Light Dilaton at Fixed Points and Ultra Light Scale Super Yang Mills

We investigate the infrared dynamics of a nonsupersymmetric SU(X) gauge theory featuring an adjoint fermion, Nf Dirac flavors and an Higgs-like complex Nf x Nf scalar which is a gauge singlet. We first establish the existence of an infrared stable perturbative fixed point and then investigate the spectrum near this point. We demonstrate that this theory naturally features a light scalar degree of freedom to be identified with the dilaton and elucidate its physical properties. We compute the spectrum and demonstrate that at low energy the nonperturbative part of the spectrum of the theory is the one of pure supersymmetric Yang-Mills. We can therefore determine the exact nonperturbative fermion condensate and deduce relevant properties of the nonperturbative spectrum of the theory. We also show that the intrinsic scale of super Yang-Mills is exponentially smaller than the scale associated to the breaking of conformal and chiral symmetry of the theory.


I. INTRODUCTION
Understanding strong dynamics constitutes a continuous challenge. One facet of strong dynamics which is attracting much interest is the phase diagram of strongly interacting nonsupersymmetric gauge theories as a function of the number of flavors, colors and matter representations. The investigation of the phase diagram for nonsupersymmetric gauge theories, with-out fundamental scalars, for any matter representation, and several gauge groups, started in [1][2][3] and it was further investigated in [4][5][6][7][8][9][10][11][12][13]. Besides the possibility of infrared fixed points recently it has been discovered that nonsupersymmetric gauge theories with fermionic matter at large number of flavors develop ultraviolet fixed points [10]. Furthermore, as soon as it was discovered that theories with an extremely low number of matter flavors were able to lead to large distance conformality and potentially become excellent candidates of models of dynamical electroweak symmetry breaking, the question whether these theories could feature light composite scalars was addressed in [14][15][16]. These papers rekindled the long debate on the existence of a light scalar compared to the intrinsic scale of the theory. Their phenomenological relevance resides on the identification of this state with the composite Higgs in models of dynamical electroweak symmetry breaking featuring near conformal dynamics or the inflaton in models of successful minimal composite conformal inflation [17].
A variety of arguments has been used in favor of the existence of a light scalar, not associated to Goldstone bosons linked to the spontaneous breaking of global symmetries. Furthermore, there has been used a number of methods ranging from the use of the saturation of the trace and axial anomaly [18] to supersymmetry [14,[19][20][21], alternative large N limits [16,22] as well as gauge-gauge duality [23][24][25][26]. Other interesting attempts to investigate the dynamics associated to light scalars in near conformal field theories appeared in the literature [27][28][29][30].
The suggestions of potential light dilatons for theories with near conformal invariance resided mostly on nonperturbative analysis and the use of critical behavior which should be backed up by either lattice or exact analytical computations. Here we investigate the perturbative and nonperturbative dynamics of a particular gauge theory, similar to the one investigated in [26], which allows us to clearly identify the dilaton and determine its properties using perturbation theory and even to be able to determine exactly some nonperturbative quantities arising at low energies. Specifically, the theory we investigate here is an SU(X) gauge theory with N f Dirac massless flavors, one adjoint Weyl fermion and a complex scalar singlet with respect to the gauge interactions, but bifundamental with respect to In Section II, as a partial motivation for our new model and investigation, we provide a critical review of an analysis similar to ours [31], but for a different gauge theory. We introduce the model in Section III, where we also establish the existence of interacting infrared fixed points in perturbation theory. Then in Section IV, we determine the perturbative spectrum of states due to the Coleman-Weinberg (CW) phenomenon [32,33]. Here we provide the dilaton mass and identify the relevant physical state coupled to the dilatation current. We determine the region of stability of the CW potential and its geometric interpretation in Section V. We then discover that at energies much lower than the vacuum expectation value of the Higgs, the low energy theory is pure N = 1 SYM in Section VI. Here, we also show that the intrinsic renormalization group invariant scale of low energy SYM is exponentially smaller than the Higgs vacuum expectation value. We determine the gluino condensate as a function of this scale.
In the Appendix we provide the explicit com-putations for the CW potential and conclude in Section VII.

MODEL
An analysis similar to the one performed here but for an entirely different theory has been done by Grinstein and Uttuyarat (GU) [31].
This pioneering work has triggered our interest in further exploring near conformal dynamics in perturbative regimes. To better appreciate the model we will investigate in the next sections we start by commenting the large N and number of flavors limits of the GU microscopic theory: where the fermions ψ, χ transform according to the fundamental representation of the SU(N) gauge group and carry flavor index j = 1, . . . , N f /2. The scalar fields φ 1 , φ 2 are real scalars and are gauge and flavor singlets.
The investigation of Banks-Zaks infrared fixed points requires that use of perturbation theory. This is achieved by arranging the value N f /N to be proportional to (1 − ) with a small expansion parameter around the asymptotically free boundary. However, to make the parameter arbitrary small and continuous requires a well-defined large N limit of the theory. This is achieved by rescaling the couplings as follows: Using these opportunely rescaled couplings, the GU fixed point [31] reads with j = 1, 2, 3 and x = N f /N. It is evident, in these variables, that the nontrivial infrared fixed point in the quartic couplings λ j is lost in the large N limit. In order to claim the existence of such a fixed point in λ j one therefore must keep 1/N corrections. However, the order of the limits become important when taking this route. In fact, consider first the exact large N (and therefore large N f ) limit. The infrared fixed point does not lead to quartic self-interactions and therefore one cannot observe chiral symmetry breaking induced by loop corrections [31].
For N large but finite, needed for having a nonzero λ * j , the counting in the loop expansion and the 1/N needs to be addressed carefully given that terms such as:

III. THE THEORY AND ITS FIXED POINTS
The gauge theory we investigate is, with the field content reported in Table I. Here of the coupling constants g, y H , u 1 , u 2 were derived in [26,36]. Possible fixed points will be perturbative for N f near 9 2 X, where the first coefficient of the gauge coupling beta function vanishes. It is therefore most convenient to fix X and define the small expansion parameter  [34,35], the complex scalar fields H ij were not canonically normalized, why some results will deviate from those papers by numerical factors.
through x ≡ N f /X = 9 2 (1 − ). As in any Banks-Zaks analysis and x are considered continuous throughout the analysis. It is convenient to work with rescaled coupling constants that do not scale with X and N f . These read Then dropping 1/X-terms the perturbative beta functions are given purely in terms of the perturbative rescaled couplings and the small pa- In this form, the beta functions are free of any explicit X and N f dependency. This is an important feature which assures the smallness of to be arbitrary in all our results. Notice that the quartic couplings do not contribute to the running of the gauge and Yukawa couplings to this level in perturbation theory. To leading order in , the system of RGEs has two real fixed points (FPs): , a H = 2 3 (7) The FP corresponding to the upper sign for z 1 , in the equation above, is all-directions (infrared) stable while the other FP has one unstable direction coinciding with the z 1 -axis. This is illustrated in Fig. 1 where we have kept gauge and Yukawa couplings at their FP values in Eq.
(7) for = 0.1. There is another solution with which however leads to complex values of z 1 and therefore is discarded. It is important to observe that the interacting FPs disappear if the adjoint fermion λ m is removed from the spectrum, since then the coefficients of β(a g ) will change. In particular, the perturbative regime would be moved to x ≈ 11 2 leading to a noninteracting FP as will be shown explicitly in Section VI.

IV. LIGHTEST SCALAR AS THE DILATON
In this section we will derive the scalar spectrum. We follow [37] and summarize below the key points: 1) If the one-loop effective potential V eff = V 0 + V 1 can be calculated at some renormalization scale M 0 for which the tree-level term V 0 vanishes and is a minimum then one-loop perturbation theory can be used to show that V 1 causes V eff to be negative and stationary. Therefore the global symmetries of the theory will be broken via the CW potential.
2) The theory possesses, at least, one real light scalar corresponding to the field in the direction in scalar field space along which the potential develops the ground state. It arises because V 1 breaks scale invariance of V 0 along this potential direction in field space. The mass of this state, to be identified with the dilaton, is given by with v as defined below 2 .
It was shown in Ref. [35] that in the fermions, the CW induced chiral symmetry breaking occurs when either condition A or B below applies: where the superscript 0 refers to the quantities evaluated at the renormalization scale M 0 .
2 Quantum effects induce scalar masses of the order of the cut-off. Following the CW analysis, however we subtract these masses away [32]. As for [31] we are not solving the hierarchy problem.
In the large X (and thus large N f ) limit only case A exists. We therefore restrict our analysis to case A. In this case the vacuum expectation value (vev) of H will be given by with v a real constant, while the factor 1/ 2N f ensures the correct normalization of the field in the δ i j direction. This expectation value breaks chiral symmetry to the diagonal subgroup, i.e.
We now provide, see the Appendix for further details, the scaling of v with N f [37]. The effective potential in the δ i j direction at the renormal- where Φ is canonically normalized such that at the minimum of the effective potential Φ = v.

This value is
Thus, all physical masses do not scale with The right-hand side of this expression is easily computed and one finds, up to terms that vanish via the equations of motion 3 The dilaton mass m D is defined by the matrix where f D is the dilaton decay constant.
The pseudo-Goldstone boson of case A is parametrized by: with φ a real scalar field. Expanding H on its mass eigenstates around the vacuum of Eq. (9) we have: where π i j parametrizes the N 2 f Goldstone bosons, and h i j parametrizes the N 2 f − 1 massive eigenstates. It follows from orthogonality of this mass eigenbasis that π i j must be hermitian and h i j must be hermitian and traceless. Then it is easy to check that only the field φ contributes linearly to the trace anomaly, i.e.
where the ellipses stand for terms that contribute to the matrix element beyond tree-level.
From this, we realize that the dilaton state |D  3 In general, the trace anomaly involves wave function renormalization as well, however, as noted in [31], these where it is clear that we can identify the mass terms vanish by using the equations of motion.
of Eq. (10) with the dilaton mass, i.e. m 2 d = m 2 D , and f d = v.
We discuss the remaining spectrum associated to the fermion in the adjoint representation, which remains massless, in a dedicated section where we use the power of supersymmetry to induce interesting features.

V. GEOMETRICAL PICTURE
To gain more insight into the dilaton mass derived above we now turn to the geometrical interpretation of the CW induced breaking of chiral symmetry using the method of Ref. [33].
To the one-loop order the adjoint fermion λ m does not contribute to the effective potential, and we can adopt the results of Ref. [36], where the method of Ref. [33] was applied to the model without λ m . The conditions for a minimum of the effective potential using the RG improved CW potential are (see also the Appendix): Eq. (17a) is a necessary condition for the min- The RG flow between the FPs does not generate a stable vacuum. This is so due to the fact that the flow runs parallel to the z 1 -axis and therefore the stability line is not crossed inside the shaded region. In this case the effective potential is not bounded from below. In Fig. 2 we kept gauge and Yukawa couplings at their FP.
All three curves (thick black, elliptic and dashed orange) intersect at the point: trivially massless. The adjoint fermion λ m is important to achieve a nontrivial dynamics. Furthermore this model was also motivated by the recent gauge-gauge duality proposal [25,26].
We now turn to the low energy SYM nonperturbative properties of the theory. We can immediately determine exactly the renormalization group invariant scale of SYM indicated with Λ SYM which is [40]: Moreover: The low energy spectrum of SYM is constituted by a chiral superfield featuring a complex scalar This is highly consistent with having assumed no contribution from the gluonic condensate to the trace of the energy momentum tensor.

VII. CONCLUSIONS
We introduced a computable model featur-

Appendix A: Effective Potential
In this appendix we clarify the origin of some of the expressions used in the main text.
The Coleman-Weinberg (CW) effective potential is defined by expanding the effective action around the classical field configuration φ c , which in our case is: The effective potential obeys the renormalization group equation (RGE) where the coefficients β and γ are respectively the beta function of the couplings and anomalous dimension of the scalar field H, i.e.
with Z H the scalar wave function renormalization constant. Following Coleman and E. Wein-berg [32] it is convenient to use the dimensionless four point function: For a generic n point function the RGE is: which specialized to V (4) reads: Using the fact that V (4) is dimensionless it can only depend on the couplings and the dimensionless variable: Making use of the further relation the RGE for V (4) reads: whereβ ≡ β A useful renormalization condition is the Coleman and E. Weinberg one: Then the leading log solution of the RGE reads: We say that this is the renormalization group improved tree-level effective potential. From this expression, it is now easy to derive the conditions in Eq. (17) for a non-trivial minimum of the classical field away from the origin [33,41].
Note that this expression is consistent with a one-loop analysis of the running of the coupling constants u 1 and u 2 .
To understand better the relation between the effective potential and the RG equations of the couplings and their relation to the dilaton mass, it is more appropriate to consider Gildener and S. Weinberg's approach to the CW potential [37].
One chooses the renormalization scale M in such a way that the tree level potential vanishes at this scale, i.e.
We are, therefore, expanding the potential perturbatively around the non-trivial vacuum of the classical field. We will be focusing on the case where u 2 > 0. In Ref. [35] was shown that H will have a minimum along its diagonal direction, i.e.
Then, it follows that the one-loop effective dimensionless potential evaluated for this value of H reads: