Dilaton and Massive Hadrons in a Conformal Phase

As the number of fermion fields is increased, gauge theories are expected to undergo a transition from a QCD-like phase, characterised by confinement and chiral symmetry breaking, to a conformal phase, where the theory becomes scale-invariant at large distances. In this paper, we discuss some properties of a third phase, where spontaneously broken conformal symmetry is characterised by its Goldstone boson, the dilaton. In this phase, which we refer to as conformal dilaton phase, the massless pole corresponding to the Goldstone boson guarantees that the conformal Ward identities are satisfied in the infrared despite the other hadrons carrying mass. In particular, using renormalisation group arguments in Euclidean space, we show that for massless quarks the trace of the energy momentum tensor vanishes on all physical states as a result of the fixed point. This implies the vanishing of the gluon condensate and suggests that the scale breaking is driven by the quark condensate which has implications for the cosmological constant. In addition form factors obey an exact constraint for every hadron and are thus suitable probes to identify this phase in the context of lattice Monte Carlo studies. For this purpose we examine how the system behaves under explicit symmetry breaking, via quark-mass and finite-volume deformations. The dilaton mass shows hyperscaling under mass deformation, viz. $m_{D} = {\cal O}(m_q^{1/(1+\gamma^*)})$. This provides another clean search pattern.


Introduction
It is well-known, since the seminal work of Ref. [1], that gauge theories in d = 4 show very different infrared (IR) behaviour depending on the matter representation, the number of flavours N f and colours N c . As the matter content is varied, these theories undergo a transition between a QCD-like phase where chiral symmetry is spontaneously broken, and hadron confinement takes place, and a phase where conformal symmetry is exhibited by the scaling of the correlation functions in the IR. 1 The latter phase is referred to as the "conformal window". Recent results are summarised in Ref. [4].
In this work we would like to investigate some properties of a third phase where conformal symmetry is spontaneously broken, leading to the appearance of a Goldstone boson (GB), the dilaton. 2 The dilaton has been widely studied in the literature as a candidate model for a composite version of the Higgs [6,7] with various effective Lagrangians [8][9][10][11][12], or as a driving field theory version force of inflation [13]. In this work we focus on the dilaton as the catalyst to the massive hadronic spectrum; indeed the massless pole corresponding to the dilaton allows for the conformal Ward identity (WI) to be satisfied even in the presence of massive states in the spectrum. In particular the trace of the energy momentum tensor (EMT) vanishes on physical states φ i , φ 2 |T µ µ (x)|φ 1 → 0 as shown in Sec. 3.3.

Axial and Dilatation Ward Identities
It is well-known that the pion decay constant F π is the order parameter of spontaneous chiral symmetry breaking. The dilaton decay constant F D plays the analogous role for the spontaneous breaking of dilatation or scale symmetry. It seems beneficial to treat them in parallel here. The decay constants are defined as 3 Γ (ab) 5µ (q) = 0|J a 5µ (0)|π b (q) = iF π q µ δ ab , where the Noether currents associated to the broken symmetries are respectively J a 5µ (x) = q(x)T a γ µ γ 5 q(x) and J D µ (x) = x ν T µν (x), where F π ≈ 92 MeV in QCD and T a is a generator of the broken axial flavour symmetry SU (N F ). The divergences of the currents are given by the explicit and anomalous symmetry breaking; using 4 ∂ · J a 5 (x) = 2m q P a (x) = 2m qq (x)T a iγ 5 q(x) , one obtains i −1 q µ Γ (ab) 5µ (0) = 2m q 0|P a (0)|π b (q) = F π m 2 π δ ab sym → 0 , (1.3) These equations vanish in the symmetry limit m q → 0. For the dilaton WI (1.4) this is not obvious as there is anomalous breaking of scale symmetry in addition. However in Sec. 3.3 we prove, using renormalisation group (RG) arguments in Euclidean space, that the equation holds. When expressed in terms of hadronic quantities, the divergences of the Noether currents are given by products of decay constants times masses, as shown on the right-hand side of Eqs. (1.3), (1.4); their vanishing occurs through m π,D → 0 as required 2 Throughout we will not distinguish conformal and scale (dilatation) invariance. It is widely believed that scale invariance implies conformal invariance in a wide class of theories in four dimensions, see e.g. Ref. [5] for a review. 3 The second equation below is consistent with 0|Tµν |D(q) = cf. also Eq. (1.4). 4 All our conventions are specified in App. A. The trace anomaly [14][15][16] contains further equation of motion terms which vanish on physical states and are not of interest to our work. particle non-Goldstone ( η F D ,π 1+γ * ≥ 1) Goldstone Table 1. Overview of how the important parameters entering the explicitly and anomalously broken Ward identities behave in the conformal dilaton phase. The scale Λ stands for a generic hadronic scale which in QCD is usually referred to as Λ QCD . The behaviour of m D,π and F D ,π (η F D ,π /(1 + γ * ) ≥ 1) under mass-deformation will be discussed in Sec. 3.1. The quantity γ * is the mass anomalous dimension at the IR fixed point. by the Goldstone nature of the pions and the dilaton. The decay constants are the order parameters and do not vanish. Heuristically one has SSB: the signal of spontaneous symmetry breaking (SSB), is equivalent to (1.1). 5 For the non-GBs, which we denote by π a and D , it is just the opposite, the WIs (1.3), (1.4) are satisfied by a zero decay constant as the hadronic masses are non-zero. For the D this is a subtle statement in view of the anomalous breaking of scale symmetry but in the end this is implied by the WI which holds for higher states cf. the remark above. An overview of the parametric behaviour in the conformal dilaton phase is given in Tab. 1 and the precise mass scalings are discussed in Sec. 3. Equipped with the broad picture we summarise the characteristics of the three phases before getting to the heart of the paper.

Overview of the Extended Conformal Window
Let us summarise the different phases of gauge theories. First, we know from the Banks-Zaks analysis [1] that there is a conformal phase for N f ≈ 16 and N c = 3 and probably well below. The range in N f before conformal symmetry is (dynamically) broken is known as the conformal window and its determination is the topic of ongoing efforts of continuum [18][19][20][21] and lattice Monte Carlo studies [22][23][24][25][26][27][28][29][30][31][32] (cf. [33] for a recent review). In N = 1 supersymmetric gauge theories this boundary is known exactly. Below the conformal window chiral symmetry is spontaneously broken and quark confinement takes place.
In particular this happens in QCD where N c = 3, N f = 3 (three light flavours) and quarks are in the fundamental representation of SU (N c ). What we are advertising here is that there might be a third phase embedded in the conformal window where conformal symmetry is spontaneously broken. It would seem reasonable to assume that this phase lives on the boundary of the conformal window as sketched in Fig. 1. 5 There is a subtlety with this argument in that the norm of the state created in Eq. (1.5) is proportional to square root of the spatial volume. This can be seen by considering the 2-point function of the currents and integrating over the spatial parts. A careful treatment for chiral symmetry can be found in Ref. [17]. The boundary with the QCD region is a matter of debate. The light-blue conformal dilaton phase is the one discussed in this paper. We wish to emphasise that this is just schematic and that the region of this phase could be rather different (should it exist at all). In this paper we discuss its logical possibility and speculate in Sec. 4.2 that QCD itself could be of this type.
The paper is organised as follows. In Sec. 2 we define the gravitational form factors and show how the dilaton restores the dilatational WI. In Sec. 3 matter mass and finite volume effects are discussed. Specific search strategies for the conformal dilaton phase with lattice Monte Carlo simulations are assembled in Sec. 4 along with a discussion on whether the dilaton could be the f 0 (500) or the Higgs in QCD or the electroweak sector. The paper ends with discussion and conclusions in Sec. 5. Apps. A, B deal with conventions and the spin-1/2 form factors.

Gravitational Form Factors of Spin-0
The gravitational form factors parametrise the matrix elements of the energy momentum tensor (EMT) between physical states; they can serve as quantum corrections to external gravitational fields [34], or as probes of the nucleon structure [35,36]. The spin-1/2 case is discussed in App. B and the spin-1, parameterised in [34], amounts to an interplay between F 1 and F 2 at zero momentum transfer. Here we focus on the spin-0 case since it illustrates all the important points without unnecessary complications. The dimensionless gravitational form factors for a generic scalar hadron, denoted by φ, are defined as follows where q ≡ p − p is the momentum transfer, P ≡ 1 2 (p + p ) and q µ T (φ) µν = 0, as required by translational invariance. This parameterisation is well suited for m φ = 0, that is the non-GB sector, which is the case we aim to examine. Further note that the limit q 2 → 0 of (2.1) is still well defined, despite the pole in G 2 , because for diagonal form factors the limit implies q µ → 0 at the same time. Since the EMT is related to the momentum, , by the usual conserved current procedure, the form factor G 1 must satisfy where we use the conventional state normalisation φ(p )|φ(p) = 2E p (2π) 3 δ (3) (p − p ). Note that (2.2) holds equally for massless hadrons (e.g. Goldstone bosons) such as the pion or the dilaton. The second structure is related to the improved energy momentum tensor which renders the free scalar field conformal in dimension other than two [37]. Everything in this section, up to now, was completely general. In the next section we discuss the conformal IR phases with particular emphasis on the dilaton case.

The Gravitational Form Factors in the Conformal Phase
In Sec. 3.3 we show that T (φ)µ µ (p, p ), as defined in (2.1), vanishes when there is an IR fixed point. This yields one constraint on the form factors for any spin and in particular for spin-0 this results in The most straightforward solution is the one of unbroken conformal symmetry for which m 2 φ = 0 leads to a trivial solution. This is the classic conformal window scenario. However, there is another possibility for m 2 φ = 0: the second term cancels the first one. In particular this implies G 2 (0) = 2/(d − 1), taking into account Eqs. (2.3) and (2.2). And this is where the dilaton pole and spontaneous breaking of scale symmetry come into play. In summary one has the three phases depicted in Fig. 1 6,7 Conformal Window: In order to avoid confusion it seems crucial to state that in this scenario the usual relation 2m 2 φ = φ|T µ µ |φ does not hold, cf. above, as it would either not allow for hadron masses or the dilation WI to be obeyed. 8 The possibility of such a scenario was mentioned prior to the discovery of the trace anomaly [41] but not worked out, for example in terms of hadronic parameters. The doing thereof is the topic of the next section. 6 The second relation can be seen as a cousin of the Goldberger-Treiman relation for the nucleons. The analogy is not strictly close as there the partially conserved axial current (PCAC) gives a non-vanishing term on the RHS (which though vanishes in the limit mq → 0). This results in gA = 1 + O(m 2 π ) ≈ 1.23 (e.g. [38]) and not an exact relation like G1(0) = 1.
7 This is the only, straightforward, logical possibility as the J P C = 0 ++ state does not contribute to the G1 form factor and a composite massless J P C = 2 ++ is forbidden by the Weinberg-Witten theorem [39]. 8 This implies that the gluon condensate definition [40], which departs from this relation, does hold in QCD-like but not in the conformal dilaton phase.

Verification of Dilatation Ward Identity at q 2 = 0 via the LSZ formalism
It is advantageous to represent the form factor G 2 in terms of a subtracted dispersion relation where 0 + indicates that the single dilaton has been removed from the integral. From (2.3) we infer the low energy theorem G 2 (0) = 2/(d − 1) which we are able to verify explicitly, using the LSZ formalism (e.g. [38,42,43]), as this point corresponds to the on-shell process φ → φD. The effective Lagrangian for the φ → φD process is L eff = g φφD To achieve our goal two steps are needed. First we need to determine g φφD in terms of other parameters and then we apply the LSZ reduction to extract Dφ|φ and match to (2.6). The g φφD coupling can be determined by writing an effective Lagrangian for the dilaton, where the field e D(x)/F D plays the role of a conformal compensator, see e.g. [44]. Namely, terms in the Lagrangian which scale like √ −gL → e −nα √ −gL under dilatations g µν → e −2α g µν can be made invariant by adding a prefactor e −nD/F D , where D → D − αF D under scale transformations. 9 Applied to the mass terms this gives the following appropriate effective Second the matrix element in (2.6) can be obtained in another way, directly from the form factor (2.1), by using the EMT as an interpolating operator of the dilaton. We are interested in the dilaton appearing in the (q µ q ν − q 2 η µν )-structure for which it is straightforward to write down a projector P 2 , such that P µν 2 (q µ q ν − q 2 η µν ) = 1 and P µν 2 P µ P ν = 0. The on-shell matrix element then follows from is, by footnote 3, the corresponding LSZ factor. Identifying the two equations one gets, using (2.7) Here gµν = ηµν and g = det(gµν ) denotes the determinant and therefore √ −g → e −dα √ −g under Weyl transformation. Note that our sign convention of FD is opposite as compated to some of the literature, e.g. [7], in order to preserve the analogy with the pion decay constant (1.1). Hence the change of sign in formulae with FD as compared to these works. In the case where the transformation parameter is chosen to be a local function one often refers to these transformations as Weyl scaling. The term below is Weyl invariant.
which satisfies (2.4) when G 1 (0) = 1 is taken into account. This matches (2.3) in the q 2 → 0 limit and thus shows that a dilaton phase seems a logical possibility indeed. The interplay of the dilaton residue and the vanishing of the trace of the EMT is an encouraging result.

Quark Mass-Deformation
We turn now to the question of how the hadronic quantities change when the quark mass is turned on. At a scale q 2 Λ, introduced in Tab. 1, all states except the dilaton and the pion decouple from the spectrum and we essentially have a conformal theory with a dilaton and pions. This situation is similar to the mass-deformed conformal window scenario extensively discussed in our previous papers [45][46][47], provided that m q Λ (as otherwise the quarks would decouple). The result that is sufficient for this section is that a matrix element of an operator O, of scaling dimension ∆ O = d O + γ O , between physical states φ 1,2 in the vicinity of the fixed point behaves like 10 where we have assumed zero momentum transfer (p = p ) for the time being. Above d O and γ O = − d d ln µ ln O stand for the engineering and anomalous dimensions respectively. The relation (3.1) has limited applicability in our case because of the presence of the additional scale Λ, a point we will return to in the next section. We can only apply it to the dilaton and the pion mass. Starting from (3.1) one can obtain a differential equation, using the trace anomaly, which leads to [46] Alternatively this result can be obtained following other techniques [45] which correspond to setting Λ = 0 in Sec. 3.2.
It is also of interest to investigate the scaling of F D ,π which can be done by using the dilaton WI (1.4) applied to D , π where the matrix element proportional to the β-function has been neglected, as it is subleading for m q ≈ 0. The statement η F D ,π m 2 D ,π /(1 + γ * ) ≥ 1 then follows from the assumption 10 In our previous work this was shown to hold on the lowest state in each channel, except for the masses where it was shown in generality [46]. However, our arguments at the end of Sec. 3.3 shows that it holds for all states.
that the matrix element 0|qq|D , π (q) is finite for m q → 0. Since m D = O(Λ) it then follows that These observations are interesting per se and complete Tab. 1 but we would like to understand how (2.3) is altered. We may use the same WI as above but applied to a diagonal matrix element and conclude The first scaling follows from the hyperscaling relation (3.1) and the second one, once more, from the assumption that the matrix element φ|qq|φ is finite as m q → 0. The correction to the form factor constraint (2.3) then follows from the correction to the onshell coupling (2.7) and leads to since G 1 (0) = 1 in general. Hence the scaling correction will come from the second term in (3.6) so that the RHS can match (3.5). An interesting question is how this changes when the momentum transfer is non-zero. We may assess this question by expanding in q 2 and demanding that the expansion converges which amounts to determine the scaling of the derivatives. First we note, cf. Sec. 3.3 for more details, that T (φ)µ µ (q 2 ) = 0 for m q → 0 and thus we may apply the RG analysis in Sec. 3 of our previous work [47] as applied to the pion form factor. We infer that where the first factor is just the previous result in (3.5) and Λ m sets the new scale. Note that the relative coefficients, unlike η T φ itself, of the form factor derivatives follows the straightforward hyperscaling law as they are not affected by the dynamical scale Λ to be assessed in the next section. Hence T In some sense the scale Λ m defines the deep IR for which the TEMT reveals its IR fixed-point in the presence of an explicit quark mass m q .

Scaling in the Presence of a Dynamical Scale Λ
Let us now revisit the RG scaling for field correlators in the case where scale invariance is spontaneously broken. We closely follow the derivations in our previous studies [46], allowing for the dependence on an extra scale Λ that is dynamically generated as a result of the spontaneous breaking. If it was not for the scale Λ one would directly conclude that η T φ = 2 and η F D m 2 D = 3. In this section we shall see why this conclusion does not hold in the presence of the dynamical scale Λ.
We consider both 2-point and 3-point functions in Euclidean space, which are defined respectively as where p 0 = iE p = i p 2 + m 2 φ , and with Φ an interpolating field for the particle φ. The theory is assumed to be defined in a finite volume of linear size L and at a scale µ, in the neighbourhood of a RG fixed point, located at δg =m q = 0. The couplings δg andm q are both dimensionless. If necessary, dimensionful couplings are rescaled by the appropriate powers of the scale µ. The spatial volume is V = L d−1 .
Using unitarity and usual RG scaling arguments -see e.g. our previous publications [45][46][47] for details -we obtain where we have used the identification (2π) (d−1) δ (d−1) (0) ↔ V and ∆ O and ∆ Φ are the scaling dimensions of the operators O and Φ. The quantities y g and y m ≡ 1 + γ * are the critical exponents that characterise the running of the couplings determined by the linearised RG equations in the vicinity of the fixed point. φ(p) is the lightest state in the spectrum with the same quantum numbers as Φ(x) and energy E p = p 2 + m 2 φ . The ellipses represent the contributions from excited states in the spectrum, which are exponentially suppressed.
The scaling formula for the 2-point function can be used as usual to derive the scaling of the masses of the hadronic states. Setting p = 0, and b ymm q = 1 yields (3.12) We may parameterise the large-t behaviour as where both functions, F and f , can and will overturn the hyperscaling behaviour found in (3.1) for masses and matrix elements. Specifically we may read off the behaviour of the φ-mass m φ ∝m 1/ym q F(m −1/ym q Λ) . (3.14) We are interested in the scaling of the masses as the fermion mass m q → 0, which corresponds to m −1/ym q Λ → ∞. We can then distinguish two different regimes where κ is a constant and the first case is an alternative derivation of the mass scaling quoted earlier. The first regime corresponds to the conformal scaling already discussed in our previous study [46]. Interestingly, the second regime yields the scaling with Λ that is expected in the theory with spontaneously broken symmetry and a dilaton. We further note that, using arguments about the finiteness of matrix elements in the m q → 0 limit (as done in Sec. 3.1), it may be possible to make further statements about the function f (x, µ) as x → ∞. We refrain from doing so as it does not add anything to the key messages of this paper.

Dilaton Ward Identity in the Vicinity of the IR Fixed Point
A similar analysis for the 3-point function allows us to derive a crucial result for the WI in the neighbourhood of a fixed point. Once again we start from the RG equation, Combining these expressions, we obtain the matrix elements from taking the large-time limits of correlators. In particular, we have and similarly (3.20) Eqs. (3.17) and (3.18) are the master formulae needed in order to understand the IR behaviour of the dilatation WI and the scaling of finite-volume effects. From these formulae one infers that evaluating the correlation functions at infinite time separation is the same as evaluating them at finite time with other dimensionful parameters appropriately rescaled. Now, taking the infinite-volume limit first, we are able to show that the on-shell WIs are insensitive to the anomalous breaking in the presence of an IR fixed point, provided that the explicit breaking of scale invariance due to the mass is tuned to zero. In order to prove this statement, we are going to consider the anomalous contribution in Eq. (1.2) due to the gauge field, form q = 0, namely This matrix element can be obtained from the large-t behaviour of the correlator and Φ D is a generic interpolating operator that has an overlap with the dilaton field but not the vacuum (e.g. Φ D → Φ D − Φ D |0 0| is a realisation thereof). Starting from the infinite-volume theory and setting L −1 = 0, we obtain from Eq. (3.17) Note that we need to keep a finite, non-vanishing mass, or a non-vanishing spatial momentum, in order to guarantee the exponential fall-off of the correlator. We see from the expression above that in the neighbourhood of an IR fixed point, the coupling g is irrelevant (that is the critical exponent is negative y g < 0). Assuming that the matrix element of G 2 does not diverge in the IR, the matrix element therefore vanishes Above g * is the value of the coupling at the IR fixed point. Eq. (3.23) shows that the anomalous breaking does not contribute to the WI between the vacuum and the dilaton state. The only assumption needed is that the gluonic matrix element remains finite when µ → 0. The order of the limits is relevant here: the mass of the matter fields guarantees that the dilaton is massive and its correlators decay exponentially, then in the large-time limit the contribution from the running of the gauge coupling vanishes. A similar argument applied to the 3-point functions shows that the matrix element of the anomalous breaking term between two one-particle states also vanishes in the presence of an IR fixed point. Note that these statements are true not only for the lowest state as one may choose an interpolating operator which has no overlap with the lowest state. This is particularly clear in the finite volume formulation where the fields can be represented in form of a discrete spectral sum. In summary we thus have that 11 in the m q → 0 limit where φ 1,2 are any physical states and the equation also holds for the vacuum expectation value if Φ is chosen to have overlap with the vacuum. Colloquially speaking, the physical matrix elements "see" the TEMT at large distances and since there is an IR fixed point this means effectively that T µ µ → 0 between physical states. This is an important result of our paper and in agreement with statements found in [7]. In order to delimit this result we stress that correlation functions with T µ µ -insertions are generically non-vanishing. For example in the context of the flow theorems they constitute the main observables [48][49][50]. It seems worthwhile to clarify that the TEMT does not need to vanish on quark and gluon external states since they are not (asymptotic) physical states even in the absence of confinement. This is the case since quarks and gluons can emit soft coloured gluons and thus colour is not a good asymptotic quantum number. Another aspect, that has the same root, is that quark and gluons correlation functions can have unphysical singularities on the first sheet.

Finite Volume Scaling
Finally, by keeping the size of the system L finite, the solutions of the RG equations presented above allow us to quantify the scaling of the correlators in finite (but sufficiently large) volumes. Choosing a reference scale L 0 and setting b = L/L 0 , we obtain for the 2-point function (3.9) This equation allows us to derive the scaling of the energy and of the matrix elements with the size of the lattice; ignoring the contribution of the irrelevant coupling, we obtain as already discussed in our previous studies [45]. It is interesting to emphasise that in a finite volume the anomalous contribution to the WI from the irrelevant coupling is proportional to L L 0 yg . Hence, the finite volume explicitly breaks the scale symmetry by 11 Since the trace of the EMT is a RG invariant this implies 0|G 2 (µ)|0 = 0, 0|G 2 (µ)|φ1(p) = 0 and φ2(p )|G 2 (µ)|φ1(p) = 0 for any scale µ > 0. By continuity it is then also implied for µ = 0. This implies that the gluon condensate is not the operator that breaks the dilatation symmetry spontaneously.
acting as an IR regulator and this is reflected in the WI for O = T µ µ . Once again, because y g < 0, the breaking term vanishes when L → ∞, which is consistent with the fact that SSB cannot occur in a finite volume, as otherwise tunnelling rates prohibit SSB [51].

Conformal Dilaton Signatures
In Sec. 4.1 we discuss concretely how the conformal dilaton phase can be searched for on the lattice and in Sec. 4.2 we comment on the ideas that the f 0 (500) in QCD and the Higgs could be dilatons from the perspective of this paper and the newly obtained scaling formula for its mass.

Lattice Monte Carlo Simulations
In order to discriminate a conformal dilaton phase from the QCD or unbroken conformal phase we propose the following two strategies. We would think that the first test is more spectacular but it might be more costly as the form factor necessitates 3-point functions whereas masses (and decay constants) can be extracted from 2-point functions.

The
Higgs and the f 0 (500) as Pseudo-Dilatons In this section we briefly discuss whether the Higgs or the f 0 (500) are (pseudo)-dilatons in the electroweak and the QCD sector respectively. Our work is distinct from other approaches in the scaling formula for the dilaton (4.1) and we mainly focus on this aspect. We would like to stress that the 0|T µ µ |0 = 0 for m q = 0, in the context of an IR fixed point, is of course of interest to the cosmological constant problem. Moreover, if masses are added for quarks and techniquarks, they would decouple in the deep IR. The question of IR conformality is then shifted to the pure Yang-Mills sector. Whereas lattice studies indicate that pure Yang-Mills is confining, it is, to the best of our knowledge, an open question whether these theories show an IR fixed point not. If this was the case then the Higgs sector and QCD would give a vanishing contribution to the cosmological constant.
• The Higgs boson could in principle be a dilaton e.g. [7] as it couples to mass via the compensator mechanism (2.7). At leading order in the low energy effective theory it is equivalent to the coupling of the Higgs. The basic idea is similar to technicolor (cf. [52,53] for reviews) in that a new gauge group is added with techniquarks q which are in addition coupled to the weak force such that the techniquark condensate breaks electroweak symmetry spontaneously, this usually implies F π = v ≈ 246 GeV. Whereas technicolor would be classed as a Higgsless theory the same is not true in the dilaton case as it takes on the role of the Higgs. Unlike technicolor the generation of fermion mass terms is not aimed to be explained dynamically.
In our scenario the dilaton is a true GB in the m q → 0 limit and acquires its mass by explicit symmetry breaking where Λ is the hadronic scale of the new gauge sector. Eq. (4.2) suggests that a mass gap between m D and Λ can be reached by making m q small. Whether or not Λ can be sufficiently large, 12 in order to avoid electroweak and LHC constraints, is another question and beyond the scope of this paper. The crawling technicolor scenario in [7], based on the dilaton, is different in that the techniquarks are assumed to be massless and the dilaton/Higgs acquires its mass by the hypothesis that the IR fixed point is not (quite) reached. According to [7] the dilaton mass is then governed and made small by the derivative of the beta function.
• In this work, cf. Fig. 1, we have distinguished the conformal dilaton phase from QCD but one might ask the question whether they are one and the same. Could it be that the so called f 0 (500) (cf. [54] for a generic review on this particle), with pole on the second sheet at m f 0 = 449(20) − i 275 (12) MeV [55], is a dilaton with mass m f 0 (500) ∝ m 1/(1+γ * ) q ? The first thing to note is that if one assumes the Gell-Mann Oakes Renner relation, F 2 π m 2 π = −2m q qq (e.g. [38]) and qq = O(Λ 3 ) then the mass scaling relation (3.2) implies γ * = 1. This is a logical possibility that is deserving of further studies. In QCD, where m s m u,d , it is not immediate how to apply the mass scaling relation (3.2). The f 0 (500) surely has a strange quark component and its mass scale can be considered to be of O(m K ). This is the case in terms of the actual masses and in scale chiral perturbation theory [56][57][58]. For more details about this EFT approach we refer the reader to a series of works by Crewther and Tunstall [56][57][58]. Our approach is though different in that we consider 12 By large, we mean larger than the naive estimate Λ ≈ 4πFπ = 4πv ≈ 3 TeV. the gluonic part proportional to the β-function as subleading. Setting this aside, there are interesting consequences for K → ππ and the famous ∆I = 1 2 rule. Such a scenario is also welcomed in dense nuclear interactions, combined with hidden local symmetry [59,60].

Discussions and Conclusions
In this paper we have analysed the possibility of a conformal dilaton phase, in addition to the QCD and conformal phase (cf. Tab. 2 for comparison), where hadrons carry mass but the theory is IR conformal. The mechanism whereby this can happen is that conformal symmetry is spontaneously broken and it is the corresponding Goldstone boson, the dilaton, that restores the dilatation Ward identity (2.3). More generally, we have shown, using renormalisation group arguments in Euclidean space that the trace of the EMT vanishes on all physical states (3.24). This implies the vanishing of the gluon condensates and suggests that the scale breaking is driven by the quark condensate (cf. footnote 11). This is an important result of our paper with consequences. For example, it imposes an exact constraint on the gravitational form factors (2.3). At zero momentum transfer we have shown that this constraint is satisfied, (2.9), using the effective Lagrangian (2.7). As far as we are aware this is a new result.  Table 2. Comparison table between the three different phases discussed in this paper. GB stands for Goldstone boson. The states in the second column are physical states and zero quark mass is assumed. The columns three to five indicate the scaling when a quark mass is turned on. Above O had is a generic hadronic observable and the conformal window scaling law has been discussed in (3.1). and the masses and the decay constant for the Goldstone boson singled out in the least two columns. The notation is the as in Tab. 1 where further information on excited states in the conformal dilaton phase can be found.
Such phases can be searched for in lattice Monte Carlo simulations for which we have proposed concrete signatures in Sec. 4.1. First, the test of the exact constraint on the gravitational form factor at zero momentum transfer in (2.9) and the scaling of the dilaton mass (4.1), as compared to all other ones. It will be interesting to see whether this new perspective can resolve some of the debates in the lattice conformal window literature.
Moreover we have speculated in Sec. 4.2 whether a pseudo-dilaton is present in QCD and or the electroweak sector in terms of the f 0 (500) and the Higgs boson. If the former were true then this would suggest that the conformal dilaton-and the QCD-phase are one and the same. 13 In our view the study of whether the Higgs is a composite dilaton is deserving of further attention also because it has the potential to ameliorate the cosmological constant problem as emphasised earlier.
Finally we wish to comment on the different (pseudo) Goldstone bosons, the pions, the η and the dilaton associated with breaking of the SU A (N f ), the U A (1) and the dilatation symmetry. In all three cases the quark masses are a form of explicit symmetry breaking. This is manifested by the corresponding WIs (1.2) and ∂ · J 5 = 2m q P + g 2 16π 2 GG for the U A (1)-case. The axial non-singlet case stands out in that there is no anomalous piece and it is indeed the case, as well-known, that in the m q → 0 limit the pions become true Goldstone bosons. In the axial singlet case the anomalous piece does contribute to the large η mass which constitutes the resolution to the U A (1)-problem [62]). It is noted that the η becomes a Goldstone boson in the N c → ∞ limit as the anomalous term is 1/N c suppressed. On the other hand, according to our analysis, the anomalous breaking of the scale symmetry does not affect the dilaton mass in the m q → 0 limit and it thus is, remarkably, a genuine Goldstone boson.