Locality in Theory Space

Locality in Theory Space

Yonatan Kahn    and Jesse Thaler ykahn@mit.edu jthaler@mit.edu
Abstract

Locality is a guiding principle for constructing realistic quantum field theories. Compactified theories offer an interesting context in which to think about locality, since interactions can be nonlocal in the compact directions while still being local in the extended ones. In this paper, we study locality in “theory space”, four-dimensional Lagrangians which are dimensional deconstructions of five-dimensional Yang-Mills. In explicit ultraviolet (UV) completions, one can understand the origin of theory space locality by the irrelevance of nonlocal operators. From an infrared (IR) point of view, though, theory space locality does not appear to be a special property, since the lowest-lying Kaluza-Klein (KK) modes are simply described by a gauged nonlinear sigma model, and locality imposes seemingly arbitrary constraints on the KK spectrum and interactions. We argue that these constraints are nevertheless important from an IR perspective, since they affect the four-dimensional cutoff of the theory where high energy scattering hits strong coupling. Intriguingly, we find that maximizing this cutoff scale implies five-dimensional locality. In this way, theory space locality is correlated with weak coupling in the IR, independent of UV considerations. We briefly comment on other scenarios where maximizing the cutoff scale yields interesting physics, including theory space descriptions of QCD and deconstructions of anti-de Sitter space.

Keywords:
Field Theories in Higher Dimensions, Technicolor and Composite Models
institutetext: Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

MIT-CTP 4346

1 Introduction

Locality is a fundamental guiding principle when constructing quantum field theories to describe physical systems. Locality appears in many different guises, from the causal structure of Lorentz-invariant theories to the analyticity of the -matrix. Theories with compact dimensions offer an interesting context in which to think about locality, since for a low-energy observer, locality in the compact dimensions is qualitatively different from locality in the noncompact ones. From an ultraviolet (UV) or top-down perspective, various mechanisms exist to ensure compact locality. In the usual picture of dimensional reduction, locality in the UV is assumed, and interactions in the compact dimensions remain local after geometric compactification. In models of dimensional deconstruction ArkaniHamed:2001ca (), a UV-complete four-dimensional gauge theory condenses at low energies to yield a theory with a compact fifth dimension, and five-dimensional locality is ensured by the irrelevance of nonlocal operators before condensation. A deeper mechanism exists in the AdS/CFT correspondence Maldacena:1997re (); Gubser:1998bc (); Witten:1998qj (), where bulk locality emerges from the large- limit of the boundary CFT Heemskerk:2009pn (); Heemskerk:2010ty (); Fitzpatrick:2010zm (); Sundrum:2011ic (); Fitzpatrick:2011hu ().

From an infrared (IR) or bottom-up perspective, however, compact locality is baffling. In the far IR, a compact dimension can be described by a tower of Kaluza-Klein (KK) modes, and locality simply enforces certain constraints on the spectrum and interactions of these modes. If there are spin-1 degrees of freedom, as will be the case in this paper, there is a cutoff scale where longitudinal scattering of the massive spin-1 KK modes becomes strongly coupled. From an IR point of view, there is no apparent reason to exclude additional nonlocal interactions, and one might even expect nonlocal terms could render the theory better behaved in the IR. Indeed, in the local case, it is precisely the interactions among different KK levels which partially unitarize KK scattering, pushing above the naive expectation from considering the KK modes as independent massive vectors. It is therefore plausible that including nonlocal interactions with the correct sign could yield a similar interference effect, possibly driving the cutoff scale higher than in the local case.

Figure 1: Local cyclic -site moose diagram, known also as “theory space,” corresponding to the latticization of compactified five-dimensional Yang-Mills. The link fields transform as bifundamentals of the gauge groups, represented by shaded circles at either end of the link. When each link field acquires a vacuum expectation value, the moose describes interacting massive gauge bosons, one massless gauge boson, and one Goldstone winding mode. Examples of nonlocal interactions are shown in Figure 2.

In this paper, we present a system where precisely the opposite is true: insisting on the highest possible cutoff scale implies locality in the compact dimension. We study the case of a deconstructed five-dimensional Yang-Mills theory in a flat geometry, described by a “theory space” cyclic moose diagram as in Figure 1. This four-dimensional theory has an intrinsic cutoff scale , and maximizing is correlated with locality in theory space. This gives a purely low-energy perspective on why compact locality is special, in the sense that local theories are the most weakly coupled in the IR. Strictly speaking, our analysis only holds for small nonlocal perturbations, and we cannot exclude the possibility that large nonlocal terms could lead to a larger value of . While unitarity violation in higher-dimensional gauge theories has been investigated before SekharChivukula:2001hz (); Chivukula:2002ej (); DeCurtis:2003zt (), to our knowledge the only studies of extra-dimensional nonlocality have been in a gravitational context Schwartz:2003vj ().111When discretizing gravity, nonlocal interactions are necessary to have a local continuum limit Schwartz:2003vj (), which is not the case for gauge theories.

More concretely, in order to isolate the effects of nonlocality and remove trivial rescalings of , we introduce a dimensionless ratio

(1)

which normalizes the cutoff scale, where is an average mass for the spin-1 modes whose precise definition will be given in Section 3. After adding a nonlocal gauge kinetic term with small coefficient and nonlocal length scale , we show that behaves to lowest order in as

(2)

where is the compactification radius and is the number of sites in the moose. Surprisingly, terms which would contribute to scattering amplitudes at linear order in are (miraculously) absent due to a group-theoretic cancellation, thanks to the fact that the dominant scattering channel is a gauge singlet. This results in a leading dependence in Eq. (2) which is quadratic, such that locality at is a special value. We specialize to for technical reasons, but the result that maximizing implies locality holds any gauge group where the singlet channel dominates the scattering matrix; in particular, it holds for general , up to possible corrections that are subleading in .

Beyond cyclic theory space, the ratio is interesting in at least two other settings. First, as is well known, the pions and mesons of quantum chromodynamics (QCD) can be described by a three-site moose, but LABEL:Georgi:1989xy showed that “nonlocal” interactions with a negative nonlocal coefficient are required to match known QCD phenomenology. We find that maximing does indeed favor a nonlocal interaction, but one whose sign depends on the choice of normalization factor . Second, when deconstructing spaces with nontrivial geometry, such as warped anti-de Sitter (AdS) geometries in Randall-Sundrum scenarios Randall:1999ee (); Randall:1999vf (), maximizing implies an “-flat deconstruction” with all link decay constants equal. Such a deconstruction was first noted in LABEL:Abe:2002rj, and used in LABEL:SekharChivukula:2006we as a convenient simplification for sum rule computations, but here we show that it leads to a deconstructed theory with a maximal domain of validity (preferable to the usual deconstruction of AdS with equal lattice spacings Randall:2002qr (); Falkowski:2002cm ()). Furthermore, the same gauge singlet channel dominates even in warped geometries, giving circumstantial evidence that maximizing the cutoff also implies locality in AdS.

The remainder of this paper is organized as follows. In Section 2, we set the notation and conventions for the moose diagram which describes the deconstructed theory, and define the nonlocal perturbations in Eq. (21). In Section 3, we motivate the form of in Eq. (1) using dimensional and scaling arguments to define the average mass scale in Eq. (28). In Section 4, we describe the coupled-channel analysis of gauge boson scattering to define the cutoff scale and carry out the calculation of for the local moose. We derive our main result in Section 5, showing that maximizing implies locality in a cyclic -site moose with nonlocal terms. In Section 6, we briefly illustrate the phenomenological consequences of our ratio for a three-site QCD moose and for warped deconstructions, and speculate about locality in AdS space. We summarize our results in Section 7 and suggest ways to extend our analysis beyond tree-level.

2 Moose notation for theory space

In the following two subsections, we define our notation for the moose diagram representing theory space, and review the phenomenology of dimensional deconstruction. A reader familiar with these results may wish to skip to Section 2.3 where we introduce nonlocal interactions in theory space.

2.1 Local Lagrangian

We define a local -site cyclic moose with gauge group by the Lagrangian

(3)

as shown in Figure 1. The -valued field strengths live on the sites , and the link fields live on the links between sites and .222For ease of notation, we freely raise and lower site indices on gauge and link fields. There is a periodic identification of sites and links by . The sites are shaded to represent gauge symmetries, as opposed to global symmetries which will be important in our discussion of the QCD moose in Section 6.1.

Our normalization for the generators is

(4)

The gauge transformations of the link fields are

(5)

where are in the fundamental representation, so transforms in the antifundamental of the site to its left and the fundamental of the site to its right. This behavior is represented by the directional arrows on the link fields in Figure 1. The covariant derivative is defined by

(6)

where we use canonical normalization for the gauge fields so the gauge couplings appear explicitly.

Writing

(7)

the Lagrangian (3) becomes a nonlinear sigma model in 3+1 dimensions describing the interactions of the “pions” with themselves and the gauge fields . Pursuing this analogy with low-energy QCD, we refer to as decay constants. With the normalization convention (4), we obtain canonically normalized kinetic terms for the fields after expanding the Lagrangian as a power series in the . The remainder of the Lagrangian involving consists of derivative interactions suppressed by powers of , all of which are nonrenormalizable operators. Thus, the nonlinear sigma model has some UV cutoff, which we will take to be the scale of tree-level unitarity violation ; we will define precisely in Section 4.1. Keeping only the leading terms with coefficients , one can estimate the scale of unitarity violation by “naive dimensional analysis” Manohar:1983md (),

(8)

Note that is determined by the minimum of the because and are decoupled for , so each link has its own scale of unitarity violation, and unitarity violation for the whole moose is dominated by whichever one occurs first.

With the parameterization (7), each gets a vacuum expectation value (vev) and the link fields spontaneously break the gauge symmetry at each site, with acting as Goldstone bosons which are eaten by the gauge fields . The remaining unbroken gauge symmetry is the diagonal subgroup, whose gauge coupling (in a notation suggestive of KK decomposition) is given by

(9)

Since this is a cyclic moose, there is also an uneaten linear combination of Goldstone modes , which we will sometimes refer to as the winding mode.

The mass-squared matrix for the now-massive gauge fields arises from the second term in Eq. (3) by setting . Setting the gauge couplings to a common value ,

(10)

This matrix has a zero eigenvalue for all choices of , with eigenvector , corresponding to the diagonal subgroup mentioned above. If we further restrict the decay constants to be equal (), we have an analytic expression for the mass spectrum of the cyclic moose:

(11)

With the exception of Section 6.2 where we explore warped spaces, we will always set the decay constants and gauge couplings equal, and , corresponding to a discrete translation invariance along the moose.

2.2 Five-dimensional interpretation

The Lagrangian (3) can be interpreted as a (4+1)-dimensional lattice gauge theory where only the compact fifth dimension has been latticized, with the providing the fifth component of the five-dimensional gauge field in the continuum limit where the lattice spacing goes to zero. The continuum limit is ordinary five-dimensional Yang-Mills

(12)

where and

(13)

Using this interpretation, known as dimensional deconstruction, we obtain a dictionary between parameters in the Lagrangian and parameters in the latticized theory ArkaniHamed:2001ca ():

(14)
(15)
(16)
(17)

The last of these relations identifies the gauge coupling of the diagonal subgroup ( in Eq. (9)) with the the effective four-dimensional gauge coupling of the KK zero mode of the five-dimensional theory after compactification.

The five-dimensional interpretation of the deconstruction can be confirmed in several ways. First, the dictionary preserves the usual relation between four- and five-dimensional gauge couplings after compactification:

(18)

Second, we can examine the mass spectrum of Eq. (11) in the continuum limit, and with fixed:

(19)

for small integers . This is precisely the KK spectrum of modes on a circle of circumference , as one would expect from compactification of a fifth dimension. In this picture, the uneaten linear combination of Goldstone bosons corresponds to the nontrivial Wilson loop around the compact extra dimension, hence the name “winding mode” for .

In the framework of dimensional deconstruction, locality in the latticized dimension is built into Eq. (3) through locality in theory space. Indeed, the field strengths are decoupled at different sites , and the only couple to nearest-neighbor gauge fields and through Eq. (6). In the continuum limit, becomes a covariant derivative along the latticed direction, which is also local. In the original application of dimensional deconstruction, LABEL:ArkaniHamed:2001ca derived these local interactions by starting with a moose with additional fermions charged under a “color” gauge group, which confines to give the fields as fermion bilinears. However, in our analysis we take Eq. (3) as our starting point, with the as “fundamental” fields rather than composite operators.

2.3 Nonlocal terms

The aim of this paper is to study nonlocality in theory space, which we will incorporate by perturbing the local moose (3) by a gauge-invariant operator , where is small and dimensionless and is inserted for normalization.333While the squares of the decay constants must be positive for the kinetic terms to have the correct sign, there is no such restriction on the sign of , although must be real. For to be nonlocal, it must connect distant sites and , and hence transform in the of and the of . We could simply define a new link field connecting these sites, but we are interested in comparing theories with the same number of four-dimensional degrees of freedom. Moreover, the theory with this extra field has a pathological local limit, since as its decay constant goes to zero, disappears but unitarity is violated immediately due to Eq. (8).444Even if fields like were present in the original Lagrangian, we could always decouple them with the plaquette operator . As we take , becomes massive and decouples from the low-energy spectrum.

Instead, we choose to consider

(20)

for , and we will refer to this as a “hopping” term with .555The factor of 2 is purely conventional and simplifies some formulas in what follows. To preserve the discrete translation invariance in the compact dimension, we sum over all sites:

(21)

Figure 2 shows an example of a 6-site moose diagram with such nonlocal terms for both and . The goal of Section 5 will be to study the effect of on the cutoff scale . For simplicity, we will only consider perturbing the local moose by a nonlocal term with a single value of (unlike in Figure 2) so that we can study the effect of nonlocality as a function of the nonlocal length scale. We will abbreviate and display all results as a function of .

Figure 2: A graphical representation of the nonlocal terms (21), showing an cyclic moose with nonlocal terms corresponding (red long-dashed lines) and (blue short-dashed lines). We draw the nonlocal terms attached to links, rather than sites, to emphasize their construction in terms of link fields as in Eq. (20).

To construct the continuum version of this operator, we define the characteristic nonlocal length scale by

(22)

The deconstructed fields and are related to the continuum five-dimensional gauge field by

(23)
(24)

where is the coordinate along the fifth dimension and is the path-ordering symbol. These relations imply that in the continuum limit,

(25)

so our nonlocal operator is a nonlocal gauge kinetic term, connecting and by means of a Wilson line along the fifth dimension,

(26)

In the five-dimensional picture, Eq. (20) is the natural nonlocal operator to consider, as compared to operators like . The continuum analogue of the latter is , which does not have any simple interpretation. Moreover, such an operator is redundant, because

(27)

so this operator already contains kinetic terms for through .666This redundant nonlocal operator will appear in the three-site QCD moose in Section 6.1, and in the discussion of KK matching in Appendix A.3. As a result, to lowest order in the effect of would be to simply shift the decay constant , a trivial change which can be absorbed by a redefinition of the . In addition, after subtracting off the kinetic terms from , we are left precisely with terms like , but for all values of . To isolate a single nonlocal length scale , we will only consider the nonlocal operator (21).

We are always free to add additional terms (local or nonlocal) which have more than two derivatives, as long as they respect the symmetries present in the original local Lagrangian. However, in a momentum power-counting scheme, it is consistent to truncate the effective Lagrangian to order provided we restrict the analysis to tree-level, as we will do in this paper. Indeed, two derivatives are the minimum required to have any nontrivial nonlocal interaction, and these nonlocal terms in Eq. (21) are the same order in momentum as the nonlinear sigma model kinetic terms appearing in the local Lagrangian. If we wanted to study the effects of higher-derivative terms, then by Weinberg’s power-counting theorem Weinberg:1978kz (), we would also have to consider one-loop amplitudes of two-derivative terms. Such effects are certainly important, but we leave an analysis to order to future work.777This power-counting also explains why we do not consider nonlocal terms like ; as is well-known from chiral perturbation theory, these terms count as in momentum.

3 Normalizing the cutoff scale

We argued in the introduction that the cutoff scale gives useful information about the weak-coupling domain of validity of a theory. Since can be raised or lowered by trivial rescaling of the mass scales of the problem, one would prefer to study a dimensionless ratio between the cutoff scale and some other physical mass scale . In this section, we define the normalization which appears in our dimensionless ratio,

(1)

The precise definition of will be given in Section 4, but for the arguments in this section, the dimensional analysis estimate (8) will suffice.

Our goal is to find a definition for that depends only on physical observables and has a well-defined continuum limit. We will use the freedom to define such that is sensitive mainly to the degree of nonlocality, and is as insensitive as possible to local aspects of the four-dimensional moose theory and its five-dimensional interpretation. We will further insist that the effects of nonlocality show up only in , not in , such that truly tests how locality affects the UV cutoff alone, not on how locality affects the spectrum of the theory.

One possibility is to simply take , the decay constant of the moose, but this is problematic for several reasons. First, does not have a nice continuum interpretation, nor is well-defined in mooses with unequal decay constants. More significantly, cannot be unambiguously defined in terms of a physical observable. In the local theory, can be made physical by identifying it with the 4-pion scattering amplitude, but with the addition of nonlocal terms, there is no unambiguous definition of , since new scattering channels open up which were not present in the original Lagrangian.888There is an unambiguous definition of the decay constant for the uneaten Goldstone mode , but can be adjusted at will by adding a kinetic term for this mode, , which does not affect the gauge boson mass matrix. Thus, does not represent a physically relevant mass scale in the problem, and is not suitable for a normalization.

Another option is to use a mass scale derived from the mass matrix in Eq. (10). We have to be a bit careful, since any such mass is proportional to . If we take , would be sent to infinity, but this tells us nothing about locality as we have artificially squashed the whole mass spectrum far below the cutoff. To correct for this, we should normalize by .999We choose rather than in anticipation of Section 6.2, since is well-defined even for unequal through Eq. (9). A seemingly natural choice for the mass scale is , the (normalized) lightest nonzero mass eigenvalue. This is appealing since is essentially the spacing between the highest and lowest mass scales in the theory. However, is not terribly useful in isolating the effects of nonlocality on , since the gauge boson mode corresponding to is already “nonlocal,” with nonzero components at most sites even in the local moose.

Instead, the choice for that we will use is

(28)

which is a sort of “average” mass. As we will show in Section 5.1, this choice is appealing since it is independent of the nonlocal perturbation (up to discretization effects). Here, we will show that also has a nice continuum limit. Consider the following scaling of the (local) deconstructed parameters:

(29)

This scaling preserves all the continuum parameters , , and according to the deconstruction dictionary in Section 2.2, but changes according to Eq. (8). For arbitrary , we can approach the continuum limit while making the scale of unitarity violation for the deconstructed theory arbitrarily high.101010This contradicts the fact that the five-dimensional theory has an intrinsic cutoff , but this apparent paradox is resolved once we realize that for large enough , the deconstructed theory goes non-perturbative. This necessitates a careful definition of the continuum limit, discussed in Appendix A.1. Consequently, we want to choose so that the ratio is independent of . Indeed, has the correct scaling to compensate for the scaling of :

(30)

In contrast, does not scale with , since is the same for both the deconstructed theory and the continuum theory (see Eq. (19)).

We thus arrive at our final definition of in the deconstructed theory,

(31)

By construction, both the numerator and denominator are proportional to , so is independent of the (local) deconstructed parameters , , and .111111Our parametric analysis of can also be applied to the continuum five-dimensional theory; see Appendix A.2. As desired, is directly sensitive to nonlocality mainly through the change in .

4 Scattering in theory space

In this section, we show how to determine the cutoff scale, defined as the scale of tree-level unitarity violation, by studying tree-level -wave scattering amplitudes. We then compute the ratio for the local moose. In principle, we could perform these calculations directly in the continuum five-dimensional gauge theory, but in practice, it is much easier to use the deconstructed language of theory space and compute scattering amplitudes for ordinary four-dimensional gauge fields.

We will make one further simplification by considering amplitudes in the Goldstone equivalence limit. In this limit, longitudinal gauge boson scattering is dominated by the eaten Goldstones, and these Goldstone modes are simply the nonlinear sigma model fields of the moose. This limit is justified as long as we consider small enough values of such that corrections to the amplitudes of order are small. The reason for considering this limit is that when we investigate the effect of nonlocality on , we will find that there is no change to to first order in . This fact is difficult to see in the full KK theory, but is straightforward to derive for the Goldstones alone.

4.1 Partial wave unitarity

The leading unitarity-violating pieces of massive gauge boson scattering amplitudes arise from the scattering of longitudinal polarization states. In the limit of high-energy scattering, for sufficiently small values of the gauge coupling , the Goldstone equivalence theorem Cornwall:1974km (); Vayonakis:1976vz () allows us to replace longitudinally-polarized gauge bosons by the corresponding (eaten) Goldstone modes when computing amplitudes.

Because the scattering amplitudes depend nontrivially on the gauge indices, computing the scale of unitarity violation for a given scattering channel does not give the full information about the scale of unitarity violation for the whole collection of Goldstone modes. Instead, we use a coupled-channel analysis Lee:1977yc () where we compute the whole scattering matrix and define the scale of unitarity violation using its largest eigenvalue .

The conventional normalization of the scattering matrix for partial waves labeled by is (following the notation of LABEL:Chang:2003vs),

(32)

where is the scattering amplitude between properly normalized in and out states and , and is the -th Legendre polynomial. As in earlier applications of this coupled-channel analysis Lee:1977yc (); Chang:2003vs (); Cahn:1991xf (); SekharChivukula:2006we (), the -wave piece provides the strictest bounds. Expressing the amplitudes in terms of the Mandelstam variables , , and , the integration in Eq. (32) for amounts to the replacement in the amplitude and simply contributes a factor of 2 from .

For unitarity to hold, the largest eigenvalue of the -wave partial amplitude must satisfy121212The precise numerical value of the unitarity bound is not important for our analysis. If we wished, we could use this freedom to suppress the effects of unknown four-derivative terms, whose amplitudes grow as . For example, imposing (corresponding to the scale of “half-unitarity violation”) would suppress such terms by a factor of . In our analysis, all that matters is the parametric dependence of on , which is unchanged by such manipulations. , leading to the condition on the amplitude131313Note that will be real for the tree-level amplitudes we consider in this paper, so is an absolute value and not a complex modulus.

(33)

where is the largest eigenvalue of the matrix expressed as a function of the Mandelstam variable (after the replacement ). For the Goldstone equivalence limit we consider in this paper, the eigenvalues will be linear in , so we can define . By solving for , we obtain the scale of unitarity violation:

(34)

Since the matrix (expressed as a function of ) is related to the -wave piece by numerical factors, we can work directly with and get the scale of unitarity violation from Eq. (34).

4.2 Local moose scattering matrix

To compute the gauge boson scattering matrix for the local moose in the high-energy limit, we can derive Feynman rules directly from the part of the Lagrangian, expressed in terms of the Goldstone fields . Because we work in the high-energy limit, we need not work in the basis of Goldstones which are eaten by the gauge boson mass eigenstates; rather, we work in a basis which makes the global symmetries of the scattering matrix manifest.141414In fact, LABEL:Chivukula:2002ej computed all amplitudes in the mass eigenstate basis, but found that only a coupled-channel analysis reproduced the correct unitarity behavior. The dominant channel is precisely the one we find in the Goldstone basis.

Figure 3: Feynman rule for the 4-point interaction contributing to singlet-channel scattering . By convention, all momenta are ingoing. For the case of , the coefficient of the amplitude is independent of the choice of indices so long as ; if the 4-point vertex vanishes identically.

The reflection symmetry of the Lagrangian (3) implies the absence of terms odd in , and hence the leading-order interaction is a 4-point term,

(35)

The Feynman diagram contributing to scattering is shown in Figure 3. Notice that for the local Lagrangian, and are totally decoupled for .

Now, consider the matrix for an SU(2) moose, which will be our starting point for introducing nonlocality in the following section. Because all of the link fields are decoupled, consists of copies of the matrix for one link, allowing us to drop link indices in what follows. The Goldstone modes are triplets of , so the two-particle scattering states decompose into irreducible representations of as . Borrowing the language of chiral perturbation theory, the pions are in the isospin representation, and scattering takes place between two-particle states of definite total isospin .

By Bose symmetry, the -wave scattering amplitude vanishes for the antisymmetric states. The remaining nonzero eigenvalues can be calculated analytically by diagonalizing :

(36)

so the largest eigenvalue (strictly speaking, the eigenvalue of largest magnitude) is in the sector. We will refer to the corresponding eigenvector as the gauge singlet state151515The additional normalization factor of accounts for the fact that in the isospin basis, the final state particles for each isospin channel are identical.

(37)

As we will see, the fact that the gauge singlet is associated with the largest eigenvalue is the key property which implies the vanishing of the leading-order nonlocal contribution to the scattering matrix. Using in Eq. (34) gives a scale of unitarity violation

(38)

Note that this is stronger than the naive dimensional analysis bound (8) by a factor161616This leads to a very rough estimate of the effects of four-derivative terms at the cutoff. The NDA bound is the scale at which one-loop effects are comparable to tree-level effects, so at the actual cutoff, one can estimate the size of tree-level four-derivative terms as being suppressed relative to the local two-derivative terms by a factor of . of . Furthermore, the largest amplitude for any individual scattering channel is a factor of 2 smaller than that of the gauge singlet; performing the coupled-channel analysis strengthens the unitarity bounds considerably. Including the results for from Eq. (30), the normalized ratio for the local theory is

(39)

5 Nonlocality in a cyclic moose

We now turn to the main result of this paper, computing the effects of nonlocality on the ratio in a cyclic -site moose. We focus on the case of gauge group because the group theory objects and which accompany 3- and 4-point Feynman diagrams have no simple expressions except for . Nonetheless, the dependence of on the strength of the nonlocal term and the nonlocal length scale is independent of the choice of gauge group, and we comment on the case of general in Appendix B.4.

Consider perturbing the local cyclic Lagrangian (3) (with equal decay constants ) by the nonlocal term discussed in Section 2.3,

(21)

where is defined in Eq. (20). In general, such a perturbation will change the values of and from their values in the local theory; we denote the nonlocal values by and . In what follows, we will do a perturbation theory analysis for and to order . We will also only be interested in the leading behavior in , since in the continuum limit, we set and take .171717Strictly speaking, is not the correct continuum limit, as there is a maximum value of such that the moose reproduces five-dimensional physics; see Appendix A.1.

The calculations below quickly become technical, so we will summarize the main points.

  • The normalization factor is not affected by the nonlocal term up to discretization errors, so all dependence on nonlocality is contained in .

  • The largest eigenvalue of the scattering matrix does not change to first order in . This surprising fact is the most important result in this paper, since it implies that is a special value. This result depends only on the particular structure of the gauge singlet state corresponding to the largest unperturbed eigenvalue, and not on the equality of decay constants which we have assumed for convenience.

  • The leading shift in the largest scattering eigenvalue is always positive and proportional to , and hence the shift in goes like , showing that locality yields a local maximum of independent of the sign of .

For larger values of beyond perturbation theory, additional scattering channels become relevant and we do not have a closed form solution for . While we cannot rule out the possibility that the global maximum of might occur with strong nonlocal terms, our numerical studies suggest that is indeed the global maximum of .

5.1 The mass matrix with nonlocality

The effect of the nonlocal term (21) on the spin-1 mass matrix is straightforward. Any individual hopping term contains new covariant derivatives and . After setting to its vev , the contribution of to the mass matrix is

(40)

where the upper-left corners of the blocks are inserted at the positions and . Unless is equal to 0 or modulo , the perturbation does not affect the diagonal elements, and hence is unchanged. Thus, the scale is identical to the local case, as long as :

(41)

As desired from Section 3, is insensitive to the nonlocal perturbation, allowing to depend on nonlocality only through . The case (mod ) can be considered as a discretization error arising from the fact that an -site moose for finite has a minimum nonlocal length scale , where is the lattice spacing.

5.2 Perturbation theory for scattering matrix eigenvalues

In general, numerical methods are needed to compute the exact eigenvalues of the scattering matrix as a function of the nonlocal coefficient . Here, we will resort to perturbation theory in , considering the effect of small nonlocal terms. By “perturbation theory,” we mean finding the largest eigenvalue of the scattering matrix, using techniques familiar from nonrelativistic quantum mechanics.181818Of course, throughout this paper we are using ordinary perturbation theory appropriate to quantum field theory to compute scattering amplitudes from tree-level Feynman diagrams. Taking the finite-dimensional matrix as our “Hamiltonian,” we can calculate the scattering matrix eigenvalues order-by-order in in terms of the unperturbed scattering eigenstates from the local Lagrangian.

As we will see below, the nonlocal terms change the kinetic matrix for the fields, requiring field redefinitions to compute scattering between canonically normalized states. Before the field redefinitions, only the nonlocal terms carry factors of , but expanding the field redefinition matrix as a series in , both the local and nonlocal terms can contribute to tree-level scattering amplitudes at all orders in . To keep track of in the total amplitude, we have to apply perturbation theory to the somewhat unusual case of a matrix perturbed by contributions to all orders in . To second order in , the scattering matrix and its largest eigenvalue are

(42)
(43)

The fact that the nonlocal term still preserves the cyclic symmetry determines the structure of the eigensystem of the scattering matrix. Indeed, the eigenstates of the full scattering matrix must respect the cyclic symmetry, and in particular there is no mixing of states with different eigenvalues under the operator which implements translations from site to . Thus we can work in the subspace of cyclically-invariant states with eigenvalue zero under this operator. An orthonormal basis of two-particle singlet states spanning this subspace is

(44)

For instance, in this notation we have

(45)

where are the gauge singlet states introduced in Eq. (37), now with a link index. In the local theory, all of the singlets are degenerate with eigenvalue

(46)

so this is also the eigenvalue of . By the arguments above, the eigenstate of the nonlocal theory with the largest eigenvalue must contain , plus order contributions from :

(47)

With the correct basis in hand, the lowest-order contribution to at first order in is

(48)

However, as we show in Section 5.4, this contribution vanishes identically for , due to a trace-related cancellation peculiar to the gauge structure of the singlet state and not because of any symmetry argument one can make at the Lagrangian level. Indeed, first-order shifts are generic for the other eigenvalues belonging to the and components of , and as we will see the vanishing of the shift for the singlet requires a detailed look at the structure of the relevant Feynman rules. The fact that the singlet state is associated with the largest eigenvalue is a necessary condition for locality to maximize ; otherwise, could be larger or smaller than depending on the sign of .

Because of the first-order cancellation, we must go to second order in . The first-order eigenvector coefficients for Eq. (47) are

(49)

which is the usual formula from ordinary perturbation theory. Note that the usual energy denominator is replaced by just because the states for have zero eigenvalues in the unperturbed matrix. The second-order eigenvalue shift is

(50)

which is the usual formula plus an extra term , which in this context contains contributions from both the local and nonlocal terms. As we will see below, the contribution from the eigenstates is essential, giving the leading behavior in and illustrating once again the necessity of doing a coupled-channel analysis rather than focusing on single scattering channels.

5.3 The scattering matrix with nonlocality

The nonlocal term (21) has several interesting effects on the scattering matrix and, by extension, on the scale of unitarity violation :

  • breaks the reflection symmetry , so 3-point terms are present, unlike in the local Lagrangian.

  • Despite only transforming under the gauge symmetries corresponding to sites and , the operator contains all the link fields between sites and , and hence several new intermediate states in 3-point diagrams are available to contribute to the gauge singlet scattering channel. These will cause the matrix elements to scale like , though will turn out to give the dominant contribution to .

  • As mentioned above, since contains terms, it changes the kinetic matrix of the fields. One must perform a field redefinition in order to recover canonical kinetic terms corresponding to the physical fields which participate in scattering.

Before the necessary field redefinitions, the operators present in the Lagrangian are as follows, writing for for ease of notation:

  • Local 4-point terms, independent of :

(35)
  • Nonlocal 3-point terms, proportional to :

(51)
  • Nonlocal 4-point terms, proportional to :191919We have abbreviated the third line because it does not contribute to any relevant amplitudes in our calculation. The omitted terms simply involve various permutations of the group and site indices.

(52)

Lorentz indices are suppressed everywhere since the Lorentz structure of all terms is identical, summation over the group indices is assumed, and the group theory factors for are ()

(53)
Figure 4: Diagrams contributing to the singlet scattering matrix at order , . The first term is a local 4-point vertex with a field redefinition insertion on one of the legs, and the second term is a nonlocal 4-point vertex. These diagrams vanish for , such that there is no order contribution to singlet scattering. For large , these diagrams yield the dominant contribution to singlet scattering at order through terms in Eq. (50) proportional to .

To take into account the field redefinition, we show in Appendix B.1 that to order , the wavefunction factor is

(54)

Note in particular that the diagonal elements have no order- components. After the field redefinitions, we have the following contributions to the gauge singlet piece of the scattering matrix:

  • Order : local 4-point diagrams only, with no field redefinitions (Figure 3). These contribute only to .

  • Order : local 4-point diagrams with order- field redefinitions and nonlocal 4-point diagrams with no field redefinitions (Figure 4). Since vanishes, these only contribute to , which dominates .

  • Order : local 4-point diagrams with order- field redefinitions and nonlocal 4-point diagrams with order- field redefinitions (Figure 5), diagrams with two nonlocal 3-point vertices and no field redefinitions (Figure 6). These contribute to , which give a subleading (in ) correction to . Note that the first appearance of 3-point terms comes at order .

Figure 5: 4-point vertices contributing to the singlet scattering matrix at order , . The first term is a local interaction with two order- field redefinitions, the second term is a local interaction with one order- redefinition, and the third term is a nonlocal interaction with one order- redefinition.
Figure 6: Contribution of 3-point (nonlocal) terms to the singlet scattering matrix at order , .

5.4 Leading effects on the normalized cutoff

We now have all of the ingredients to calculate to order . First, we can show that the first-order shift vanishes for the singlet state as long as . As mentioned above, we must consider the local terms with field redefinitions and the nonlocal terms separately. The local terms (35) are all even in , and so after field redefinitions according to (54), the order- pieces are odd in , and thus do not contribute to the matrix element between singlet states. The relevant nonlocal terms in Eq. (5.3) would be of the form for fixed group indices and , but these terms vanish identically due to cyclicity of the trace.202020This argument breaks down when , because contains the redundant kinetic terms discussed in Section 2.3, which add extra terms to the second line of Eq. (5.3) and destroy the trace-related cancellation. This gives a contribution to the scattering matrix which goes like , where is the “winding number” around the fifth dimension. The structure of the field redefinition matrix also changes for , giving a further local contribution. This can be seen explicitly by setting the appropriate indices equal in the second line of Eq. (