aa r X i v : . [ h e p - t h ] D ec WIS/12/11-DEC-DPPA
The Constraints of Conformal Symmetry on RG Flows
Zohar Komargodski
Weizmann Institute of Science, Rehovot 76100, IsraelInstitute for Advanced Study, Princeton, NJ 08540, USAIf the coupling constants in QFT are promoted to functions of space-time, the depen-dence of the path integral on these couplings is highly constrained by conformal symmetry.We begin the present note by showing that this idea leads to a new proof of Zamolod-chikov’s theorem. We then review how this simple observation also leads to a derivationof the a -theorem. We exemplify the general procedure in some interacting theories in fourspace-time dimensions. We concentrate on Banks-Zaks and weakly relevant flows, whichcan be controlled by ordinary and conformal perturbation theories, respectively. We com-pute explicitly the dependence of the path integral on the coupling constants and extractthe change in the a -anomaly (this agrees with more conventional computations of the samequantity). We also discuss some general properties of the sum rule found in [1] and studyit in several examples.Dec 2011 . Introduction, a New Proof of the c-Theorem, and Summary Promoting various coupling constants to background fields has always been a usefultool in the analysis of QFT. The applications of this idea are too numerous to list ex-haustively. For example, one may recall the classification of terms in the pion Lagrangianby Gasser and Leutwyler [2,3]. In addition, many of the seminal realizations regardingthe dynamics of SUSY gauge theories can also be understood by studying the dependenceof various observables on coupling constants. The NSVZ beta function [4] is one suchexample, and Seiberg’s realization regarding the power of holomorphy [5] is another.Let us recall why this is such a powerful idea. Generically, when various parametersin the Lagrangian are set to zero one finds that the symmetries of the theory are enhanced.If one reintroduces the coupling constants, this enhanced symmetry breaks explicitly tothe actual symmetry of the theory. However, we can always assign transformation rules tothe coupling constants such that, if the field transformations are accompanied by trans-formations of the coupling constants, the full enhanced symmetry is preserved. One caneasily see that, for instance, the expectation values of operators have to be consistent withthe enhanced symmetry. In addition, when one integrates over the fluctuating fields, oneobtains a functional of the background parameters. This functional of the backgroundparameter ought to respect the full extended symmetry.The statements above can be slightly generalized. When the coupling constants areset to zero, some of the extended symmetries of the Lagrangian could have quantumanomalies. Then, under a transformation of the fields by this extended symmetry onepicks up the anomaly. Introducing the couplings back to the Lagrangian and lettingthem transform under the the extended symmetries, one does not create new sourcesfor the violation of the extended symmetry. In other words, performing the extendedsymmetry transformation on the theory, one still picks up exclusively the anomaly. Then,for instance, if we path integrate over the dynamical fields and remain with a functionalof the background parameters, it must be true that this functional of the backgroundparameters reproduces the anomaly.Imagine any renormalizable QFT (in any number of dimensions) and set all the massparameters to zero. The extended symmetry includes the full conformal group. If the Our discussion hereafter applies to scale invariant theories which are also conformal. It isnot known whether all scale invariant theories are conformal (under some assumptions the answerin two dimensions is positive [6]) and it is not clear which of our results henceforth, if any, wouldapply to a scale invariant theory which is not conformal (if such an example existed). The readerinterested in this topic is referred to some recent literature on related matters [7,8,9,10,11]. k + 2, there may begravitational anomalies [14]. We will completely ignore gravitational anomalies here.Upon introducing the mass terms, one violates conformal symmetry explicitly . Thus,in general, the conformal symmetry is violated both by trace anomalies and by an oper-atorial violation of the equation T µµ = 0 in flat space-time. As we explained, on generalgrounds, the latter violation can always be removed by letting the coupling constants trans-form. Indeed, replace every mass scale M (either in the Lagrangian or associated to somecutoff) by M e − τ ( x ) , where τ ( x ) is some background field (i.e. a function of space-time).Then the conformal symmetry of the Lagrangian is restored if we accompany the ordinaryconformal transformation of the fields by a transformation of τ . To linear order, τ ( x ) al-ways appears in the Lagrangian as ∼ R d d x τ T µµ . Setting τ = 0 one is back to the originaltheory, but we can also let τ be some general function of space-time. The variation of thepath integral under such a conformal transformation that also acts on τ ( x ) is thus fixed bythe anomaly of the conformal theory in the ultraviolet. This procedure allows us to studysome questions about general RG flows using the constraints of conformal symmetry.One such question is the dependence on τ at very low energies. In other words, weintegrate out all the high energy modes and flow to the deep infrared. Since we do notintegrate out the massless particles, the dependence on τ is regular and local. As we haveexplained, the dependence on τ is tightly constrained by the conformal symmetry. Sincein even dimensions the conformal group has trace anomalies, these must be reproduced bythe low energy theory. The conformal field theory at long distances, CFT IR , contributesto the trace anomalies, but to match to the defining theory, the dilaton functional has tocompensate precisely for the difference between the anomalies of the conformal field theoryat short distances, CFT UV , and the conformal field theory at long distances, CFT IR .To warm up, let us see how these general ideas are borne out in two-dimensionalrenormalization group flows. First, let us study the constraints imposed by conformalsymmetry on the action of τ (remember τ is a background field everywhere in this paper).An easy way to analyze these constraints is to introduce a fiducial metric g µν into thesystem. Weyl transformations act on the dilaton and metric according to τ → τ + σ , g µν → e σ g µν . If the Lagrangian for the dilaton and metric is Weyl invariant, upon settingthe metric to be flat, one finds a conformal invariant theory for the dilaton. Hence, the task2s to classify local diff × Weyl invariant Lagraignains for the dilaton and metric backgroundfields.It is convenient to define b g µν = e − τ g µν , which is Weyl invariant. At the level oftwo derivatives, there is only one diff × Weyl invariant term:
R pb g b R . However, this is atopological term, and so it is insensitive to local changes of τ ( x ). Therefore, if one startsfrom a diff × Weyl invariant theory, upon setting g µν = η µν , the term R d x ( ∂τ ) is absent.The key is to recall that unitary two-dimensional theories have a trace anomaly T µµ = − c π R . (1 . c = 1.) One must therefore allow the Lagrangianto break Weyl invariance, such that the Weyl variation of the action is consistent with (1.1).The action functional which reproduces the two-dimensional trace anomaly is S W Z [ τ, g µν ] = c π Z √ g (cid:0) τ R + ( ∂τ ) (cid:1) . (1 . τ that survives even after the metric is taken to be flat. This is ofcourse the familiar Wess-Zumino term for the two-dimensional conformal group.Consider now some general two-dimensional RG flow from a CFT in the UV (with cen-tral charge c UV and a CFT in the IR (with central change c IR ). Replace every mass scaleaccording to M → M e − τ ( x ) . We also couple the theory to some background metric. Per-forming a simultaneous Weyl transformation of the dynamical fields and the backgroundfield τ ( x ), the theory is non-invariant only because of the anomaly δ σ S = c UV R d x √ gσR .Since this is a property of the full quantum theory, it must be reproduced at all scales.An immediate consequence of this idea is that also in the deep infrared the effective actionshould reproduce the transformation δ σ S = c UV R d x √ gσR . At long distances, one ob-tains a contribution c IR to the anomaly from CFT IR , hence, the rest of the anomaly mustcome from an explicit Wess-Zumino functional (1.2) with coefficient c UV − c IR . In par-ticular, setting the background metric to be flat, we conclude that the low energy theorymust contain a term c UV − c IR π Z d x ( ∂τ ) . (1 . × Weyl invariant terms, and there is no a3riori reason for them to be universal (that is, they may depend on the details of the flow,and not just on the conformal field theories at short and long distances).Zamolodchikov’s theorem [15] follows directly from (1.3). Indeed, we consider the par-tition function of the (
Euclidean ) theory in the presence of two insertions of the background τ ( x ), as in figure 1.Fig.1: The partition function of the Euclidean theory with two insertions of the backgroundfield with momentum k .From this general object we can extract c UV − c IR by expanding around k = 0, readingout the term quadratic in momentum, and matching to (1.3). Reflection positivity thusimmediately leads to c UV > c IR . (1 . τ to matter must take the form τ T µµ + · · · , where the corrections have more τ s. To extract the two-point function of τ with two derivatives we must use the insertion τ T µµ twice. (Terms containing τ can be lowered once, but they do not contribute to thetwo-derivative term in the effective action of τ .) As a consequence, we find that (cid:10) e R τT µµ d x (cid:11) = · · · + 12 Z Z τ ( x ) τ ( y ) h T µµ ( x ) T µµ ( y ) i d xd y + · · · = · · · + 14 Z τ ( x ) ∂ ρ ∂ σ τ ( x ) (cid:18)Z ( y − x ) ρ ( y − x ) σ h T µµ ( x ) T µµ ( y ) i d y (cid:19) d x + · · · . (1 . y integral is x -independent Z ( y − x ) ρ ( y − x ) σ h T µµ ( x ) T µµ ( y ) i d y = 12 η ρσ Z y h T µµ (0) T µµ ( y ) i d y . (1 . Z d xτ τ Z d yy h T ( y ) T (0) i . (1 . τ τ is ( c UV − c IR ) / π , and so by comparingwe obtain ∆ c = 3 π Z d yy h T ( y ) T (0) i . (1 . c > π from ours.)This finishes our discussion of two-dimensional flows. The same ideas can be appliedto four-dimensional flows. The following presentation of the proof of the a -theorem iscompletely equivalent to the one in [1]. The only slight pedagogical difference being thatwe treat τ as a background c-number field and never path integrate over it. In this wayone avoids having to introduce the, artificial, large decay constant, and therefore one doesnot need to expand in 1 /f . This difference is purely pedagogical as far as the dynamicsof QFT in 3+1 dimensions is concerned, but this slight twist in the logic allows to exhibitthe harmony between two and four dimensions.One starts by classifying local diff × Weyl invariant functionals of τ and a backgroundmetric g µν . It has been shown that the result up to four derivatives is (setting the back-ground metric to be flat) Z d x (cid:16) α e − τ + α ( ∂e − τ ) + α (cid:0) τ − ( ∂τ ) (cid:1) (cid:17) , (1 . α i are some real coefficients. However, in the quantum theory Weyl invariance mustbe violated because of the a - and c -trace anomalies T µµ = aE − cW µνρσ , (1 . E is the Euler density and W µνρσ is the Weyl tensor. It turns out that the c -anomaly does not lead to a Wess-Zumino term, so the c -anomaly disappears when thebackground metric is flat. The a -anomaly, however, does lead to a Wess-Zumino term5see [17],[1]). Since the total anomaly in the infrared must match that of the definingtheory, the Wess-Zumino term again comes with a universal coefficient S W Z = 2( a UV − a IR ) Z d x (cid:0) ∂τ ) τ − ( ∂τ ) (cid:1) . (1 . a UV − a IR . A more transparent way to discern the WZ termfrom the term proportional to α in (1.9) is found by switching to a new variableΨ = 1 − e − τ . (1 . Z d x (cid:18) α Ψ + α ( ∂ Ψ) + α (1 − Ψ) ( Ψ) (cid:19) , (1 . S W Z = 2( a UV − a IR ) Z d x (cid:18) ∂ Ψ) Ψ(1 − Ψ) + ( ∂ Ψ) (1 − Ψ) (cid:19) . (1 . α disappearsand only the last term in (1.14) remains. Therefore, by computing the partition functionof the QFT in the presence of four null insertions of Ψ one can extract directly a UV − a IR . ) ) ) ) Fig.2: Four insertions of the background field Ψ with P i k i = 0 and k i = 0. The blobrepresents the quantum matter fields.Indeed, consider all the diagrams with four insertions of a background Ψ with momenta k i , such that P i k i = 0 and k i = 0 (see figure 2). Expanding this amplitude to fourth order6n the momenta k i , one finds that the momentum dependence takes the form s + t + u with s = 2 k · k , t = 2 k · k , u = 2 k · k . Our effective action analysis shows that thecoefficient of s + t + u is directly proportional to a UV − a IR .In fact, one can even specialize to the so-called forward kinematics, choosing k = − k and k = − k . Then the amplitude of figure 2 is only a function of s = 2 k · k . a UV − a IR can be extracted from the s term in the expansion of the amplitude around s = 0.Continuing s to the complex plane, there is a branch cut for positive s (corresponding tophysical states in the s -channel) and negative s (corresponding to physical states in the u -channel). There is a crossing symmetry s ↔ − s so these branch cuts are identical.To calculate the imaginary part associated to the branch cut we utilize the opticaltheorem. See figure 3. The imaginary part is manifestly positive definite. ) ) ) ) ~ dX Fig.3: The imaginary part is given by calculating all the connected diagrams involving twoinsertions of the background field and any final state. One then squares the amplitude forthe transition to this particular final state and sums over all possible final states.Using Cauchy’s theorem we can relate the low energy coefficient of s , a UV − a IR , toan integral over the branch cut. Fixing all the coefficients one finds a UV − a IR = 14 π Z s> Im A ( s ) s . (1 . Im A ( s ) can be evaluated by means of the optical the-orem, figure 3, and hence it is manifestly positive. Since the integral converges by powercounting (and thus no subtractions are needed), we conclude a UV > a IR . (1 . It is interesting to compare the proof of the a -theorem and the argument regardingtwo-dimensional flows. As we have emphasized repeatedly, they both rely on the samesimple idea of promoting the masses to functions of space-time. In addition, in botharguments the key is to identity the anomalous Wess-Zumino-like term in the generatingfunctional for the background dilaton. This allows us to isolate a special term in the dilatonfunctional which only depends on the anomalies in the UV and IR CFTs, and not on theparticular flow. The main difference is, however, in the way positivity is established. In twodimensions, one invokes reflection positivity of a two-point function (reflection positivity isbest understood in Euclidean space). In four dimensions, the Wess-Zumino term involvesfour dilatons, so the natural positivity constraint comes from the forward kinematics (andhence, it is inherently
Minkowskian ).Let us say a few words about the physical relevance of (1.16). Such an inequalityconstrains severely the dynamics of quantum field theory, and in favorable cases can beused to establish that some symmetries must be broken or that some symmetries mustbe unbroken. In a similar fashion, if a system naively admits several possible dynamicalscenarios one can use (1.16) as an additional handle. (For example, an interplay between a -maximization [23] and the a -theorem has been used in [24] to throw light on the possibledynamics of various N = 1 theories. ) The a -anomaly is also closely related to the entan-glement entropy across spheres [26,27,28] and it would be very interesting to understandthe inequality (1.16) in these terms. (In two dimensions it has already been shown [29]that one can derive Zamolodchikov’s theorem via the ordinary inequalities that entangle-ment entropy obeys.) A concrete relation to the entanglement entropy is also likely to begeneralizable to 2+1 dimensions, where a plausible suggestion of a universal monotonicproperty of renormalization group flows now exists [27,30].Note that even though we have already shown that the a -theorem holds, and further,one can easily construct a universal monotonically decreasing function that interpolates For instance, in the context of the pion Lagrangian see [18], and in the closely related realmof electroweak physics the analysis is carried out in [19]. There are also applications for super-symmetric theories [20], and for fermion scattering [21]. A refreshing point of view on the natureof these constraints was given in [22]. An additional constraint on a class of supersymmetric theories has been proposed recentlyin [25]. It would be very nice to understand it in the language of the effective action for backgroundfields. a UV and a IR (e.g. by cutting the integral (1.15) at intermediate scales; ob-viously there are infinitely many ways of doing this), one fundamental question aboutfour-dimensional renormalization group flows is still outstanding. That is, whether onecan say something akin to the gradient flow property in two dimensions. The propertyof gradient flow in two dimensions follows directly from the fact that there is a two-pointfunction in the sum rule (1.8). The situation in four dimensions is more complicated; onedeals with a four-point function which lives naturally in Minkowski space. We hope toelucidate the object that appears instead of the familiar gradient flow soon.In the remaining of this note we consider some interacting models in four space-timedimensions. We being from generic weakly relevant flows and continue to the Banks-Zaks [31] fixed point. In both cases we perform the path integral over the quantumfields and extract the dependence on the coupling constants. Consistency requires thedependence on the coupling constants assumes the constrained form (1.9),(1.11). Weverify that this is indeed the case and calculate a UV − a IR in these examples. In bothcases our result for a UV − a IR agrees with other, more conventional, methods of extractingthe change in the a -anomaly. These computations are summarized in sections 2,3.In section 4 we discuss in more details the sum rule (1.15). We first explain howthe integrand Im A ( s ) /s behaves at small and large s in a general quantum field theory.We then examine the sum rule explicitly in the Banks-Zaks flow (and also revisit the freemassive scalar).
2. Weakly Relevant Flows
Consider a four-dimensional CFT in which there exists a primary operator O of dimen-sion ∆ = 4 − ǫ , with ǫ >
0. Let us deform the action by this operator, δS = R d xλ O ( x ).The beta function for the dimensionless coupling g = λµ − ǫ takes the form dgd log µ = − gǫ + 12 C Ω g + · · · , (2 . · · · stand for corrections of higher order in g , Ω d − ≡ π d/ / Γ( d/ C isthe coefficient in the OPE O ( x ) O ( y ) ∼ x − y ) − ǫ + C O ( x )( x − y ) − ǫ + · · · . (2 . ǫ << C >
0, there exists an IR fixed point with g ∗ = 2 ǫ/C Ω . The smallness of g ∗ guarantees that this fixed point can be controlledby conformal perturbation theory around g = 0 and that we can henceforth drop thecorrections in (2.1). 9 .1. Promoting Coupling Constants to Functions of Space-Time We restore conformal invariance by promoting the coupling constant g to a functionof space-time. A convenient way to do this is to replace the perturbation g ( µ ) µ ǫ O by δS = Z d xg ( F ( x ) µ ) µ ǫ O ( x ) , (2 . F = e τ . (2 . e τ transforms like a coordinate under scaletransformations. In other words, the renormalization group transformation taking thetheory from the scale µ to µ ′ can be compensated by the shift of the dilaton, such thatthe physical coupling remains intact.Note that to linear order in the dilaton, (2.3) means that we have ∼ τ β ( g ) µ ǫ O , whichis the expected coupling to the trace of the energy-momentum tensor. The prescrip-tion (2.3),(2.4) is valid under any circumstances (the generalization to CFTs deformed byseveral relevant operators is obvious), but we can compute the path integral explicitly andextract the dependence on τ only in special cases.In the weakly relevant flows we are discussing here the coupling g stays small through-out the RG evolution and so we can expand the (Euclidean) partition function as follows Z = Z e − S CF T (cid:20) − µ ǫ Z g ( F ( x ) µ ) O ( x ) d x + 12 µ ǫ Z g ( F ( x ) µ ) g ( F ( y ) µ ) O ( x ) O ( y ) d xd y − µ ǫ Z g ( F ( x ) µ ) g ( F ( y ) µ ) g ( F ( z ) µ ) O ( x ) O ( y ) O ( z ) d xd yd z + · · · (cid:21) . (2 . O s, evaluated inthe conformal field theory. We can drop the term linear in O because the expectationvalue of O vanishes in the CFT. Thus we get Z = Z e − S CF T (cid:20) µ ǫ Z g ( F ( x ) µ ) g ( F ( y ) µ ) O ( x ) O ( y ) d xd y − µ ǫ Z g ( F ( x ) µ ) g ( F ( y ) µ ) g ( F ( z ) µ ) O ( x ) O ( y ) O ( z ) d xd yd z + · · · (cid:21) . (2 . µ ǫ Z g ( F ( x ) µ ) g ( F ( y ) µ )( x − y ) − ǫ d xd y − C µ ǫ Z g ( F ( x ) µ ) g ( F ( y ) µ ) g ( F ( z ) µ )( x − y ) − ǫ ( x − z ) − ǫ ( y − z ) − ǫ d xd yd z + · · · . (2 . g , we only integrate over domains wherethe distances between points are not parametrically different from µ . In this way, we willeventually complete integrating over the whole space by employing RG transformations.(Instead of this conventional RG trick, one can also explicitly sum the leading contributionsfrom all orders in perturbation theory.)One can simplify (2.7) drastically if one observes that expanding in τ is closely relatedto the expansion in ǫ . This is due to the following chain rule identity ∂∂x g ( F µ ) = β ( F µ ) ∂∂x log F = β ∂τ∂x . (2 . g is of order ǫ , while β is of order ǫ . In general, we pay a factor of ǫ for everyadditional τ that we extract.Let us now show how terms in the effective action for τ arise from the integrals in (2.7).We shall analyze the term µ ǫ R g ( F ( x ) µ ) g ( F ( y ) µ )( x − y ) − ǫ d xd y and later explain why the othersare negligible to leading nontrivial order in the ǫ expansion. Expanding in derivatives wehave Z g ( F ( x ) µ ) g ( F ( y ) µ )( x − y ) − ǫ d xd y = Z ( g ( F ( x ) µ )) ( x − y ) − ǫ + 18 g ( F ( x ) µ ) g ( F ( x ) µ )( x − y ) − ǫ + 1192 g ( F ( x ) µ ) g ( F ( x ) µ )( x − y ) − ǫ + · · · ! d xd y (2 . y . We focus on the four-derivative term (namely the one appearingwith coefficient 1 /
192 in (2.9)). Performing the y integral over an infinitesimal energy slice( µ + dµ ) − < | x − y | < µ − , plugging back to (2.7), and expanding in ǫ we obtain thefollowing contribution to the effective action − Ω Z d xg ( F ( x ) µ ) g ( F ( x ) µ ) d log µ . (2 . ǫ . Indeed,focusing on the leading contribution in ǫ , (2.10) is equivalent to − Ω β ( g ) d log µ Z d x ( τ ) . (2 . C in (2.7) andall the higher corrections contribute only at a higher order in ǫ . Hence, (2.11) represents thegenuine four-derivative effective action for the dilaton background field to leading orderin ǫ . We can now compare (2.11) to the most general allowed four-derivative effectiveaction for the dilaton (1.9). Even though there are three equations in two variables, thereis a solution with α = 2( a UV − a IR ). Hence, the difference between the a -anomalies isobtained by dividing the coefficient in (2.11) by a factor 2 and integrating over all µ ∆ a = − Ω Z β ( g ( µ )) d log µ . (2 . g , ranging from the fixedpoint in the ultraviolet with g = 0 down to the CFT in the IR with g ∗ = 2 ǫ/C Ω ∆ a = − Ω Z g ∗ β ( g ) dg = 11152 ǫ C Ω . (2 . a -anomaly from the Wess-Zumino term for thebackground coupling constant. A more familiar definition of the a -anomaly is via thepartition function over the four-sphere. In the next subsection we perform the path integralover the four-sphere explicitly and compare to (2.13). Since the a -anomaly is proportional to the Euler density while the c -anomaly is pro-portional to the Weyl tensor squared, the a -anomaly can be isolated by computing thepartition function over S .As long as the perturbation is weak, one can compute the partition function by ex-panding in the coupling (in the spirit of (2.5)). This computation has been done in [32]and in a slightly different context originally in [33]. Keeping only the significant terms inthe weak coupling expansion, [32] found δF ( λ ) = − λ R ) ǫ π d + d − Γ( − d + ǫ )Γ( d +12 )Γ( ǫ ) + λ C π d +1) / R ǫ Γ( d ) Γ( − d + ǫ )Γ( ǫ ) , (2 . R is the radius of the d -dimensional sphere. To compute the a -anomaly we need tostudy the coefficient of log R . dd log R δF ( λ ) = − λ ǫ (2 R ) ǫ π d + d − Γ( − d + ǫ )Γ( d +12 )Γ( ǫ ) + λ C ǫ π d +1) / R ǫ Γ( d ) Γ( − d + ǫ )Γ( ǫ ) . (2 . d is even, the Gamma functions are singular around ǫ = 0, so one has to exercise a littlecare when expanding in ǫ . The leading order result is dd log R δF ( λ ) = ( − ) d/ ( d/ − λ ǫ π d + d − Γ( d +12 ) + λ C π d/ Γ( d ) ! . (2 . µ . The relation between the bare and physical coupling is λµ − ǫ = g + Cπ d/ ǫ Γ( d/ g + O ( g ) . (2 . dd log R δF ( g ) = ( − ) d/ π d d/ − ( d/ d − (cid:18) − g ǫ + g C Ω d − (cid:19) . (2 . R derivative of the partition function and the a -anomalyis through the Euler characteristic of the four-sphere, R S E √ gd x = 64 π . This allowsus to extract the formal quantity ∆ a ( g )∆ a ( g ) = π (cid:18) g ǫ − g C Ω d − (cid:19) , (2 . g = g ∗ this describes thephysical change in the a -anomaly as we flow from g = 0 at short distances to the IR fixedpoint g = g ∗ . Substituting g = g ∗ in (2.19), one verifies that ∆ a ( g ∗ ) coincides perfectlywith the total change in the a -anomaly computed via the Wess-Zumino procedure in (2.13). Since there is no logarithmic dependence on the radius in odd-dimensional conformal fieldtheories, the computation in [32] proceeds quite differently. . The Banks-Zaks Fixed Point In some respects, the Banks-Zaks fixed point [31] is quite similar to the weakly relevantflows analyzed in the previous section. However, it is sufficiently different to necessitatea separate treatment. The first marked difference is that in the Banks-Zaks fixed pointone perturbs by a marginal (and not a slightly relevant) operator. The weakly coupledfixed point is then achieved due to a balance between the one- and two-loop contributionsto the beta function. The more conceptual difference is that the the Banks-Zaks fixedpoint is at infinite distance away from the free theory (distances are measured using theZamolodchikov metric). Indeed, the perturbation of the free Yang-Mills theory by turningon a gauge coupling amounts to adding the operator F µν with a coefficient that scales like1 /g . By contrast, the cases we studied in the previous section can be realized by addingto an existing CFT a well-defined (relevant) operator with an arbitrarily small coefficient.We define the gauge coupling g in the usual way, through the Lagrangian − g Tr( F µν ),where Tr( T A T B ) = δ AB in the fundamental representation. Then, in terms of α = g π the beta function is dαd log µ = β α + β α + · · · , (3 . SU ( N ) gauge theory with N f Dirac fermions in the funda-mental representation) − β = 113 N − N f , − β = 343 N − N f (cid:18) N + N − N (cid:19) . (3 . N limit this simplifies to − β = 113 N − N f , − β = 343 N − N f N . (3 . N, N f → ∞ , while ǫ = N − N f N ≪
1. In this case the betafunction further reduces to dαd log µ = − N ǫα + 25 N α + · · · . (3 . α ∗ = ǫ N .Along the flow one can choose the effective theory at a scale µ to take the form L = − g ( µ ) Tr( F µν ) + N f X i =1 ψ i γ µ D µ ψ i . (3 . g representsboth the self coupling of the gauge fields as well as the fermion-fermion-gauge vertex. If thebeta function of the gauge coupling g in (3.5) is zero then the theory is clearly conformal.This is why the beta function of g chosen in this way coincides with the beta function (3.4)to the order we are interested in.We can render this theory conformal by simply writing (as before) L = − g ( F µ ) Tr( F µν ) + N f X i =1 ψ i γ µ D µ ψ i , (3 . F = e τ . We can extract the dependence on τ by expanding the Lagrangian around τ = 0 L = − g ( µ ) Tr( F µν ) + N f X i =1 ψ i γ µ D µ ψ i − τ β λ Tr( F µν ) − τ ˙ β λ Tr( F µν ) + · · · . (3 . λ = 1 /g and accordingly β λ ≡ dd log µ g . Expanding the functional integral in τ , wefind that the leading nontrivial term is quadratic in τ (cid:28)Z d xd y τ ( x ) β λ Tr( F µν ( x )) τ ( y ) β λ Tr( F µν ( y )) (cid:29) . (3 . τ corresponds to an expansion in ǫ (as can already be seen in (3.7)),it is easy to see that all the corrections to (3.8) are suppressed by more powers of ǫ .To obtain the answer to the leading nontrivial order, we ought to evaluate the corre-lator (cid:10) Tr( F µν ( x ))Tr( F µν ( y )) (cid:11) in free-field theory. The answer is (cid:10) Tr( F µν ( x ))Tr( F µν ( y )) (cid:11) = 12 N g π ( x − y ) . (3 . ds ∼ g − dg . Hence, the distanceto g = 0 diverges logarithmically.) Plugging this into (3.8) we find3 N π g β λ Z d xd y τ ( x ) τ ( y ) 1( x − y ) . (3 . d log µ is N π Ω g β λ Z d x τ ( x ) τ ( x ) d log µ . (3 . ǫ , we canextract the change in the a -anomaly by comparing (3.11) to (1.9),(1.11). Integrating overall energy scales one arrives at ∆ a = N π Ω Z ∞ λ ∗ dλλ β λ . (3 . λβ λ = 4 N ǫ π − π ) N λ − . (3 . λ = ∞ . The integral evaluates to∆ a = N ǫ π . (3 . a -anomaly of the Banks-Zaks fixed point has been computed in [34],directly from the definition of a . To compare, we normalize (3.14) with respect to the a -anomaly of a free scalar field ( a scalar = π ) ) and obtain∆ aa scalar = 85 N ǫ . (3 . − ǫ (with ǫ > ǫ is small, a fixed point that can be studied in conformal perturbationtheory always exists. This scenario is embedded naturally in many known supersymmetricmodels. One general way to realize it is to take any theory with a chiral operator whosedimension approaches 2 from below and couple it to an external, free, singlet field.16 . Comments on the Sum Rule (1.15) Consider some conformal field theory describing the ultraviolet, and suppose thistheory is deformed by relevant operators of dimensions ∆ i . The mass terms in front ofthese operators are promoted to functions of space time by M → M e − τ ( x ) . Therefore, thecoupling of the spurion to the matter theory at high energies is always through terms withpositive powers of various mass terms. Thus, dimensional analysis implies that Im A ( s )must behave, at large s , like lim s →∞ Im A ( s ) ∼ s − ǫ , (4 . ǫ >
0. In terms of the dimension of the relevant operator in the UV, ∆ i , ǫ =min(4 − ∆ i ). Hence, the sum rule is convergent in the UV. Analogous arguments apply for the low s properties of the imaginary part (one onlyneeds to replace the relevant operators by irrelevant ones, because the IR CFT is ap-proached by irrelevant operators which become less and less important at low energies).One therefore finds that the integral (1.15) is IR convergent. We will now study some examples of the imaginary part, Im A ( s ). To warm up,consider the free massive scalar field. One replaces the mass by a space-time dependentfunction according to M → M e − τ . In [1] we have performed the path integral overthe scalar field and showed that the generating functional for τ is of the form (1.9),(1.11),with a UV − a IR = π ) , the a -anomaly for a free scalar field. Let us now compute theimaginary part explicitly and examine the sum rule (1.15). Rewriting the term M e − τ Φ in terms of Ψ we get − M Φ + M ΨΦ − M Ψ Φ .In the case of a free massive field, the diagrams contributing to the four-dilaton ampli-tude are all one-loop diagrams. Their imaginary part is obtained by cutting them with the This argument leaves one possiblity unaccounted for – when there is a marginally relevantoperator. For instance, for a free Yang-Mills theory deformed by the gauge coupling one wouldget that the high energy behavior of the imagery part is ∼ s g ( s ) ∼ s / log ( s ). This is sufficientfor the convergence of the sum rule (1.15). In this section we will see that this is indeed the casefor the Banks-Zaks flow. Here the assumption that the IR theory is conformal (and not just scale invariant) appearsvital. If the IR theory were only scale invariant, there would be a derivative coupling of the dilatonto the so called “virial current,” a non-conserved vector operator of dimension three . By contrast,when the IR theory is conformal the dilaton couples only to irrelevant (or marginally irrelevant)operators in the CFT. Related ideas have been emphasized in [36]. ) ) )+ + ) ) ) ) ) ) ) ) ) Fig.4: Diagrams contributing to the imaginary part of the four-dilaton amplitude in thetheory of a free massive scalar.The formula for the imaginary part is thus (one must not forget an extra factor of in the optical theorem due to the fact we have identical particles in the final state) Im A ( s ) = | k | E c.m | ǫ µxyν k µ k ν | π ) Z d Ω |M ( k , k → k , k ) | . (4 . z direction.) The imaginary part is manifestly positive, as required by unitarity.To compute the amplitude we sum over the diagrams in figure 4 M = − iM − i M ( k − k ) − M − i M ( k − k ) − M . (4 . k = ( E, , , E ), k = ( E, , , − E ), we can parameterize the outgoing momentaas k = ( E, k ⊥ cos φ, k ⊥ sin φ, | k | cos θ ), k = ( E, − k ⊥ cos φ, − k ⊥ sin φ, −| k | cos θ ), where k ⊥ = | k | sin θ , and the on-shell condition implies E − M = | k | . Thus the amplitudetakes the form 12 M = − iM + 2 iM E E sin θ + M E cos θ . (4 . Im A ( E ) = M √ E − M (4 π ) E c.m Z d Ω − E M sin θ + cos θ ! . (4 . θ and get Im A ( E ) = M √ E − M πE Z − d cos θ − E M sin θ + cos θ ! = M πE (cid:18) EM + E M (cid:19) p E − M + (cid:18) − E M (cid:19) tanh − √ E − M E !! . (4 . O (1), in perfect agreement with the anticipated result.To extract the change in the a -anomaly we now integrate this over all energies via oursum rule (1.15) ∆ a = 14 π Z ∞ s =4 M ds Im A s = 132 π Z ∞ E = M dE Im A ( E ) E . (4 . a = π ) . This is precisely the correctanswer.Let us now consider the imaginary part of the four dilaton amplitude in the context ofthe Banks-Zaks flow. The vertices connecting the dilaton with the matter fields are givenby the Lagrangian (3.7). First, we rewrite (3.7) in terms of Ψ = 1 − e − τ L = − g T r ( F µν )+ N f X i =1 ψ i γ µ D µ ψ i − β λ (cid:0) Ψ + Ψ / · · · (cid:1) T r ( F µν ) −
14 ˙ β λ (Ψ + · · · ) T r ( F µν ) . (4 . Im A to the transition of two insertions of Ψ to anything. Clearly, theleading contribution to such transitions is given by the vertex − β λ Ψ T r ( F µν ) in (4.8).All the other vertices would contribute more factors of ǫ . More explicitly, this vertex takesthe form β λ Ψ ( ∂ µ A aν − ∂ ν A aµ )( ∂ µ A aν − ∂ ν A aµ ), where we neglected contributions from thenon-Abelian terms since they entail more powers of ǫ . See figure 5.The imaginary part is thus given by (where the quantum numbers in the final stateare ( a, ǫ (1) ), ( b, ǫ (2) ), and one also keeps in mind that the particles in the final state areidentical) Im A ( E ) = 132(2 π ) Z d Ω X final states (cid:12)(cid:12)(cid:12) M ( k , k → ( k , a, ǫ (1) )( k , b, ǫ (2) )) (cid:12)(cid:12)(cid:12) . (4 . M ( k , k → ( k , a, ǫ (1) )( k , b, ǫ (2) )) = g β λ (cid:16) ( k · k )( ǫ (1) · ǫ (2) ) − ( k · ǫ (2) )( k · ǫ (1) ) (cid:17) δ ab . (4 . µ . In this case, again, the renormalization group allows us to write the answerwithout summing explicitly infinitely many diagrams. (Of course, this is precisely whatthe renormalization group is made for.) A simple consistency check can be performedright away; we can verify that upon replacing the polarization vectors by momentum theamplitude vanishes. (1) ( k ) ) (2) ( k ) ) Fig.5: Leading diagram contributing to the imaginary part of the four-dilaton amplitudein the Banks-Zaks flow.We now square the amplitude and sum over all the possible final states, that is, allthe possible polarizations and color states. We find X final states |M| = 2 g N β λ ( k · k ) . (4 . Im A ( E ) = N π g ( E ) β λ ( E ) E . (4 . a -anomaly very much like in (4.7)∆ a = N π Z ∞ dEE g ( E ) β λ ( E ) = N π Z ∞ λ ∗ dλλ β λ . (4 . cknowledgments We would like to thank O. Aharony, M. Buican, T. Dumitrescu, H. Elvang, D. Freed-man, J. Hung, D. Jafferis, M. Kiermaier, I. Klebanov, J. Maldacena, R. Myers, S. Pufu,B. Safdi, N. Seiberg, J. Sonnenschein, S. Yankielowicz, and S. Zamolodchikov for many il-luminating discussions directly and indirectly pertaining to this project. We are espeicallyindebted to A. Schwimmer for asking many incisive questions throughout this work. In ad-dition, we are grateful to S. Theisen for providing us with very helpful unpublished notes.Z.K. is supported by NSF PHY-0969448, by a research grant from Peter and PatriciaGruber Awards, and by the Israel Science Foundation (grant eferences [1] Z. Komargodski, A. Schwimmer, “On Renormalization Group Flows in Four Dimen-sions,” [arXiv:1107.3987 [hep-th]].[2] J. Gasser and H. Leutwyler, “Chiral Perturbation Theory To One Loop,” Annals Phys. , 142 (1984).[3] J. Gasser and H. Leutwyler, “Chiral Perturbation Theory: Expansions In The MassOf The Strange Quark,” Nucl. Phys. B , 465 (1985).[4] V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, “Exact Gell-Mann-Low Function Of Supersymmetric Yang-Mills Theories From Instanton Calculus,”Nucl. Phys. B , 381 (1983).[5] N. Seiberg, “Naturalness Versus Supersymmetric Non-renormalization Theorems,”Phys. Lett. B , 469 (1993) [arXiv:hep-ph/9309335].[6] J. Polchinski, “Scale and Conformal Invariance in Quantum Field Theory,” Nucl. Phys.B , 226 (1988).[7] Y. Nakayama, “No Forbidden Landscape in String/M-theory,” JHEP , 030(2010). [arXiv:0909.4297 [hep-th]].[8] D. Dorigoni and V. S. Rychkov, “Scale Invariance + Unitarity → Conformal Invari-ance?,” arXiv:0910.1087 [hep-th].[9] I. Antoniadis and M. Buican, “On R-symmetric Fixed Points and Superconformality,”Phys. Rev. D , 105011 (2011) [arXiv:1102.2294 [hep-th]].[10] J. F. Fortin, B. Grinstein, A. Stergiou, “Scale without Conformal Invariance: Theo-retical Foundations,” [arXiv:1107.3840 [hep-th]].[11] T. L. Curtright, X. Jin and C. K. Zachos, “RG flows, cycles, and c-theorem folklore,”[arXiv:1111.2649 [hep-th]].[12] M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav. , 1387 (1994)[arXiv:hep-th/9308075].[13] S. Deser and A. Schwimmer, “Geometric classification of conformal anomalies in ar-bitrary dimensions,” Phys. Lett. B , 279 (1993) [arXiv:hep-th/9302047].[14] L. Alvarez-Gaume and E. Witten, “Gravitational Anomalies,” Nucl. Phys. B , 269(1984)..[15] A. B. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a2D Field Theory,” JETP Lett. , 730-732 (1986).[16] A. Cappelli, D. Friedan, J. I. Latorre, “C Theorem and Spectral Representation,”Nucl. Phys. B352 , 616-670 (1991).[17] A. Schwimmer and S. Theisen, “Spontaneous Breaking of Conformal Invariance andTrace Anomaly Matching,” Nucl. Phys. B , 590 (2011) [arXiv:1011.0696 [hep-th]].[18] T. N. Pham, T. N. Truong, “Evaluation of the Derivative Quartic Terms of The MesonChiral Lagrangian from Forward Dispersion Relation,” Phys. Rev.
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