Lattice gauge theory beyond the Standard Model
Abstract and Keywords
This chapter gives an overview of methods and recent studies in the application of lattice gauge theory to physics beyond the standard model. It focuses on theories in which electroweak symmetry is broken spontaneously by new strong interactions. An introduction is given to some common features of such theories, described in terms of the chiral Lagrangian. Apparent difficulties in reconciling QCD-based models with precision experimental results provide the motivation for study of more general Yang-Mills theories. They are examined using the running coupling, which exhibits a perturbative infrared fixed point when enough light fermions are included, defining a "conformal window" in which confinement is lost and chiral symmetry is restored. Some analytic approaches to studying the conformal window are discussed, but the lack of an adequate method in the presence of a strongly-coupled fixed point leads to the consideration of non-perturbative lattice methods. Direct determination of the running coupling and its evolution is discussed, as well as the use of thermal phase transitions to distinguish confining from infrared-conformal theories. Finally, the direct simulation of the spectrum and other properties for theories outside the conformal window is described.
Lattice gauge theory has been very successful in deepening our understanding of the strong nuclear interactions. During the past two years, stimulated to some extent by the start-up of the Large Hadron Collider, interest is growing in applying lattice methods to new, strongly interacting theories that could play a role in extending the standard model. This chapter focuses on the use of lattice methods to study strongly coupled gauge theories with application to models of dynamical electroweak symmetry breaking.
Other applications of lattice methods to beyond-Standard-Model (BSM) physics are also being pursued, for example to supersymmetric theories. Since supersymmetry is inextricably connected to the Poincare symmetry group, which is broken to a discrete subgroup on the lattice, the construction of a supersymmetric theory on the lattice is quite challenging. However, some progress has been made; recently, Catterall and collaborators (Catterall et al, 2009) have demonstrated a formulation that in d space-time dimensions preserves 1 out of 2d supersymmetries exactly on the lattice, analogous to the preservation of a subset of chiral symmetry in the staggered fermion formulation. In addition to supersymmetric extensions of the Standard Model, supersymmetric theories are often studied in the context of dualities. In particular, the AdS/CFT duality (Maldacena, 1998; Witten, 1998) provides a link between gauge theory and quantum gravity, which opens up the possibility of exploring gravity on the lattice.
Although not a direct study of physics beyond the Standard Model, precision flavor physics calculations on the lattice have become more commonplace. The dominant systematic error for many such quantities comes from QCD contributions, and it is possible that a lattice computation could reduce the error enough to reveal disagreements with Standard Model predictions, leading to important constraints on new physics. For a comprehensive review, see the chapter of Laurent Lellouch.
12.1.1 Electroweak symmetry in the Standard Model
The Standard Model of particle physics describes all of the known fundamental forces (excluding gravity) as arising from the gauge symmetry group
Since the carriers of the weak force are massive, the symmetry group 𝔖must be broken into a subgroup at low energies. This is accomplished by the yet-unknown Higgs sector of the theory. In the simplest version, a complex-doublet scalar field (the Higgs field) breaks the gauge symmetry by acquiring a non-zero vacuum expectation value. The breaking pattern is
As a fundamental scalar field with a quartic self-interaction, the mass of the Higgs boson is subject to additive renormalization; if the loop momenta are allowed to run up
(p.701) to a very high scale, extremely delicate fine tuning of the bare Higgs mass is required to keep the renormalized mass near the electroweak scale. Also, the physical Higgs boson has yet to be detected experimentally.
The Standard Model also includes QCD, based on the gauge group SU(3)C, which gives rise to the strong nuclear force. In fact, the QCD sector exhibits another well-known example of spontaneous symmetry breaking; the chiral symmetry SU(2)L × SU(2)R is broken into a diagonal subgroup by a non-zero vacuum expectation value for the chiral condensate, . As the left-handed quarks are charged under the electroweak gauge group, this sector also makes a (small) contribution to electroweak symmetry breaking. However, the bulk of the breaking must come from another source; this is the unknown Higgs sector.
12.1.2 The electroweak chiral Lagrangian
Whatever the Higgs sector is, it must include the three Nambu-Goldstone boson (NGB) fields that provide the longitudinal components of the W and Z gauge bosons. If the new physics in addition to the NGB fields is heavy enough to be integrated out, it leaves an effective low-energy theory, consisting of the transverse gauge bosons, the NGB bosons, and the quarks and leptons. The gauge-boson and NGB sector of this theory is described by a non-linear chiral Lagrangian, whose lowest-dimension operators are (Appelquist and Wu, 1993)
D μ is the gauge-covariant derivative,
where and B μ are the SU(2)L and U(1)Y gauge fields, respectively, and τ a are the usual Pauli matrices. The kinetic terms for the gauge bosons are defined in the standard way, with
In addition to the SU(2)L and U(1)Y gauge symmetries, the Lagrangian Eq. 12.3 obeys a global SU(2)R ”custodial” symmetry.
The Lagrangian is non-linear, containing an infinite series of interaction terms between the π a fields; we will work here to first order in π a. Expanding Eq. 12.5 and multiplying by the conjugate,
At this point it is manifestly obvious that if we make the standard field redefinitions
(where Z μ and A μ are rescaled so that their kinetic terms will have the standardnormalization), then the πa field is completely removed from the Lagrangian at thisorder, and the remaining terms from the interaction are simply mass terms for theshifted gauge fields. The fourth gauge degree of freedom Aμ, corresponding to thephoton, remains massless as it should. The Lagrangian (again, to lowest order in πa)is now given by
This gives us an immediate prediction for the W/Z mass ratio :
in agreement with experimental observation.
(p.703) This success is a consequence of the extra, “custodial” SU(2)R symmetry built in to the scalar sector of the EW chiral Lagrangian terms so far. The minimal electroweak model with a single elementary Higgs doublet field possesses the same custodial symmetry and therefore yields the same prediction for . QCD as described by chiral perturbation theory also gives the same prediction, due to the same custodial symmetry, but the Lagrangian above describes a distinct set of fields from the physical pions given by QCD, and at a very different energy scale ΛQCD ≪ few. Going beyond leading order in the low-energy expansion, there are several additional operators that can be added to the EW chiral Lagrangian (12.3). First, one can define some building blocks that respect the SU(2)× U(1) electroweak symmetry:
and B μv, and W μv that were defined above. There are 11 new terms that can be added that are dimension four and CP-invariant (Appelquist and Wu, 1993):
In addition, there is another dimension-two operator that can be constructed:
This operator does not respect the custodial SU(2) symmetry, with the value of β 1 encapsulating any contributions from physics above the cutoff 4πf that break custodial symmetry.
An attractive UV completion of the EW chiral Lagrangian is technicolor. One adds to the Standard Model gauge group a new interaction SU(N TC), and N TF massless flavors of technifermions. The physics of chiral-symmetry breaking is fixed in terms of the fundamental scale ΛTC of the theory. In a technicolor model, there is no fundamental Higgs particle, with the degrees of freedom that are “eaten” by the W and Z generated as composite Nambu-Goldstone bosons of the spontaneous breaking of chiral symmetry. In the simplest version, the technicolor sector is essentially a copy of the QCD sector: N TC = 3, with two flavors of technifermions (U, D) possessing the same quantum numbers as the up and down quarks, except that color charge is (p.704) exchanged for technicolor charge. In a more general model with. NTD technidoublets, the scale of electroweak symmetry breaking fEW ~ 246 GeV/ . With a single technidoublet, by analogy to QCD (where ΛQCD ~ 300 MeV and f π ~ 93 MeV), the fundamental scale Λ TC (roughly the scale below which the technicolor gauge interaction becomes “strong”) should be somewhere around 1 TeV.
12.1.3 Fermion mass generation and extended technicolor
To replace the Higgs boson in the breaking of electroweak symmetry one also needs a replacement mechanism to give mass to the various standard model particles. One approach is to add a set of additional gauge interactions that couple the standard model and technicolor-sector fields to one another; this framework is known as extended technicolor (Eichten and Lane, 1980; Dimopoulos and Susskind, 1979). In the Higgs case, the masses are given by Yukawa couplings to the Higgs scalar field; here the mass terms must arise through some coupling to the technifermions.
One possible construction takes the extended technicolor gauge group to be SU(3 + N TC)ETC, containing the technicolor gauge group SU(N TC). All of the matter fields are charged under the ETC gauge group. To give rise to the generational structure observed in the Standard Model, ] scales Λ1,Λ2,Λ3:
The diagram through which a Standard Model quark in generation i becomes massive is depicted in Fig. (12.1). The technifermion condensate , while at momentum scales much smaller than Λi the ETC gauge boson propagator will contribute roughly . In full, the mass arising from this diagram is given by (Appelquist et al, 2004)
If the anomalous dimension γ(μ) of the condensate is small, then η ~ 1. The observed up-type quark masses can then be reproduced correctly by the following assignment of the ETC breaking scales, taking ΛTC = 400 GeV since. NTD = 4 in this model:
In models with additional technifermion content, ΛTC can be lower, reducing the above scales in turn. The scale Λ3 associated with generation of the top-quark mass can be as low as 1 TeV, and the lack of scale separation between TC and ETC renders a simple analysis in terms of effective four-fermion interactions invalid. (It should be further noted that the large splitting between the top and bottom quark masses can lead to a large contribution to Δρ (Appelquist et al., 1984; 1985), and that the large effective coupling to the top quark can increase the decay rate, allowing some models to be ruled out by existing experimental data (Chivukula et al, 1992). Models that attempt to address these and other issues with the top-quark sector do exist, but we will not discuss them here.)
In order to reproduce fully observed Standard Model physics, an extended technicolor model must include not just the particle masses, but also the mixing effects encapsulated in the CKM matrix. This can be accomplished by adding mixing terms between the ETC gauge bosons (Appelquist et al, 2004). However, these same interactions will also lead to the generation of flavor-changing neutral currents (FCNCs), which are tightly constrained by experiment. In particular, the contribution to the kaon long-short mass difference is given roughly by
which forces the ETC scale Λ2 ≳ 1300 TeV (Lane, 2000), which in turn forces a quark mass that is far too low.
A way out of this contradiction is through enhancement of the technifermion condensate, which appears in the masses but not in the FCNCs. Enhancement here would come in the form of a large contribution from η the RG running of the condensate between the TC and ETC scales. In particular, if a theory has the property that the anomalous dimension γ ~ 1 over the full range of scales to be integrated over, then
which can provide a large enough enhancement to give the correct masses while escaping FCNC bounds.
12.1.4 The S parameter
(p.706) There is one further problem that we must keep in mind, relating to precision electroweak measurements. In particular, we are concerned with the S, T, and U parameters (Peskin and Takeuchi, 1992), which describe the presence of physics beyond the standard model; by definition, the parameters are 0 in the Standard Model once the Higgs mass is fixed. Contributions to S are generally the greatest concern for technicolor theories; although the top sector in such models can lead to a large T parameter (equivalent to large Δρ) as discussed above, a positive T is much easier to reconcile with current experimental bounds than a positive S.
S is related to the difference between vector-vector and axial-axial vacuum polarization functions ∏(q2), evaluated in the limit q 2 → 0. For a general theory described by the electroweak chiral Lagrangian, S is related simply to the parameter α 1 of Eq. (12.16).
We can write down a dispersion relation for the contribution to S of a technicolor theory in terms of the imaginary parts of the vacuum polarizations, R(s) = — 12π Im ∏′(s):
In addition to R V and R A, this expression contains some additional terms that serve to regulate the integral in the infrared, and subtract off the Standard Model Higgs boson contribution to S at some chosen reference mass m h.
For a technicolor model that is a scaled-up version of QCD, one can use QCD experimental input to determine the functions R V(s) and R A(s) (Peskin and Takeuchi, 1992), yielding the estimate
at reference Higgs mass m h = 1 TeV. Current experimental bounds favor S 〈 0 at the same reference Higgs mass (Amsler et al, 2008), so there is considerable tension here. Furthermore, naive scaling estimates and counting pseudo-NGB loop contributions both indicate that S should only increase with N TF and. N TC, suggesting that more elaborate TC models might be in sharper conflict with experimental bounds on S.
However, it should be kept in mind that the estimate Eq. (12.25) relies on QCD phenomenology. In a more general strongly coupled theory, QCD-based estimates do not necessarily apply. In particular, S is sensitive to differences between the vector and axial spectrum of the technicolor theory. In a QCD-like theory, such differences are large, leading to a large contribution as above. If, for example, a given technicolor theory exhibits approximate parity doubling - degeneracy between the vector and axial sectors - then the contribution could be reduced (Appelquist and Sannino, 1999).
We next move on to consider the structure of more general Yang-Mills gauge theories. Our primary motivation is to search for theories that share with QCD the properties of asymptotic freedom and spontaneous chiral symmetry breaking, allowing them to form the basis for a technicolor model, and yet possess different enough
(p.707) dynamical properties that estimates based on QCD phenomenology are no longer valid constraints.
12.2 The conformal window and walking
12.2.1 Perturbative RG flow in Yang Mills theories
As a strongly coupled gauge theory QCD is far from unique. The number of colors, number of light fermion flavors, and the gauge-group representation of the fermion fields can all be varied over a wide range of choices while maintaining the crucial property of asymptotic freedom. However, the infrared physics of such theories can be strikingly different from QCD. Consider a Yang-Mills theory with local gauge symmetry group SU(N C), coupled to N f massless Dirac fermion flavors:
The fermions are taken to live in a representation R of the gauge group. The scale dependence of the renormalized coupling g = g(μ) is determined by the β-function, which we can expand perturbatively:
with α(μ) = g(μ)2 /4π. The universal values for the first two coefficients are
where T(Rv) and C 2(R) are the trace normalization and quadratic Casimir invariant of the representation R, respectively. The Casimir invariants for a few commonly used representations of SU(N) are shown in Table 12.1; a more exhaustive list of invariants and other group-theory factors is given in (Dietrich and Sannino, 2007). So long as N f /N c 〈 11/(4T(R)) so that β0 〉 0, the theory is asymptotically free. One may continue on to higher order in this expansion, at the cost of specifying a renormalization scheme; in the commonly used scheme, the next two coefficients are known (van Ritbergen et al, 1997).
If N f is taken sufficiently large that β1 〈 0, but not so large that β0 〈 0, then in addition to the trivial ultraviolet fixed point α = 0, the two-loop β-function admits a second fixed-point solution,
Table 12.1 Casimir invariants and dimensions of some common representations of SU(N): fundamental (F), two-index symmetric (S 2), two-index antisymmetric (A 2), and adjoint (G).
As N f is decreased, the value of the fixed-point coupling increases, quickly causing the two-loop perturbative description to break down. Even so, it is clear that the properties of confinement and chiral symmetry breaking must be recovered at sufficiently small N f as we approach QCD and the quenched (N f = 0) limit. Theories within the range
where marks the transition point between conformal and confining infrared behavior, are said to lie in the “conformal window”. Perturbation theory is almost certainly unreliable to describe physics in the vicinity of the infrared fixed point as N f approaches , so study of the transition region requires some non-perturbative approach, which in turn demands the choice of a renormalization scheme. The question of scheme dependence of any results should always be kept in mind when dealing with quantities such as the running coupling, although any physical, measurable predictions should certainly be independent of the scheme.
12.2.2 Infrared conformality and walking behavior
Theories that lie inside the conformal window, although interesting in their own right, are not generally useful in describing electroweak symmetry breaking due to the lack of (p.709) chiral-symmetry breaking (although it is possible to trigger chiral-symmetry breaking even in the conformal window by including explicit fermion-mass terms, as in (Luty, 2009; Evans et al., 2010).) Our focus will be on theories that lie outside the window but close to the transition, that may show interesting and novel dynamics while still breaking chiral-symmetry breaking spontaneously.
Suppose that within some scheme, there is a critical coupling α c, which when exceeded will trigger the spontaneous breaking of chiral symmetry. Now consider a theory with a beta function (in the same scheme) such that the coupling is approaching a somewhat supercritical fixed point α ⋆ 〉 α c. As the coupling gets near a ⋆, the magnitude of the β-function approaches zero, and the running slows down. However, when α c is exceeded, confinement and chiral-symmetry breaking set in. The fermions that were responsible for the existence of the fixed point develop masses and are screened out of the theory, causing the coupling to run as in the N f = 0 theory below the generated mass scale. The situation is depicted in Fig. (12.2).
This idea is known as “walking technicolor”. The “plateau” depicted in Fig. (12.2) results in a separation of scale between the UV physics (in the context of ETC models, ΛETC) where the coupling runs perturbatively and the IR scale (ΛTC) at which confinement sets in. This dynamical scale separation is exactly what is needed in order to address the FCNC problem in extended technicolor. The conflict there was between trying to simultaneously match the Standard Model particle masses and suppress FCNC-generating effects, both of which are tied to the same ETC scale Λi. With walking, the ultraviolet-sensitive condensate can pick up a large additional contribution from the scales between ΛTC and ΛETC, allowing recovery of Standard Model masses without violation of precision electroweak experimental bounds.
While walking technicolor offers a solution to the difficulties with technicolor models, the existence of such a theory is speculative. The onset of walking is a strong-
A quasi-perturbative approach sometimes cited in the old literature is the “ladder gap equation” or “rainbow gap equation” method, which requires the use of a truncation that is not justifiable at strong coupling. In addition, the standard use of Landau gauge in this calculation raises further questions of gauge dependence. We summarize the approach here on a cautionary note.
One begins with the standard Schwinger-Dyson equation for the fermion propagator. In order to make progress, the “rainbow approximation” is then applied, which replaces the full gluon propagator and gluon-fermion vertex with their tree-level values. At weak coupling, the Feynman diagrams that are dropped in this approach are suppressed by small α, so that it can be regarded as an approximation to the full equation. However, if the coupling is sufficiently strong then the procedure becomes an unjustified truncation.
Proceeding under the “rainbow approximation”, one may show that a non-zero solution for the fermion dynamical mass Σ(p 2) exists once the coupling strength exceeds a certain value, α C = π/(3C 2(R)). If the corresponding effective potential, again constructed using the same “rainbow” truncations, is evaluated at the extremal solution for Σ(p 2), it can be seen that this non-zero solution is energetically preferred to Σ(p 2) = 0, implying that chiral symmetry is spontaneously broken.
Because the estimated critical coupling is strong (α c C 2(R) ~ 1), the truncation used in order to determine it is unjustified. In spite of this, the further exercise of combining the estimated α c with some determination of the fixed-point coupling strength α ⋆(N f) is often performed in order to “estimate” . If the two-loop perturbative value of α ⋆ is used, then setting yields the values at N c = 2, and at N c = 3. There is little reason to expect these values to be robust against higher-order perturbative corrections.
12.2.3 The thermal inequality
We turn now to a more general, conjectured constraint on the structure of field theories and the value of known as the ACS thermal inequality (Appelquist et al, 1999). This inequality connects the UV and IR degrees of freedom directly. In this way the thermal inequality is similar in spirit to the t’Hooft anomaly-matching condition (‘t Hooft et al, 1980).
The basic proposal is simple: the number of relevant infrared degrees of freedom should be not larger than the number of ultraviolet degrees of freedom. In other words, the RG “blocking” transformation that takes us from UV to IR should never increase the number of degrees of freedom. Consider the thermodynamic degrees of freedom, which are encapsulated by the free energy of the theory:
In a free field theory with N s scalars, N v vector particles and N f Dirac fermions (all massless), the free energy is given by
(p.711) More generally, one can define a function f(T) that “counts” the relevant degrees of freedom:
For the case that the limits of f(T) as T → 0 and T → ∞ are both finite, the proposed constraint is
In the context of two-dimensional conformal field theories, Zamolodchikov’s c-theorem (Zamolodchikov, 1986) makes the analogous statement rigorous: the free energy as determined by the central charge c is always reduced under RG flow from an ultraviolet to an infrared fixed point.
As an example, consider the case of an SU(N C) gauge theory with N f fermion flavors in the fundamental representation, with N f low enough that the theory is in the chirally broken phase. In the ultraviolet, the relevant degrees of freedom are the gluons, of which there are , and the N f N c fermions. Assuming that the SU(N f) × SU(N f) chiral symmetry is broken spontaneously to SU(N f), the relevant degrees of freedom describing the theory in the infrared will be the massless Goldstone bosons. Thus,
For N f ≳ 4N c, the inequality would be violated, and the chiral symmetry must therefore be unbroken. Assuming this picture is correct, one finds the restriction , which, curiously, is nearly saturated by the gap-equation estimate in the fundamental representation noted above.
As a final observation, we note that at N c = 2, the reality or pseudoreality of the group representations can lead to very different patterns of chiral-symmetry breaking from the standard case at N c 〉 3 (Peskin, 1980). In particular, for N f fermions in the pseudoreal fundamental representation, the global chiral symmetry group is now SU(2N f), and the breaking pattern preserving the largest symmetry group is SU(2N f) → Sp(2N f). The dimensions of the groups SU(N) and Sp(N) are N 2− 1 and N(N + 1)/ 2, respectively, so we find that
(p.712) Then, assuming the chiral symmetry to be broken,
This estimate is in sharp contrast with the gap-equation calculation, which puts just below 8. A lattice study in the range 5 〈 N f 〈 8 for the N c = 2 case could thus be quite interesting.
12.3 Lattice studies of the conformal transition
Lattice field theory provides an ideal way to study the conformal transition, and more generally the properties of Yang-Mills theories at various N f. Lattice simulations are truly non-perturbative, although the continuum limit must be taken carefully to recover information about continuum physics (the ability to take this limit being made possible by the asymptotic freedom of the theories in which we are interested.) Lattice simulations allow broad investigation; a large number of different observables can be computed simultaneously on a single set of gauge configurations.
Although the dynamics of the theories we are interested in are different from QCD, at the level of the Lagrangian the changes needed to investigate the conformal window are relatively minor. The addition of extra degenerate fermion flavors simply requires the insertion of multiple copies of the chosen fermion action into the simulation code. Modification of the color gauge group or fermion representation is less trivial, but the required changes should represent a small fraction of the overall size of the simulation code, with many of the algorithms and constructions used in QCD remaining applicable.
The relatively low barrier of entry to simulations of these theories can make the undertaking of such studies quite attractive for current practitioners of lattice QCD. However, as in the continuum, great care must be taken to avoid relying too heavily on intuition and experience based on working with QCD; new techniques will be needed to deal with the presence of widely separated scales. Furthermore, the expense of the simulations themselves often requires state-of-the-art computational resources, with the addition of many extra degrees of freedom increasing the cost greatly.
The following subsections will detail three different approaches to lattice study of the conformal transition that have been widely used; although most of the simulations carried out to date fall into one of these three categories, the list is not exhaustive, and new approaches are under development. In Section 12.3.2 below, we describe simulation methods that focus on the extraction of a running coupling from the (p.713) lattice, with the goal of directly locating an infrared fixed point in the β-function. Section 12.3.3 discusses an alternative approach, that involves searching for a thermal phase transition that appears as a physical transition in confining theories but as a lattice artifact within the conformal window. Finally, in Section 12.3.4, we detail a different approach in which the spectrum and chiral properties of theories at various N f are computed, with the goal of determining the N f dependence of various observables as the conformal transition is approached from below.
12.3.2 Running coupling methods
The concept of the conformal window is usually first introduced by way of the running coupling and the β-function (as it was in this chapter). As such, a natural application of lattice simulation to investigate the transition is by the direct computation of a running coupling constant and determination of the β-function non-perturbatively
The first step in extracting the β-function is to select a non-perturbative definition of a running coupling that can be measured on the lattice. There are a number of such choices possible, including the standard extraction of the static potential from Wilson loops, the Schrödinger functional (Lüscher et al., 1992; Sint, 1994; Bode et al., 2000; Appelquist et al, 2008; 2009), the twisted Polyakov loop scheme (Bilgici et al., 2009b), and constructions using ratios of Wilson loops (Bilgici et al., 2009a; Fodor et al, 2009). The Monte Carlo Renormalization Group scheme provides another approach for determining RG evolution (Hasenfratz, 2009).
Regardless of the chosen definition of the running coupling, the end goal is to map out its evolution over a large range of distance scales R. If one works at a fixed lattice spacing a, then the range of available R at which the coupling can be measured is quite small, with the computational expense quickly becoming prohibitive even in QCD. The problem is exacerbated in a theory with large N f, where the size of the (β- function is small; to go from weak to strong coupling, a change in scale of many orders of magnitude is often required. To achieve the goal, then, some way must be found to match together lattice measurements of the running coupling taken at different lattice spacings and combine them into an overall measurement of continuum evolution. A technique known as step scaling (Lüscher et al, 1991; Caracciolo et al, 1995) provides a systematic approach.
Step scaling is a recursive procedure that describes the evolution of the coupling constant g(R) as the scale changes from R → sR, where s is a numerical scaling factor known as the step size. The relation between the coupling at these two scales in the continuum is defined through the step-scaling function,
The step-scaling function σ is simply a discrete version of the usual continuum β- function, both of which describe the evolution of the coupling as a function of the coupling strength. In a lattice calculation, the step-scaling function that we extract also contains lattice artifacts in the form of a/R corrections. The lattice version of the step-scaling function is denoted by Σ, and it is related to the continuum σ by extrapolation of the lattice spacing a to zero:
(We are neglecting finite-volume effects here, which must also be dealt with in a lattice simulation for most definitions of running coupling on a case-by-case basis.)
Generically, the implementation of step scaling begins with the choice of an initial value u = g 2(R). Several ensembles at different a/R are then generated, tuning the lattice bare coupling so that on each ensemble one measures the chosen value of the renormalized coupling, g 2(R) = u. Then, one generates a second ensemble at each β, but measures the coupling at a longer scale R → sR. The value of the coupling measured on the second lattice is exactly Σ(s, u, a/R). Since Σ(s, u, a/R) has now be computed for multiple values of a/R, one can carry out the extrapolation a/ R → 0 and recover the continuum value σ(s, u). Taking σ(s, u) to be the new starting value, the procedure is repeated, mapping g 2(R) → g 2(sR)…→ g 2(s n R) until the coupling is sampled over a large range of R values.
There is a natural caveat on the step-scaling procedure, especially in the context of studying theories with infrared fixed points. The procedure as outlined above depends crucially on the ability to take the limit a/R → 0. If g 2(R) is held fixed while taking the limit a/R → 0, it is important that the bare coupling , which depends on the short-distance behavior of the theory, does not become strong enough to trigger a bulk phase transition. This is satisfied automatically if the short-distance behavior is determined by asymptotic freedom, in which case vanishes as 1/ log(R/a). However, in a theory with an infrared fixed point, if g 2(R) is measured above the fixed point at , then will increase as a → 0, with no evidence that it remains bounded and therefore that a continuum limit exists. Even so, it is possible to extrapolate to small enough values of a/R to render lattice artifact corrections negligible, providing that is kept small enough to avoid triggering a bulk transition into a strong-coupling phase.
Carrying out step scaling through the iterative procedure described can be quite expensive in both computational power and real time, especially in theories where many steps are required to see significant evolution in the coupling. Each tuning of β to match the initial value u may require several attempts. Furthermore, each step depends on the value of g 2(R) taken from the previous step, removing the ability to parallelize the problem. A more efficient method is to measure g 2 (R) for a wide range of values in β and R/a, and then to generate an interpolating function. Step scaling may then be done analytically using the interpolated values. Such an interpolating function should reproduce the perturbative relation at weak coupling, but otherwise its form is not strongly constrained. One possible choice is an expansion of the inverse coupling 1/g 2(β, R/a) as a set of polynomial series in the bare coupling at each R/a:
The order n of the polynomial is arbitrary, and can be varied as a function of R/a to achieve the optimal fit to the available data.
(p.715) The remainder of this section will focus in detail on the use of one formulation of a running coupling, the Schrödinger functional (SF), which is particularly robust with respect to finite-volume effects. The SF running coupling is defined through the response of a system to variation in strength of a background chromoelectric field. It is a finite-volume method, with the coupling strength defined at the spatial box size L, so that we identify R = L and can discard finite-volume corrections. Formally, the Schrödinger functional describes the quantum-mechanical evolution of some system from a given state at time t = 0 to another given state at time t = T, in a spatial box of size L with periodic boundary conditions (Lüscher et al., 1992; Sint, 1994; Bode et al., 2001). The temporal extent T is fixed proportional to L, so that the Euclidean box size depends only on a single parameter. The initial and final states are described as Dirichlet boundary conditions that are imposed at t = 0 and t = T, and for measurement of the coupling constant are chosen such that the minimum-action configuration is a constant chromoelectric background field of strength O(1/L). This can be implemented both in the continuum (Lüscher et al, 1992) and on the lattice (Bode et al., 2000).
We can represent the Schrödinger functional as the path integral
where A is the gauge field and are the fermion fields. W and W′ are the boundary values of the gauge fields, and , are the boundary values of the ermion fields at t = 0 and t = T, respectively. The fermionic boundary values are subject only to multiplicative renormalization (Sommer, 2006), and as such are generally taken to be zero in order to simplify the calculation.
As noted above, the gauge boundary fields W, W′ are chosen to given a constant chromoelectric field in the bulk, whose strength is of order 1/L and controlled by a dimensionless parameter η (Lüscher et al., 1994). The Schrödinger functional (SF) running coupling is then defined by the response of the action to variation of η:
where (with the standard choice of gauge boundary fields for SU(3)), the normalization factor k is
The presence of k ensures that ḡ 2(L, T) is equal to the bare coupling at tree level in perturbation theory. In general, ḡ (L,T) can be thought of as the response of the system to small variations in the background chromoelectric field.
For most fermion discretizations, at this point we can take T = L in order to define the running coupling as a function of a single scale, ḡ 2 (L). However, if staggered fermions are used (as they are often in order to offset the cost of simulating additional (p.716) fermion flavors), then an additional complication arises that can be envisioned geometrically. The staggered approach to fermion discretization can be formulated as splitting the 16 spinor degrees of freedom available up over a 24 hypercubic sublattice. Clearly such a framework requires an even number of lattice sites in all directions. If all boundaries are periodic or antiperiodic, then setting T = L can be done so long as L is even. However, with Dirichlet boundaries in the time direction, the site t = T is no longer identified with t = 0, so that a total of T/a + 1 lattice sites must exist. In order to accommodate staggered fermions, T/a must be odd.
Thus, when using staggered fermions, the closest we can come to our desired choice of T is T = L ± a. In the continuum limit, the desired relation T = L is recovered. However, at finite lattice spacing O(a) lattice artifacts are introduced into observables. This is especially undesirable, since staggered fermion simulations contain bulk artifacts only at O(a 2) and above. Fortunately, there is a solution: simulating at both choices T = L ± a and averaging over the results has been shown to eliminate the induced O(a) bulk artifact in the running coupling (Heller, 1997). We define ḡ 2(L) through the average:
To be precise, we can only write ḡ -2 (L) unambiguously in the continuum; on a lattice with spacing a and bare coupling g 0 given by , the coupling that we measure can be written as ḡ 2 (β, L/a).
As a case study of many of the above concepts and formulas in use, we now show some simulation data and results from a Schrödinger functional running coupling study of the SU(3) fundamental, N f = 8 and 12 theories, using staggered fermions (Appelquist et al., 2009). We begin with the N f = 8 theory, for which data was gathered in the range 4.55 ≤ β ≤ 192 on lattice volumes given by L/a = 6, 8, 12, 16. The lower limit on β was determined to keep the lattice coupling too weak to trigger a bulk phase transition. A selection of the data, together with interpolating function fits of the form Eq. (12.45), are shown in Fig. 12.3. Note that at any fixed value of β, the coupling strength ḡ 2 (L) increases with L/a, showing no evidence of the “backwards” running that we would expect to observe in a theory with an infrared fixed point.
Although the study of Fig. 12.3 is indicative that the N f = 8 theory lies outside the conformal window, it is possible for results at fixed β to be misleading; we must take the continuum limit in order to recover information about the continuum theory. We apply the step-scaling procedure detailed above in order to extract the continuum step-scaling function σ(2, u), by extrapolation of a/L → 0 with each doubling of the scale L. A selection of extracted values for the lattice step-scaling function Σ(2, u, a/L) are shown in Fig. 12.4 for the two available steps 6 → 12 and 8 → 16. Our results for σ(2, u) will depend on the choice of continuum extrapolation, i.e. the model function for a/L dependence of Σ(2,u, a/L). As this is a staggered fermion study, the leading bulk lattice artifacts are expected to be of O(a 2), but there are additional boundary artifacts of O(a) that are only partially cancelled off by subtraction of their perturbative values. However, in this case the a/L dependence is weak, with the associated systematic error (p.717)
The resulting continuum running of ḡ 2(L) for N f = 8 is shown in Fig. 12.5. L 0 is an arbitrary length scale here defined by the condition ḡ(L) = 1.6, anchoring the step-scaling curve at a relatively weak value. The points shown correspond to repeated doubling of the scale L relative to L 0. Derivation of statistical errors uses a bootstrap technique, and is described in detail in the reference (Appelquist et al, 2009). Perturbative running at two and three loops is also shown for comparison up through ḡ 2(L) ≈ 10, beyond which the accuracy of perturbation theory is expected to degrade. The coupling measured in this simulation follows the perturbative curve closely up through ḡ 2(L) ≈ 4, and then begins to increase more rapidly, reaching values that exceed typical estimates of the coupling strength needed to induce spontaneous chiral-symmetry breaking (for example, the gap equation estimate of Section 12.2 above.) As there is no evidence for an infrared fixed point, or even for an inflection point in the running of ḡ 2(L), this study supports the assertion that the N f = 8 theory lies outside the conformal window.
We now move on to consider the N f = 12 theory. We do not show the data and interpolating fits here, but they are available in the reference. The lattice step-scaling function Σ(2,u, a/L) for selected values of u is shown in Fig. 12.6. Three steps are now available: 6 → 12, 8 → 16, and 10 → 20. As above, we choose a constant continuum extrapolation, i.e. weighted average of the three points. Again, we note the sharp contrast with the analogous plot Fig. 12.4 for N f = 8; we observe that Σ(2,u, a/L) approaches the starting coupling u as u increases.
The infrared fixed point here also governs the infrared behavior of the theory for values of ḡ 2(L) that lie above the fixed point. As discussed previously, we cannot naively apply the step-scaling procedure in this region, since we can no longer approach the ultraviolet fixed point at zero coupling strength in order to take the continuum limit. Instead, we can restrict our attention to finite but small values of a/L, small enough to keep lattice artifacts small and yet large enough so that does not trigger a bulk phase transition for ḡ 2(L) near (but above) the fixed point. With these caveats in mind, the step-scaling procedure can then be applied and leads to the running from above the fixed point shown in Fig. 12.7. The observation of this “backwards-running” region is crucial to distinguishing theories with true infrared fixed points from walking theories, in which the β-function may become vanishingly small before turning over and confining.
Having shown evidence for the existence of an infrared fixed point in the N f = 12 theory and demonstrated its absence up to strong coupling at N f = 8, we have constrained the edge of the conformal window for the case of N c = 3 with fermions in
12.3.3 Thermal transition methods
For a pure Yang-Mills gauge theory on the lattice and in the strong-coupling limit, it can always be shown that the theory confines, i.e. that Wilson loops obey area-law scaling (Osterwalder and Seiler, 1978). With the addition of unimproved Wilson or staggered fermions, analysis at strong coupling and large N c reveals behavior consistent with confinement and spontaneous breaking of chiral symmetry (Blairon et al., 1981; Kluberg-Stern et al., 1981; Kawamoto and Smit, 1981). If a theory with enough fermion content to place it inside the conformal window confines in a simulation performed at strong coupling, to be consistent with the lack of confinement in the continuum or weak-coupling limit the lattice theory must undergo a phase transition as a function of the bare coupling. This phase transition is purely a lattice artifact; in a continuum, asymptotically free Yang-Mills theory with an IR fixed point, the strong-coupling limit can never be reached, but on the lattice the bare coupling may be freely adjusted to any value.
(p.721) The deconfining phase transition is characterized by a first-order jump in the values of the Wilson line 〈|W 3〉 and chiral condensate . Within the conformal window, it is a “bulk” transition driven entirely by physics at the lattice spacing, and therefore occurs at the same bare coupling as the lattice volume is varied. A similar transition can be observed in theories outside the conformal window, with the same behavior of the observables noted above as the bare coupling is varied. However, in a QCD-like theory this deconfining and chiral-symmetry-restoring transition is driven by infrared physics; the presence of a deconfined phase in a QCD-like theory is a finite-volume or finite-temperature effect. As such, the bare coupling associated with the transition will vary as a function of the box size.
In most lattice simulations investigating the conformal window, such as the running coupling studies detailed in Section 12.3.2 above, the phase transition presents an obstacle to be avoided; the strongly coupled physics on the confining side of the transition cannot tell us anything about the continuum theories in which we are interested. However, the difference in bare-coupling dependence between the bulk phase transition within the conformal window and the thermal transition outside it provides another possible approach to distinguishing such theories from one another through lattice simulation. If the transition can be accurately located as a function of the bare coupling g 0 on a range of lattice volumes, then the presence of variation in the transition coupling with box size will indicate the existence of a continuum thermal phase transition, placing the theory outside of the conformal window.
The choice of fermion action is important in attempting such a phase-transition study. The strong-coupling results referred to above have been demonstrated only for the simplest, unimproved fermions, and are not guaranteed to hold if an improved action is used. Furthermore, even for unimproved Wilson fermions it has been observed that with N f ≥ 7 in the SU(2) case, the phase diagram becomes more complicated and extrapolation to zero-current quark mass becomes impossible (Iwasaki et al, 2004). In the examples to follow, we will restrict our attention to staggered fermions, which show no such signs of N f dependence in their observed strong-coupling behavior (Damgaard et al., 1997). However, with the use of staggered fermions away from the continuum limit, the possibility of taste-breaking effects reducing the effective number of fermion flavors should always be kept in mind.
For a finite-temperature simulation the temperature T = 1/(N t a(g 0)), where N t is the number of sites in the time direction and a(g 0) is the lattice spacing. In a theory outside the conformal window, there exists a physical deconfining transition temperature T c that is related to the critical bare coupling g 0,c by T c = (N t a(g 0,c))-1. Since T c is fixed, g 0,c should vanish as N t → ∞, with perturbative scaling of a describing the evolution with N t so long as g 0,c is sufficiently weak. The standard approach is to relate the lattice spacing to the perturbative Λ parameter; applying the two-loop perturbative β-function, one finds
with . If the temporal extent is now changed from , the expected scaling of the critical bare coupling is determined by the relation
As an example of this technique, Deuzeman and collaborators have studied the theory with N f= 8,N c = 3 and the fermions in the fundamental representation, searching for a thermal phase transition (Deuzeman et al, 2008). Their simulations were carried out using staggered fermions and an improved gauge action, and focused on finite-temperature simulations at N t = 6 and N t = 12, with multiple spatial volumes at N t = 6 used in order to study finite-volume effects. As N t is increased from 6 to 12, they observe a clear shift in the transition towards weaker bare coupling.
The precise critical coupling values determined are β c = 4.1125 ± 0.0125 at N t = 6, and β c = 4.34 ± 0.04 for N t = 12, where . Applying the scaling formula Eq. (12.51) to the value β c = 4.34 ± 0.04 at N t = 12 yields the prediction β c = 4.04 ± 0.04 for N t = 6, which is reasonably close to the observed value of 4.1125 ± 0.0125. Observing roughly the expected perturbative scaling further supports the case that the transition seen corresponds to a thermal deconfining phase transition of the continuum N f = 8 theory.
An early application of this approach to theories potentially within the conformal window was carried out by (Damgaard et al., 1997), using staggered fermions and the conventional Wilson gauge action. Their focus was on the N c = 3 theory with N f = 16 degenerate flavors in the fundamental representation, a case that is unambiguously within the conformal window due to the presence of a very small, perturbative IR fixed point. One therefore expects to find a confining phase transition at a fixed value of the bare coupling that does not change as the extent of the lattice is varied.
This is exactly what Damgaard et al. observe, in the chiral condensate (as shown in Fig. 12.8) and other observables; the phase transition that they identify occurs at a fixed critical coupling β c ~ 4.1 for all lattice volumes considered. A similar study carried out for the N f = 12 theory again shows a transition that does not vary with box size (Deuzeman et al., 2009), supporting the placement of N f = 12 within the conformal window.
Locating a phase transition and establishing that it does not appear to change as the lattice volume is varied is not definitive evidence that a theory is inside the conformal window. The ability to resolve movement of the transition coupling is limited by computational power, which restricts both the maximum lattice volume L/a and the resolution in β. Order parameters corresponding to short-distance physics, e.g. the plaquette, will show a first-order jump for a bulk transition but not for a physical one, in principle allowing us to distinguish the two cases. However, a given theory measured with a given lattice action can easily have both transitions occur very close to one another, and the two could be easily confused. To definitively place a theory with an apparent bulk phase transition inside the conformal window, some other method must be used. For the N f = 16 theory, Heller went on to measure the Schrödinger functional running coupling, demonstrating the presence of a phase in which the sign of the β- function is reversed, so that the coupling appears to run “backwards” (Heller, (p.723)
12.3.4 Study of spectral and chiral properties
If a given theory is to be considered for a candidate technicolor model, compatibility with experimental bounds generally requires certain effects to be generated by the dynamics; namely enhancement of the chiral condensate to avoid flavor-changing neutral current problems, and reduction of the S-parameter contribution compared to naive estimates. Aside from such practical concerns, determining the variation with N f of observables related to chiral symmetry and confinement are important in finding the order of the conformal transition, and novel phenomena such as walking may be revealed as the transition value is approached.
In this subsection, we discuss such studies that focus on the evolution with N f of various observables on the broken side of the conformal transition. There are several lattice groups attempting to measure correlation functions corresponding to standard QCD bound states and other quantities for theories that are ambiguously or definitely within the conformal window (Fodor et al., 2009; Jin and Mawhinney, 2009; Del Debbio et al, 2009); the interpretation of the lattice results here is much less clear, and we (p.724) will omit such theories from our discussion here, focusing on the case where chiral symmetry is definitely broken.
In order to meaningfully compare any quantity between theories with different N f, we must first identify a physical scale to hold fixed. In the context of technicolor theories, a natural choice is the pion decay constant F, which is identified with the scale of electroweak physics through the chiral Lagrangian as described in Section 12.1.2. However, the extraction of F from lattice simulations can be challenging. The rho meson mass m p is much more easily determined, due to the lack of a chiral logarithm at next-to-leading order (NLO) in a χPT-derived fit (Leinweber et al., 2001). However, in the end we are more interested in the evolution of physics with respect to F than m p. In QCD the two scales are connected, m p ~ 2πF, but it is not known a priori whether this connection will persist near the edge of the conformal window. The Sommer scale r 0 (Sommer, 1994), associated with the scale of confinement, is another possible choice with similar advantages and drawbacks to m p. In the present discussion we will assume that these scales do not evolve with respect to each other, so that holding any one constant with N f is sufficient. This assumption is supported by available data, but the choice of scale is an open question going forward.
As lattice simulations are necessarily performed at finite mass, while we are interested in the behavior of theories in the chiral limit, extrapolation of results m → 0 is crucial. Chiral perturbation theory provides a consistent way to carry out this extrapolation. The familiar expressions for the Goldstone boson mass M m, decay constant F m and chiral condensate (with the subscript denoting evaluation at finite quark mass m) are easily generalized to theories with arbitrary N f ≥ 2 by inclusion of known counting factors. The next-to-leading order (NLO) expressions for a theory with 3 colors are (Gasser and Leutwyler, 1987):
where . These expressions have also been computed at next-to-next-to-leading order (NNLO) and for fermions in the adjoint representation and the pseudoreal representations of the 2-color theory (Bijnens and Lu, 2009).
Each of the unknown coefficients α M, α F, α C also contain terms that grow linearly with N f. α C also contains a unique, N f-independent “contact term” that remains even in the absence of spontaneous chiral-symmetry breaking. This contribution is linear in m, quadratically sensitive to the ultraviolet cutoff (here the lattice spacing a -1.) This term dominates the chiral expansion of , making numerically accurate extrapolation of the condensate more difficult. Finally, the growth of the chiral log term in F M with N f forces the use of increasingly smaller fermion masses m as N f is (p.725) increased, in order to keep the NLO terms small enough relative to the leading order so that χPT is trustworthy.
The goal here is to search for enhancement of the condensate relative to the scale F. One way to proceed is to construct the ratio , and extrapolate directly m → 0; however, as noted above the presence of the contact term can make such an extrapolation difficult to carry out precisely. By making use of the additional quantity and the Gell-Mann-Oakes-Renner (GMOR) relation , incorporated into the NLO formulas shown above, we can construct other ratios at finite m that will also extrapolate to in the chiral limit: the other two possibilities are , and . Due to the contact term in should have the mildest chiral extrapolation of the three ratios.
A lattice study of the type outlined here, investigating the evolution from N f= 2 toN f = 6 in the SU(3) fundamental case, is described in (Appelquist et al, 2010). We refer to this reference for details of the simulation and analysis, and here give only selected details. The physical scales chosen to be matched are the rho mass m p and the Sommer scale ; each observable was first measured in the N f = 6 case, and then matched by tuning the bare lattice coupling at N f = 2. The resulting chiral extrapolation of these quantities is shown in Fig. 12.9, and shows good agreement, so that the lattice cutoffs are well matched between N f = 2 and N f = 6.
Determination of the presence or absence of condensate enhancement is done through comparison of the quantity between the N f = 6 and N f = 2 theories, by way of the equivalent ratio . We can directly construct a “ratio of ratios”
A direct computation of the S-parameter is also important to the study of general Yang-Mills theories with a focus on technicolor, and is well within the reach of existing lattice techniques. We will not discuss such a calculation in detail here, but include some references for further reading on the topic (Shintani et al., 2008; Boyle et al., 2010).
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