( x ) (cid:63) ψ ( x )) χ ( x ) as long as this is notbeing multiplied by any other functions of x . So we may use this form of the interactionto simply integrate out the auxiliary fields. We end up with∆ L eff = κ κ ψ (cid:63) ψ (cid:63) f ( − i∂ ) ϕ (A.8)which is exactly of the right form to account for an IR divergence in the three-point functionwhich only depends on the ϕ momentum.For the fermionic legs, we need to add fermionic auxiliary fields which split the Yukawaoperator in the other possible ways. We introduce Dirac fields ξ, ξ (cid:48) and a differentialoperator on such fields O − ( − i∂ ). Then if we add to the Lagrangian∆ L = − ξ O − ξ (cid:48) − ξ (cid:48) O − ξ + c ( ξ (cid:63)ψ(cid:63)ϕ + ψ(cid:63)ξ (cid:63)ϕ )+ c ( ξ (cid:63)ϕ(cid:63)ψ + ψ(cid:63)ϕ(cid:63)ξ )+ c ( ξ (cid:48) ψ + ψξ (cid:48) ) , (A.9)we now end up with a contribution to the effective Lagrangian∆ L eff = c c (cid:0) ¯ ψ (cid:63) O ( ψ ) (cid:63) ϕ + ¯ ψ (cid:63) O ( ψ (cid:63) ϕ ) (cid:1) + c c (cid:0) ¯ ψ (cid:63) ϕ (cid:63) O ( ψ ) + ¯ ψ (cid:63) O ( ϕ (cid:63) ψ ) (cid:1) , (A.10)where we have abused notation and now the argument of O specifies which fields it actson. These terms have the right form to correct both vertex orderings.Now that we’ve introduced interactions between auxiliary fields and our original fields,the obvious question to ask is whether we can utilize the same auxiliary fields to correct– 30 –oth the two-point and three-point actions. In fact, using two auxiliary fields with off-diagonal propagators per particle we may insert any corrections we wish. The new trick isto endow the auxiliary field interactions with extra momentum dependence.For a first example with a scalar, consider differential operators f , Φ, and add to theLagrangian ∆ L = − χ f − ( − i∂ ) χ + κ χ ϕ + κ χ ψ (cid:63) ψ + gϕ Φ( − i∂ ) χ . (A.11)We may now integrate out the auxiliary fields and find∆ L eff = gκ ϕf (Φ( ϕ )) + κ κ ψ (cid:63) ψ (cid:63) f ( ϕ ) (A.12)where we’ve assumed that f and Φ commute. If we take Φ = then we have theinterpretation of merely inserting the χ two-point function in both the two-and three-point functions. But we are also free to use some nontrivial Φ, and thus to make thecorrections to the two- and three-point functions have whatever momentum dependencewe wish. It should be obvious how to generalize this to insert momentum dependence intothe scalar lines of arbitrary n − point functions.The case of a fermion is no more challenging in principle. For differential operators O , F , we add∆ L = − ξ O − ξ (cid:48) − ξ (cid:48) O − ξ + c ( ξ (cid:63) ψ (cid:63) ϕ + ψ (cid:63) ξ (cid:63) ϕ ) + c ( ξ (cid:63) ϕ (cid:63) ψ + ψ (cid:63) ϕ (cid:63) ξ )+ c ( ξ (cid:48) ψ + ψξ (cid:48) ) + g (cid:0) ¯ ξ O − F ψ + ¯ ψ O − F ξ (cid:1) , (A.13)and upon integrating out the auxiliary fields we find∆ L eff = gc ¯ ψ F ψ + c c (cid:0) ¯ ψ (cid:63) O ( ψ ) (cid:63) ϕ + ¯ ψ (cid:63) O ( ψ (cid:63) ϕ ) (cid:1) + c c (cid:0) ¯ ψ (cid:63) ϕ (cid:63) O ( ψ ) + ¯ ψ (cid:63) O ( ϕ (cid:63) ψ ) (cid:1) , (A.14)where the generalization to n -points is again clear. 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