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MCTP-12-09

MIT-CTP-4362

PUPT-2413

SU-ITP-12/14

On renormalization group flows and the -theorem in 6d

Henriette Elvang, Daniel Z. Freedman, Ling-Yan Hung,

[1mm] Michael Kiermaier, Robert C. Myers, Stefan Theisen

Randall Laboratory of Physics, Department of Physics,

University of Michigan, Ann Arbor, MI 48109, USA

[2mm] Department of Mathematics, Center for Theoretical Physics,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

[2mm] Stanford Institute for Theoretical Physics, Department of Physics,

Stanford University, Stanford, CA 94305, USA

[2mm] Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

[2mm] Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA

[2mm] Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany

[4mm] elvang@umich.edu, dzf@math.mit.edu, jhung@perimeterinstitute.ca, mkiermai@princeton.edu, rmyers@perimeterinstitute.ca, stefan.theisen@aei.mpg.de

We study the extension of the approach to the -theorem of Komargodski and Schwimmer to quantum field theories in spacetime dimensions. The dilaton effective action is obtained up to 6th order in derivatives. The anomaly flow is the coefficient of the 6-derivative Euler anomaly term in this action. It then appears at order in the low energy limit of -point scattering amplitudes of the dilaton for . The detailed structure with the correct anomaly coefficient is confirmed by direct calculation in two examples: (i) the case of explicitly broken conformal symmetry is illustrated by the free massive scalar field, and (ii) the case of spontaneously broken conformal symmetry is demonstrated by the (2,0) theory on the Coulomb branch. In the latter example, the dilaton is a dynamical field so 4-derivative terms in the action also affect -point amplitudes at order . The calculation in the (2,0) theory is done by analyzing an M5-brane probe in AdS.

Given the confirmation in two distinct models, we attempt to use dispersion relations to prove that the anomaly flow is positive in general. Unfortunately the 4-point matrix element of the Euler anomaly is proportional to and vanishes for forward scattering. Thus the optical theorem cannot be applied to show positivity. Instead the anomaly flow is given by a dispersion sum rule in which the integrand does not have definite sign. It may be possible to base a proof of the -theorem on the analyticity and unitarity properties of the 6-point function, but our preliminary study reveals some difficulties.

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1 Introduction

There is a common paradigm in quantum field theory in which the correlation functions of a non-conformal theory approach those of a conformal theory, the CFT, at short distance and those of another conformal theory, the CFT, at long distance. The two CFT’s are viewed as end-points of the renormalization group flow (RG flow). Among the quantities characterizing a CFT are its trace anomaly coefficients, frequently called central charges, obtained by embedding the theory in a curved background metric .

Zamolodchikov’s -theorem [1] revealed a remarkable structure of RG flows of two-dimensional quantum field theories. He showed that there is a positive definite function on the space of couplings that satisfies three properties: (a) decreases monotonically along RG flows, (b) fixed points of the RG flow are critical points for , i.e., , and (c) at these fixed points, coincides with the central charge of the corresponding conformal field theory. These properties hold in any unitary, renormalizable and Lorentz invariant two-dimensional QFT. A direct consequence is that for any RG flow, the central charges of the end-point CFT’s satisfy the inequality

(1.1)

The central charge is often interpreted as providing a measure of the ‘number of degrees of freedom’ and hence the c-theorem confirms the intuition that this number should decrease along RG flows.

Zamolodchikov’s result motivated the search for a similar property in higher-dimensional quantum field theory. For the trace anomaly has two central charges and and takes the form

(1.2)

where denotes the square of the Weyl tensor of the background geometry and multiplies the quadratic combination of curvatures which is the integrand of the Euler integral invariant

(1.3)

In 1988 Cardy [2] conjectured, with evidence from several models, that for any RG flow, it is the Euler central charge and its analogue in higher even spacetime dimension that satisfies the desired inequality

(1.4)

For several reasons it was a difficult problem to prove this conjecture, although the unsuccessful 25 year effort has taught us quite a bit of interesting physics. For example, the central charges of many supersymmetric gauge theories in four dimensions can be calculated [3, 4, 5], and the Euler central charge satisfies (1.4). Furthermore, it is quite easy to prove the -theorem for QFT’s which have an AdS/CFT dual [6, 7, 8, 9].

A concise and insightful proof for any four-dimensional RG flow connecting two conformal fixed points was recently presented by Komargodski and Schwimmer in [10] and discussed further in [11, 12]. The key idea in this work is to use the dilaton field to probe the trace anomaly. If conformal symmetry is spontaneously broken, as on the Coulomb branch of SYM theory, the dilaton is a massless Goldstone boson in the spectrum of the theory. If conformal symmetry is broken explicitly by dimensionful parameters of the QFT, the dilaton is introduced as a conformal compensator. As we review in Sec. 2 below, the -theorem can be proved quite elegantly in this framework: it follows from the analyticity and unitarity of the forward 4-point scattering amplitude of the dilaton.

It is natural to ask whether the dilatonic approach of [10] can be extended to QFT’s in any even spacetime dimension. For general , the trace anomaly can be written as [13, 14]111Scheme dependent terms of the form are ignored in (1.2) and (1.5). See [15, 16] for an interesting application.

(1.5)

and it defines a number of central charges of the CFT. Each term on the right-hand side is constructed from the background geometry and has conformal weight . In particular, is the Euler density in dimensions while are conformal invariants. It is the Euler central charge which Cardy [2] conjectured to satisfy an -theorem. The question that originally motivated the present paper was whether the new insights of [10] can be applied to prove such a theorem for RG flows in dimensions.

We begin our investigation by constructing the effective dilaton action in Sec. 3. As in , the anomaly flow appears as the coefficient of the dilaton Wess-Zumino term which arises from the 6-derivative Euler density.222In this paper we assume that RG fixed point theories are conformal. A scenario in which fixed point theories are scale invariant but not conformal invariant has been discussed in the recent literature, see [17, 18] (and see also [19]), but are argued to be ruled out in [12]. A new feature appears in : there are Weyl-invariant 4-derivative terms that contribute to the dilaton scattering amplitudes. We present the general 4-, 5- and 6-point on-shell dilaton scattering amplitudes in the low-energy expansion, specifically focusing on the contributions of order and in the momenta.

The structure of the dilaton effective action and scattering amplitudes is confirmed in explicit examples. In Sec. 4 we study a free massive scalar and verify that when the massive mode is integrated out, the low-energy effective action of the dilaton (which is introduced as a conformal compensator in this case) agrees with our general result.

Encouraged by the massive scalar example, we discuss in Sec. 5 the structure of dispersion relations for the dilaton amplitudes. Unfortunately, no positivity statement or monotonic -function can be extracted from the 4-point amplitudes at order . It may be possible to derive an -theorem for RG flows in 6d from the 6-point dilaton amplitude, but its analyticity, crossing, and unitarity properties are quite complicated, e.g., see [20], and we leave such an analysis for the future.

It is difficult to find conventional interacting field theories with tractable RG flows in dimensions. In , the prime example for a theory with spontaneously broken conformal symmetry is SYM on the Coulomb branch. The analogue in is the M5-brane (2,0) theory on the Coulomb branch. It has no weakly coupled description and is not easy to treat directly. For this reason, we turn to a holographic description. This system is holographic sine qua non, but we will be testing the dilatonic formulation of the -theorem in a different way from traditional holographic -theorems [7, 6, 8]. As a warm-up to the 6d (2,0) theory, we present in Sec. 6 a detailed account of how the 4d dilaton effective action is extracted from the holographic treatment of the Coulomb branch RG flow of SYM.333We thank Juan Maldacena for this suggestion. The techniques are then extended to the Coulomb branch flow for the 6d (2,0) theory in Sec. 7. Recently, ref. [21] studied how supersymmetry determines for the 6d theory. In our analysis, we match the full dilaton effective action from the DBI action of a probe M5-brane. In particular, we extract at large including the numerical coefficient and find that it agrees with the literature [22]. Our analysis also clarifies the role of the dilaton in the case of spontaneously broken conformal symmetry: the dilaton must then be treated as a dynamical field, as opposed to the scenario of explicitly broken conformal symmetry in which the dilaton may be treated as a source [11, 12]. The difference is manifest in the dilaton amplitudes.

We conclude with a brief summary of our results and future directions in Sec. 8. Technical details are relegated to appendices: App. A collects conventions and some exact forms of 6d Weyl-invariants. App. B contains two complementary derivations of the 6d Euler anomaly action. And finally App. C summarizes the Coulomb branch flow in the D1-D5 system at large ; this is the 2d analogue of our analyses of the 4d SYM and the 6d (2,0) theory.

2 The -theorem for

We review the proof of the -theorem in four dimensions presented by Komargodski and Schwimmer [10]. Issues relevant to the analysis in higher dimensions will be highlighted. The key ingredient to understand is the role of the ‘dilaton’. This is conceptually simpler in theories in which conformal symmetry is broken spontaneously, so we begin with this case. The dilaton is then a Goldstone boson of the theory. To find its low-energy effective action, we couple the theory to the general metric and consider the effect of diffeomorphisms and Weyl transformations

(2.1)

The dilaton effective action has two classes of terms, those which are manifestly invariant under (2.1) and those which encode the Weyl anomaly. Of course, both sets of terms are also invariant under diffeomorphisms.

The anomaly term is more interesting. It is a functional whose linear variation in is the trace anomaly (1.2). We start with the simple term

(2.2)

with and as in (1.2). The variation reproduces the correct Weyl term of (1.2) because the combination is Weyl invariant and independent of . By contrast, is not Weyl invariant, so the variation of the Euler term above yields a spurious term linear in , as well as the desired contribution proportional to . However, the action above can be corrected by adding additional terms nonlinear in [23, 10] such that the final form of the anomalous dilaton Wess-Zumino action becomes

(2.3)

In the flat space limit, one can see that (2.3) generates order terms in dilaton amplitudes.

We must be careful to include in all other possible terms in the effective action which generate amplitudes of order and are consistent with the symmetries. These (Weyl diffeo)-invariant terms can easily be found by writing all independent curvature invariants in terms of the manifestly Weyl invariant ‘metric’: . Up to four derivatives, the independent invariants in a conformally flat spacetime are [10]

(2.4)

Here is a constant with mass dimension one. We now combine eqs. (2.4) and (2.3), take the flat space limit, i.e., , and obtain the total effective action of the dilaton used in [23, 10, 11]

(2.5)

Hence Weyl and diffeomorphism invariance lead to a highly constrained form for the effective dilaton action even in flat space.

Let us make two further refinements of the action in order to more easily apply it to the calculation of dilaton scattering amplitudes. The first step is to ‘complete the square’ in the terms proportional to . With the shift , we rewrite (2.5) as

(2.6)

The second step is the change of variables

(2.7)

The net result is the action

(2.8)

Let us now follow the argument of [10] to obtain the -theorem for RG flows with spontaneous breaking of conformal symmetry. The stress tensor in the flat space limit is traceless; remains as the Ward identity for global conformal symmetry at the quantum level, although there is an anomaly in curved spacetime.444A consequence of the curved space anomaly is that flat space correlation functions involving contain contact terms. The constant is related to the VEV of a relevant operator of CFT with central charges and . In the IR, i.e., at energies , where all particles with masses are integrated out, the RG flow ends at a CFT with and . The situation is similar for anomalous chiral symmetries in which the Ward identity holds in the absence of external sources whether or not the symmetries are spontaneously broken. If unbroken, the low energy spectrum of the theory contains massless chiral fermions and there is strict anomaly matching, . If spontaneously broken, the spectrum of the theory contains massless Goldstone bosons, and the strength of their low-energy self-interactions is fixed by [24]. For conformal symmetry we have the intermediate situation that the total UV and IR anomalies match [23], and the difference is the coefficient of the dilaton Wess-Zumino term in the flat-space limit of (2.3). This means that in (2.5), (2.6) and (2.8) is replaced by .

In principle, all coefficients of (2.8), including , can be calculated by evaluating the path integral for the interpolating theory whose RG flow is being studied. The first three coefficients depend on the renormalization scheme, but the anomaly coefficient is universal. In the present case of spontaneous breaking of conformal invariance, there is a moduli space of vacua and thus no potential for the dilaton. So .

When , it is clear from (2.8) that is a canonically normalized field with . It is thus well suited for the computation of scattering amplitudes and can be thought of as the ‘physical dilaton’. The on-shell condition is simply or, equivalently, for all external particles of on-shell amplitudes. It follows from (2.8) that -point on-shell scattering amplitudes of the dilaton vanish as at low energy.555Note that an -point amplitude for any set of identical massless scalars cannot have an order term at low energy because the only Bose symmetric Lorentz invariant available vanishes; for example for it is ! For example, the 4-point amplitude has low-energy matrix element

(2.9)

To prove the -theorem we consider the full amplitude which approaches at low-energy. The forward amplitude satisfies a dispersion relation in which the right and left-hand cuts are equal by crossing symmetry. The simplest way to proceed is to note that is analytic in the annular domain which is the interior of the curve in Fig. 1. Using Cauchy’s theorem and crossing, we obtain

(2.10)

Here is a temporary IR cutoff which allows us to separate the right and left cuts from the pole. In the limit we obtain the -theorem

(2.11)

where the positive sign is an immediate consequence of unitarity, . The interpretation of the sum rule is that the difference receives contributions from all scales of the interpolating QFT.

Figure 1: The contour used to derive (2.10) and (5.2). The contour surrounds the simple pole of at .

We discuss the sum rule (2.11) further below, but we first prefer to bring theories with explicitly broken conformal symmetry into the picture. This means that there are explicit scale parameters in the Lagrangian or implicit scales which appear due to dimensional transmutation. We simply denote these scales by . In this important case, Komargodski and Schwimmer [10] introduce the dilaton as a new weakly coupled dynamical field of the theory. (Here it can also be viewed as a source [11, 12].) The dilaton kinetic term of (2.5) is added with “decay constant” as a free adjustable parameter. Mass scales are made spacetime dependent by the replacement . In other words, the dilaton is added as a conformal compensator. The cosmological or potential term in (2.8) no longer vanishes. As in [12] we simply add a counter term to cancel it. The (improved) stress tensor of the theory modified in this way is traceless, so effectively we have recreated the previous situation of spontaneous breaking since acquires a VEV. With this understanding the previous discussion is applicable, and the sum rule (2.11) is derived as above.

There is, however, one important difference between the two cases. For RG flows with explicit breaking, the constant with dimension of mass is adjustable. It is chosen to be much larger than any physical scale i.e., . In this limit diagrams containing the dilaton as an internal line — or as an intermediate state in the computation of — are strongly suppressed. The dilaton then effectively acts as a source for the trace of the stress tensor in the unmodified theory. For spontaneously broken flows, the decay constant is a fixed physical constant (it is essentially the VEV), and effects of the dilaton are included in the low energy theory. Indeed, it contributes with the strength of a free massless scalar to .

It is important to show that the integral in (2.11) converges both at and at large . It is argued in [11] and [12] that these limits are controlled by operator deformations of the IR and UV CFT’s. Thus the approach to the UV is determined by the least relevant operator in the flow away from the CFT. Dimensional analysis implies the large behavior

(2.12)

with . The estimate (2.12) indicates that the contribution of the large circular contour in Fig. 1 vanishes and that the dispersion integral converges at high energy. The behavior (2.12) is confirmed in the massive scalar boson example (in ) in [11], where and thus for the mass term .

The approach to the IR should be determined by the least irrelevant operator of the CFT, and we expect the low energy behavior

(2.13)

with . This makes the integral (2.11) IR finite. To exemplify this behavior we discuss an RG flow with spontaneously broken conformal symmetry and study the contribution of the 2-dilaton intermediate state to the sum rule (2.11) at low energy. For the Coulomb branch of SYM theory, we are interested the 2-dilaton cut of diagrams of the type shown in Fig. 2. Detailed knowledge of the dilaton 4-point amplitude on each side of the cut is not required, all we need to know is that at low energy it behaves as multiplied by a real constant. Thus in this limit, unitarity tells us that

(2.14)

where and . With the power law , the sum rule (2.11) nicely converges at However, it is not clear to us that it is associated with an irrelevant operator (which would have ).

Figure 2: Computing via unitarity. The external particles are massless dilatons. In Section 2, the particles in the intermediate state are also dilatons. In Section 5, they are massive particles.

With the extension of these ideas to in view, we distinguish two aspects of the approach of Komargodski and Schwimmer [10]:

  • The Euler action, with coefficient , determines the form of the dilaton scattering amplitudes at low energy in . In a given example, one can, in principle, check this form, compute the coefficient and compare the result with a calculation of by a conventional method, such as the heat kernel method [22].

  • They show that in any 4d theory using the analyticity and unitarity properties of the dilaton amplitude. We study whether these properties are as simple and as effective in .

3 Dilaton effective action for

To discuss the -theorem in , our first task is to derive the dilaton effective action . Following the 4d approach of [23, 10], is the most general action whose Weyl-variation equals the trace anomaly. In 6d, the trace anomaly contains — in addition to the Euler -anomaly — three independent Weyl tensor terms . We confine our attention to the Euler anomaly . Working out the index contractions in the case of (A.7), one finds

(3.1)

where “+…” stands for terms that vanish in a conformally flat background; see (A).

The Weyl variation is non-vanishing, so we need to determine terms such that

(3.2)

Then produces the correct anomaly action. We present two complementary derivations of in App. B. Its flat-space limit is given in (3.11).

Any Weyl-invariant action can be added to without affecting (3.2). Hence the most general dilaton effective action includes also all possible Weyl-invariants. For our purpose we need to consider all 2-, 4- and 6-derivative Weyl-invariants.666As in Sec. 2, the cosmological term with 0-derivatives is tuned to vanish. Since is Weyl invariant, the terms we seek are found by replacing in linear, quadratic, and cubic curvature invariants. Since we are interested only in the invariants which remain independent when evaluated on conformally flat metrics,

(3.3)

the Riemann tensor can be replaced by plus Ricci-terms (see (A.2)), and we need only consider terms constructed from , , and covariant derivatives. Here and henceforth, hatted quantities refer to the conformally flat Weyl-invariant metric (3.3). We classify the possible terms of this type in Secs. 3.1-3.3. The result-oriented reader can skip ahead to (3.15) and (3.14) which give the form of that will be used in the remainder of the paper.

3.1 2-derivative Weyl-invariants

In any dimension , there is a unique 2-derivative diffeo-Weyl invariant, namely . It gives rise to the dilaton kinetic term. We present the -dimensional result in (A.12), but focus here on the case whose flat space limit (3.3) is

(3.4)

The second expression is obtained by partial integration. We write the kinetic term of the dilaton as

(3.5)

where has dimension of mass. The normalization of the kinetic term is chosen for later convenience. It follows from (3.5) that the equation of motion of is

(3.6)

It is relevant to note that is proportional to the equation of motion, as can be seen from (3.4).

3.2 4-derivative Weyl-invariants

At the level of 4-derivatives, there are three basic curvature invariants: , , and . The latter is a total derivative in the effective action, so we do not consider it further. Evaluating the two others on the Weyl-invariant metric (3.3) we find

(3.7)

The couplings and have dimension of (mass). It is clear that vanishes under the EOM (3.6), but does not. It will contribute to scattering amplitudes.

For the purpose of calculating on-shell scattering amplitudes, it is useful to rearrange (3.7) as

(3.8)

with . This follows from using straightforward algebra to show that is a simple linear combination of the two Weyl-invariants in (3.7).

3.3 6-derivative Weyl-invariants

It is simple to list the general set of 6-derivative Weyl-invariants constructed from Ricci tensors and covariant derivatives. Taking the Bianchi identity and partial integration into account we find

(3.9)

This can also be deduced from the list of eleven curvature invariants in [25] or [26]. Of those, six invariants involve the full Riemann tensor which can be replaced by the Weyl tensor plus Ricci’s; the 5 remaining invariants are (3.9).

All five invariants vanish when is on-shell; this is clear for the first 3 invariants since they are proportional to . As integrated quantities, the latter two are actually not independent from the three others for conformally flat metrics. To see this, first note that the integral of the combination777 There is a continuation of (3.10) valid for general . See eq. (3.3) of [16].

(3.10)

vanishes for conformally flat metrics. This is used to eliminate in favor of the four other invariants in (3.9). Second, the Euler density is a total derivative, so the integral of can be expressed in terms of the integral of and via (3.1).

Thus the most general Weyl-invariant 6-derivative action can be written as a linear combination of the integrals of ,  , and . Since they vanish on-shell, they will not affect the on-shell dilaton amplitudes at , but they are nonetheless useful for us in the following section, so we present their explicit forms for the conformally flat metric (3.3) in (A.15), (A.16), and (A.17).

3.4 The 6-derivative Euler action

It is this action whose Weyl variation produces the conformal anomaly. We have computed from (3.2) by two related methods, and we refer readers to the self-contained discussions in App. B. For calculations in Secs. 4-7, we need only the result in the flat space limit,

(3.11)

Amplitudes are easier to calculate after a rewriting of the action (3.11). It takes straightforward algebra to show that

(3.12)

where all hatted terms are evaluated on the conformally flat metric (3.3). The first three terms can be absorbed in the general 6-derivative Weyl-invariant terms discussed in Sec. 3.3, and we can then write the most general 6-derivative effective action as

(3.13)

Note that (3.12) requires an algebraic miracle: 7 equations in 3 unknowns have a unique solution. We refer readers who believe in mathematics rather than miracles to the end of App. B, where the formalism of Paneitz operators and -curvatures is discussed briefly.

When satisfies the equation of motion , the three Weyl-invariants in (3.13) vanish. Thus only the last term, , contributes to the scattering amplitudes and it captures the information about the -anomaly. Repeated use of the equation of motion (and partial integration) shows that . This latter is the same result obtained by applying the equations of motion in (3.11).

3.5 The dilaton effective action

The 4d argument of Komargodski and Schwimmer [10], as reviewed in Sec. 2, shows that the change in the Euler central charge in the flow from the CFT to CFT will be carried by the dilaton. Our discussion in this section can then be summarized in the flat-space limit of the dilaton effective action

(3.14)

with expressions for the 6-derivative Weyl-invariants given in (A.15)-(A.17). Dropping terms that vanish on the equations of motion and therefore do not contribute to the low-energy on-shell dilaton amplitudes of our interest, we can simplify this to

(3.15)

Compared with 4d, it is a new feature in that both a Weyldiffeo invariant 4-derivative term and the 6-derivative Euler anomaly term contribute to on-shell dilaton amplitudes.

3.6 Dilaton matrix elements from the effective action

We are interested in the on-shell matrix elements of (3.15) at 4th and 6th order in the low-energy small-momentum expansion. As discussed in Sec. 2, we first transform to the ‘physical dilaton’ field , defined in 6d by

(3.16)

The equation of motion (3.6) implies , and the on-shell condition for physical dilaton is therefore simply .

From the action (3.15) we obtain888Note that we have dropped 6-derivative terms which do not contribute to the on-shell amplitudes, for example . We have also dropped quadratic terms and which do not influence the amplitudes of interest here.

Next we extract the dilaton matrix elements.

Matrix elements at
It is easy to extract the on-shell matrix elements from (3.6). At order , the results can be read off directly from the interaction vertices. Any vertex with can be dropped. The term with contributes simply as , where . For example, for the 4-point matrix elements, the only contribution comes from which gives the vertex rule . We list the results at below for later reference:

(3.18)

In the last case, we used that for null momenta.

At the matrix elements receive contributions from the 6-derivative contact terms in (3.6), but if the dilaton is dynamical — as in the spontaneously broken case — there will also be contributions from pole diagrams with two vertices from 4-derivative terms in (3.6). We will consider these two cases separately.

Matrix elements at : explicitly broken conformal symmetry
When conformal symmetry is softly broken by relevant operators, we can regard the dilaton as a weakly coupled scalar with the scale chosen much larger than any other scale in the problem; in particular , so since pole diagrams at order are they will be suppressed. Alternatively, we can regard the dilaton as a source and in that case there are simply no pole diagrams. The on-shell matrix elements are found directly from the contact terms in (3.6) and they are

(3.19)
(3.20)
(3.21)

The simplest example of this case is the free massive scalar which we study in Sec. 4.

Matrix elements at : spontaneously broken conformal symmetry
When the conformal symmetry is broken spontaneously, the dilaton is the corresponding Goldstone boson and as such it is a dynamical degree of freedom of the theory. Therefore we must include the pole diagrams with dilaton exchanges. For example, the 4-point matrix element receives a contribution from the tree-diagram with two 3-vertices from (3.6). The value of the -channel diagram is

(3.22)

Adding to this the -channel and -channel diagrams, it is clear that the contribution from these diagrams are local and of exactly the same form, , as the contact term contribution that gave (3.19). Similarly, the 5- and 6-point matrix elements receive contributions from 2- and 3-particle channel tree diagrams. We simply list the results

(3.23)
(3.24)
(3.25)

Obviously, for , the matrix elements in (3.23)-(3.25) reduce to those of (3.19)-(3.21). The 6-point matrix elements also have terms with poles in ; the “perms” indicate the sum of 10 independent 3-particle channels. It is worth noting that the combination shows up in all three matrix elements. This plays a role in our study of the 6d (2,0) theory in Sec. 7.

4 Example of explicit breaking: a free massive scalar

As the simplest example of explicitly broken conformal symmetry, we take the CFT to be the 6d theory of a free massless scalar and deform it with a mass term operator . In the far IR, the massive field decouples and the CFT is trivial with no degrees of freedom. Hence . The -central charge in the UV is that of a conformally coupled 6d free massless scalar,

(4.1)

This value was computed in [22] using heat kernel methods. For this RG flow, we therefore have .

Following [10], we introduce the dilaton via the conformal compensator , where is a mass scale that we choose such that . Specifically, we start with

(4.2)

and introduce as a canonically normalized scalar of mass dimension 2 such that the coupled action is

(4.3)

The model is weakly coupled since the dimensionless coupling is small thanks to , and the action (4.3) has a traceless improved stress tensor. For , there is a moduli space in , and the model is conformally invariant at the origin of moduli space where . When acquires a non-zero VEV, , the conformal symmetry is spontaneously broken, and the fluctuation around the VEV, , is the associated Goldstone boson. In fact, we recognize as the physical dilaton introduced in (3.16). The action coupling to is

(4.4)

Upon integrating out the massive field , the action (4.4) should yield the dilaton effective action. To verify this, it suffices to show that the on-shell amplitudes calculated at low-energy from (4.3) agree with those of the dilaton effective action, as computed in Sec. 3.6. Thus, taking advantage of , we proceed perturbatively and calculate the 4-, 5-, and 6-point amplitudes of the dilaton from 1-loop diagrams with fields on internal lines, extracting their order and terms in the low-energy expansion.

In the action (4.4), is coupled to only through the cubic interaction term . Therefore it is quite simple to calculate the 1-loop dilaton scattering amplitudes. For example, to obtain the on-shell 4-point function of , we have to sum 3 permutations of the elementary box diagram

(4.5)

Computing this diagram and its permutations using Feynman parameters, we obtain the following low-energy expansion for the 4-point amplitude:

(4.6)

The calculation of 5-point functions requires the sum of 12 independent pentagon diagrams. The low energy contribution can be expressed in terms of the 5-point invariants

(4.7)
(4.8)

Proceeding analogously to 6-point functions, we find that the sum of 60 independent hexagon diagrams has the low energy expansion

(4.9)
(4.10)

Let us now compare these amplitudes with matrix elements of Sec. 3.6. The results (4.6), (4.7), and (4.9) completely match the matrix elements in (3.18) with a unique and consistent identification of as

(4.11)

Next, at , the amplitudes (4.6), (4.8), and (4.10) are fully consistent with the dilaton matrix elements (3.19), (3.20), and (3.21) with the identification of . This is in agreement with the expectation, as discussed below (4.1).

We conclude that the simple example of a free massive scalar confirms the structure of the dilaton effective action derived in Sec. 3. Note that it is a key point in the analysis that can be chosen freely, in particular such that . This ensures that the amplitudes can be calculated perturbatively in small . Moreover, shows that the pole exchange diagram contributions in (3.23)-(3.25) are 2-loop effects in this example, and they are arbitrarily suppressed.

5 The anomaly from dispersion relations

One of the most striking features of the approach to the -theorem in [10] is the use of a dispersion relation for the forward 4-point dilaton scattering amplitude. Unitarity then provides a quick proof that for a general RG flow in . It is an obvious question to ask if a similar approach can work in . We begin with the 4-point amplitude for an RG flow with explicit breaking, then discuss spontaneous breaking, and finally briefly comment on 6-particle dispersion relations.

Explicit breaking of conformal symmetry
The low-energy expansion of the 4-dilaton amplitude is given by (3.18) and (3.19):