Conformal gauge-Yukawa theories away from four dimensions

Journal of High Energy Physics, Jul 2016

We present the phase diagram and associated fixed points for a wide class of Gauge-Yukawa theories in d = 4 + ϵ dimensions. The theories we investigate involve non-abelian gauge fields, fermions and scalars in the Veneziano-Witten limit. The analysis is performed in steps, we start with QCD d and then we add Yukawa interactions and scalars which we study at next-to- and next-to-next-to-leading order. Interacting infrared fixed points naturally emerge in dimensions lower than four while ultraviolet ones appear above four. We also analyse the stability of the scalar potential for the discovered fixed points. We argue for a very rich phase diagram in three dimensions while in dimensions higher than four certain Gauge-Yukawa theories are ultraviolet complete because of the emergence of an asymptotically safe fixed point.

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Conformal gauge-Yukawa theories away from four dimensions

Revised: June Conformal gauge-Yukawa theories away from four Alessandro Codello 1 Kasper Lang ble 1 Daniel F. Litim 1 2 Francesco Sannino 1 3 Brighton 1 QH U.K. 1 Renormalization Group 0 Origins, University of Southern Denmark 1 Campusvej 55 , Odense, DK-5230 Denmark 2 Department of Physics and Astronomy, University of Sussex 3 Danish Institute for Advanced Study, Danish IAS, University of Southern Denmark We present the phase diagram and associated xed points for a wide class of Field Theories in Higher Dimensions; Field Theories in Lower Dimensions - Gauge-Yukawa theories in d = 4 + dimensions. The theories we investigate involve nonabelian gauge elds, fermions and scalars in the Veneziano-Witten limit. The analysis is performed in steps, we start with QCDd and then we add Yukawa interactions and scalars which we study at next-to- and next-to-next-to-leading order. Interacting infrared xed points naturally emerge in dimensions lower than four while ultraviolet ones appear above four. We also analyse the stability of the scalar potential for the discovered xed points. We argue for a very rich phase diagram in three dimensions while in dimensions higher than four certain Gauge-Yukawa theories are ultraviolet complete because of the emergence of an asymptotically safe xed point. 1 Introduction 2 3 4 5 1 Introduction QCDd at leading order QCDd at next-to-leading order Next-to-leading order Next-to-next-to-leading order There has been recent interest in the conformal structure of gauge theories in d = 4 + dimensions with special attention to QEDd [1]. Reference [2] is a nice review of the dynamics and applications of several quantum eld theories in lower dimensions such as QED3, that has been further investigated in [3]. Of particular interest are applications to phase transitions with multiple order parameters such as con nement and chiral symmetry breaking [4, 5] at nonzero nite temperature and matter density. Analyses in d = 4 + revealed intriguing possibilities such as the possible occurrence of tetracritical-type phase transitions [6]. Related recent studies appeared in [7{9]. Away from four dimensions nonabelian gauge theories have been studied in [10{14], the non-linear sigma model in [15] and scalar QED3 in [16]. Similarly quantum gravity has been investigated in general dimensions in [17{19]. Our goal is to go beyond the present knowledge by simultaneously extending the perturbative analysis beyond the leading order and by including gauge-singlet scalar degrees of freedom. The latter are in the form of complex scalar Higgs matrices that are bifundamental with respect to the global symmetry group. In this initial investigation we work in the Veneziano-Witten limit of in nite number of avors and colors to neatly uncover the salient properties. In four dimensions a similar analysis has led to the discovery of the rst nonsupersymmetric class of asymptotically safe quantum eld theories [20]. Furthermore the quantum corrected potential was determined in [21] while basic thermodynamic properties were uncovered in [22]. The nonperturbative dynamics of supersymmetric cousin theories in four dimensions was analysed in [23], further generalising or correcting the results of [24]. Here { 1 { it was nonperturbatively shown that the supersymmetric versions of the theories investigated in [20] cannot be asymptotically safe. The discovery of asymptotically safe eld theories in four dimensions has far reaching consequences for model building [25{27]. The work is organized as follows. We brie y introduce the Lagrangian for a general class of Gauge-Yukawa theories in section 2 together with the Veneziano-Witten rescaled couplings. Here we also discuss some general features of the functions of this class of theories in d = 4 + dimensions. In section 3, as a stepping stone we consider rst the interesting case of QCDd. We then upgrade it to the full Gauge-Yukawa system in section 4. We will consider the full Gauge-Yukawa system at both the next-to-leading order (NLO) and next-to-next-to-leading order (NNLO). We o er our conclusions and general considerations in section 5. 2 Gauge-Yukawa theory template Our starting point is a general class of Gauge-Yukawa theories that, in four dimensions, features non-abelian gauge dynamics, gauged vector-like fermions and complex gauge-singlet self-interacting scalars interacting with the fermions via Yukawa interactions. Speci cally we consider an SU(NC ) gauge theory featuring gauge elds Aa with associated eld strength F a , NF Dirac fermions Qi (i = 1; ; NF ) transforming according to the fundamental representation of the gauge group (color-index muted), and an NF NF complex matrix scalar eld H uncharged under the gauge group. We take the fundamental action to be the same as in [20, 28], such that the Lagrangian is the sum of the following terms: 1 2 LYM = Tr F F LF = Tr Q iD= Q LY = y Tr QLHQR + QRHyQL LU = LV = u Tr (HyH)2 v (Tr HyH)2 : Tr is the trace over both color and avor indices, and the decomposition Q = QL + QR with QL=R = 12 (1 5)Q is understood. In four dimensions, the model has four classically marginal coupling constants given by the gauge coupling g, the Yukawa coupling y, the quartic scalar coupling u and the `double-trace' scalar coupling v, which we write as: g = h = g2 NC u NF (4 )d=2 (d=2) (4 )d=2 (d=2) ; ; y = v = y2 NC (4 )d=2 (d=2) v NF2 (4 )d=2 (d=2) ; : We have normalized the couplings with the appropriate powers of NC and NF so that the limit of in nite number of colors and avours with NF =NC a real nite number (Veneziano{ 2 { (2.1) (2.2) Witten limit) is well de ned. In dimensions di erent from four the couplings are dimensionful. We will denote by i with i = (g; y; h; v) the -functions for the dimensionless version of the couplings in (2.2), which we will still call i for simplicity. We express the functions in terms of = NF NC 11 2 ; which in Veneziano-Witten limit is a continuous parameter taking values in the interval [ 11 ; 1). 2 Since we perform the -expansion around four dimensions, we require the -functions to abide the Weyl consistency conditions [28{30]. This implies that, at the LO, one only xed point, as function of the gauge coupling. One can then substitute into the gauge function. To this order we can anticipate the generic form of the associated xed points by analysing the expected two-loop gauge-beta function in terms of the gauge coupling. The situation is more involved at the NNLO. At LO the gauge -function in d = 4 + does not depend on the Yukawa coupling and reads (2.3) (2.4) (2.5) (2.6) In four dimensions, i.e. = 0, and to this order the theory displays only a non-interacting xed point. For B > 0 it corresponds to asymptotic freedom and for B < 0 to an infrared free theory. Away from four dimensions, i.e. 6= 0, we have a Gaussian xed point (G) and a non-Gaussian xed point (NG) given by g = g B g2 : gG = 0 ; C > 0. When asymptotic freedom is lost, B < 0, the theory can have an asymptotically safe UV xed point at g = B=C for C < 0 [20]. Away from four dimensions, i.e. 6= 0, at the NLO we have at most three xed points: a Gaussian xed point (G) and two non-Gaussian { 3 { xed points (NG ) given by gG = 0 NG g = B p 2C = B 2C 1 r 1 4 C ! B2 ; (2.7) 4 C. In gure 1 we show the -function of (2.6) with and without the where = B2 linear term in the case B2 Gaussian when In d = 4, perturbativity of the non-Gaussian (either IR or UV) xed point is guaranteed for jBj 1 and jCj of order unity [31]. The situation changes when we go away from fourdimensions. By inspecting (2.7) we can consider di erent regimes. The rst is the one in which vanishes more rapidly than jBj2 (proportional to 2). This regime is a slight modi cation of the four dimensional case. To be able to extend our analysis to nite values (ideally achieving integer dimensions above and below four), we need B2 > jCj. For jCj and C of order unity, we have a perturbative xed point and a non-perturbative one at g ' B=C. Only when B=C > 0 the second is physical. The rst one is physical when and B have the same sign. Also, a negative corresponds to having an UV xed point for vanishing couplings. When both nonxed points are physical, they are separated by p =C. They will thus merge g ' =B xed point = 0. A near conformal behaviour is expected for very small negative values of since the -function almost crosses zero. This situation, however, is very di erent from the four-dimensional case (see [32] for a recent review) because: i) it arises away from four { 4 { sections. the full Gauge-Yukawa theory in the next sections. g = g + 2 g : dimensions; ii) it can appear already at two-loop level while it requires at least three loops in four dimensions [33, 34]. In the case of one coupling, there is only one scaling exponent which is simply the derivative of the -function at the xed point, = 0( g): G = NG = B 2C p We will discuss the scaling exponents when we have explicit forms for B, C in the next The term linear in coupling in d = 4 + g and proportional to appears when rendering dimensionless the dimensions via the replacement i ! i with the RG scale. In addition to the simple zero at the origin gG = 0, the beta function of (3.1) has a non-trivial zero at Positivity of g requires that below (above) four dimensions we must have > 0 ( < 0). In dimensions higher than four the non-interacting xed point is IR while the interacting one is UV safe. In dimensions lower than four the UV and IR roles are inverted. Di erently from the four-dimensional case perturbativity is guaranteed by a non-vanishing value of for any nonzero and small . Larger values of , implying going away from the asymptotic freedom boundary in four-dimensions, naively seem to allow larger values of that could potentially assume integer values.1 This is, however, unsupported by a careful analysis of higher order corrections. At each new order higher powers of appear. They are organised as shown in [34, 35] and therefore in the in nite limit the couplings must be properly rede ned. We shall not consider this limit here and we will analyze our results for nite values of . We will however further test the stability of our results by comparing LO, NLO and NNLO approximations when assuming integer values of . The LO scaling exponents are G = and NG = for either signs of , and do not depend on . Clearly for positive the interacting xed point is UV safe since the scaling exponent is negative indicating a relevant direction driving away from it. 1One could naively consider arbitrary large positive values of because we can have Nf Nc. On the other hand the smallest negative value of occurs for Nf = 0 corresponding to = 11=2. { 5 { 4 3 g = g + 25 + 3 g To this order a third zero of the beta function appears of the form anticipated in (2.7) with 11 for the lowest admissible value of 2 [0; 7256 ], both B and C are positive so that the interacting are positive in dimensions higher than four, i.e. with > 0. Otherwise, B and C have opposite signs, and one of the two couplings is unphysical being negative. Furthermore by tuning the interacting xed points disappear via a merging phenomenon. The condition for this to happen is obtained by setting = 0, or more explicitly 4 3 c 2 = 4 25 + 26 3 c : the merger occurs for c ' perturbation theory to hold. It is instructive to expand g in powers of This yields a critical c. For = 0 we nd c = 0, which is the point where asymptotic freedom is lost. In the general case, we nd c = 34 13 p (169 + 100) . In d = 5 2:551 with NG g ' 0:588, which is a too large value for NG g = NG 3 4 9(26 + 75) 64 2 + O( 3) : 0:1567 0:4269 1 1 1.0457 1:2421 22.892 5:1304 Adding the two loop contribution we arrive at the gauge -function of the form (2.6) In agreement with perturbation theory we nd that the leading order term in matches the LO case, while corrections appear to order 2 . We report the NLO scaling dimensions (2.8) in table 1. The full four loop investigation in four dimensions of the phase diagram of QCD like theories has been performed in [34]. It can serve as basis, in the future, to go beyond the NLO analysis away from four dimensions. 4 with LO-QCDd. At the NLO the Yukawa coupling starts playing an important role [20] { 6 { (3.3) (3.4) (3.5) a ecting the phase diagram of the theory. The quartic couplings h and v will be relevant at the NNLO discussed in section 4.2. y vanishes identically and that corresponds to the decoupled Yukawa limit with xed points given by NLO-QCDd; the second is the interacting one for which y( g) = 6 g the form given in (2.6). For gG = 0 we have the purely Yukawa interacting xed point at y YG = 13+2 . Since ranges in the interval [ 11 ; 1) the denominator is always positive. This implies that this 2 xed point exists only in dimensions lower than four. Dimensions higher than four, i.e. > 0, can be achieved for g > 6 . For < 0, the xed point satis es y > 13j+j2 . To be explicit, by inserting (4.2) into (4.1) we obtain the e ective gauge -function: ge = g + ( 4 3 + 2 11 . Furthermore when the gauge coupling crosses =6 the Yukawa coupling becomes negative. The crossing occurs for , which at NLO reads with = 3:66 for = 1. We have therefore chosen the numerical value of = for the phase diagram shown in gure 2. For small we have = 3(25 + 36) G ● NG HJEP07(216)8 0.2 To illustrate our results we now discuss the overall phase diagram for the limiting physical cases of ve and three dimensions. In gure 2 we present the NLO phase diagram for d = 5. For negative values of we observe three xed points, the Gaussian (G), the non-Gaussian (NG ) and the Yukawa non-Gaussian (YNG ) in the ( g; y)-plane. The rst xed point is IR attractive in both directions while the second is UV attractive in both directions. The third one has a relevant direction (with negative scaling exponent) and an irrelevant one. Along the relevant direction it constitutes an asymptotically safe xed point. This phase diagram is very similar to the one considered in [20] and points towards the possible existence of a fundamental ve dimensional Gauge-Yukawa theory. For the d = 3 case, the purely Yukawa xed point (YG) occurs on the y axis but now G is an UV xed point while YNG is an IR one. Both YG and NG are of mixed character with one of the two couplings vanishing. This is a safety-free situation similar to the one uncovered in four dimensions in [37]. In the IR they also both ow to the same theory, i.e. to the xed point YNG which is attractive from both directions. For either d = 3 or d = 5 the xed points NG+ and YNG+ occur for negative g and are therefore unphysical. the form Critical exponents. The linear RG ow in the vicinity of a given xed point assumes X j i = Mij ( j j ) + subleading ; (4.7) { 8 { NLO NNLO of the scaling exponents are meant to indicate the direction to which the stream line would ow in the deep infrared. All the scaling exponents go to 1 at NLO in the limit FP G YG NG YNG FP G NG YNG g 1 1 g 1 1 4:88. The name of the scaling exponents are meant to indicate the direction to which the stream line would ow in the deep infrared. universal numbers and characterize the scaling of couplings in the vicinity of the xed point. The scaling exponents in three and ve dimensions are listed respectively in tables 2 and 3. The values of the scaling exponents give quantitative meaning to the phase diagrams we presented in the previous paragraph. 4.2 Next-to-next-to-leading order In this section, we will consider the e ect of adding NNLO-terms [36] (3) = g (2) = y h ( 1 ) = 2 g y 20 701 6 6 + 53 3 ( 1 ) = 12 h2 + 4 v( v + 4 h + y) : v 1 4 (11 + 2 )2 g y + (11 + 2 )2(20 + 3 ) y2 385 8 + 23 2 + 2 2 2 y (44 + 8 ) y h + 4 h2 (4.8) Phase diagrams and scaling exponents. We now investigate the NNLO phase diagrams and explore the e ects of the scalar self-couplings. We will comment on the reliability of the NLO physical picture that emerged previously when assuming integer values of . { 9 { G ● NGNGαg in the ( g; y)-plane the xed points and the ow is qualitatively unchanged. But the new scalar directions a ects the nature of the xed points. In d = 3, as can be seen from table 2, only G remains a complete UV-trivial xed point while YG and NG acquire two relevant directions in the scalar self-couplings. Two relevant directions are added also to YNG . We thus conclude that in three dimensions scalar self-couplings increase the dimension of the critical surface of YNG and that the IR consequently reduces predictivity. The d = 5 dimension case is quite interesting since, as it is clear from table 3, both G and YNG don't change their character: the rst remains a complete IR xed point while the second displays complete asymptotic safety with only one relevant direction. NG adds two irrelevant directions to the previous two relevant ones. For the asymptotically safe xed point YNG one observes that the scaling exponents do not change much at the NNLO compared to the NLO case. The overall picture is that there is encouraging evidence for a ve dimensional complete asymptotically safe Gauge-Yukawa theory. Stability of the scalar potential. Here we will analyze the e ect of adding loop contributions to the beta functions of the quartic couplings. In order for the scalar potential to be stable in the Veneziano-Witten limit, we need [21] h > 0; ( h + v ) Solving the subsystem of the quartic couplings ful lling this constraint, we nd a required bound on the Yukawa coupling y 4 . At the equality, we have u = Hence, stability of the scalar potential puts a lower bound on the Yukawa coupling in dimensions lower than four. At the same time, it seems that in order to have small quartic couplings, we need negative. Combining this with the NLO Gauge-Yukawa system we predict for the pure Yukawa xed point in three dimensions that is in the range v = j8j q 121 + . - 3 at NNLO (solid orange). The dotted lines are the constant (blue) and linear (orange) terms form an -expansion of . For a value of above the line, the theory has a xed point with non-zero Yukawa coupling. The insert in the upper right corner shows the lower bound on g at NLO (4.2) in blue and NNLO (4.10) in orange. [ 4:5; 5:5] while the couplings are g = 0; y 0:25; u v 0:12. For the interacting xed point, the prediction is that is in the range [1:5; 4] while the couplings are g 0:7; y 0:25; u v 0:4. We see that naively maintaining the condition for four dimensional stability of the scalar potential in three dimensions requires non-perturbative values of the xed points. Furthermore new super cially relevant operators can occur in three dimensions, which might change the stability condition (4.9). We checked, via direct NNLO computation, that pushing for large negative values of is challenging from the scalar potential stability point of view. In ve dimensions, as we shall see, the system is less constrained resulting in a stable scalar potential at the asymptotically safe xed point. The critical lower bound g has changed into . In the analysis at NLO, we found that the gauge coupling crosses its =6 at the critical value (4.4). For the NNLO system this lower bound g = 18 p p 6 20 + 93 + 54 20 93 : (4.10) If we solve the gauge beta function with Yukawa and quartic couplings set to zero, we can nd a relation in ( ) for which the equality is exactly true. This curve is shown on gure 4 together with the corresponding lower bound on g (4.10). The critical value at NNLO evaluates to = 4:98 at = 1. In the phase diagram shown in gure 3 we use = 4:88. We can now set up an expansion around . In this way, we will develop a fully interacting asymptotically safe xed point YNG in 5D with perturbative control for the three couplings ( y; u; v). For a theory with + x, where x the Yukawa coupling emerges as linear in x, while the quartic couplings ( u; v) are subleading, x2 and x4. The coe cient of u is positive, while it is negative for v. Hence, the constraints (4.9) are satis ed. Composite operators. Because our xed points are perturbative in nature, composite operators have small anomalous dimensions at the xed points and therefore are not expected to a ect the dynamics. In four dimensions this is guaranteed by either direct computation of the anomalous dimensions of relevant scalar and fermion composite operators at xed points [20, 21] or via bootstrap constraints [38]. We expect the situation to change when considering large values of , when these operators can become relevant. In this limit, however one needs to resort to alternative computational methods and the related analysis goes beyond the scope of this work. 5 In this paper we have investigated Gauge-Yukawa theories in d = 4 + dimensions within the Veneziano-Witten limit and in perturbation theory. In d = 5 the consistency between the NLO and NNLO phase diagrams points to the existence of an asymptotically safe xed point. This result extends the discovery of asymptotic safe Gauge-Yukawa theories beyond four dimensions [20]. In three dimensions we found a rich phase diagram featuring, besides the UV noninteracting xed point, several xed points with the fully interacting one attractive in two directions but repulsive in the other two, rendering the theory less predictive in the deep infrared. In this work we have been concerned with determining the calculable xed point structure of gauge-Yukawa theories respecting the underlying SU(NF ) SU(NF ) U( 1 ) global symmetry. However, conformality can be lost for su ciently large values of the couplings. This can happen either at large values of or, for xed when changing the number of avours and colours of the theory. The theory can undergo either a scalar or fermion condensation, or both. Whichever condensation phenomenon occurs, we expect the global symmetry to break spontaneously. If the scalar condensation occurs rst it will also gap the fermions via Yukawa interactions leaving behind the gauge degrees of freedom. Also the intriguing dynamical mechanisms are expected to depend on whether we consider higher or lower than four space time dimensions and deserve an independent investigation. Our analysis constitutes a step forward towards a systematic study of the phase diagram of Gauge-Yukawa theories in several space-time dimensions. Acknowledgments The CP3-Origins centre is partially funded by the Danish National Research Foundation, grant number DNRF90. This work is also supported by the Science Technology and Facilities Council (STFC) [grant number ST/L000504/1], by the National Science Foundation under Grant No. PHYS-1066293, and by the hospitality of the Aspen Center for Physics. Open Access. 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Alessandro Codello, Kasper Langæble, Daniel F. Litim. Conformal gauge-Yukawa theories away from four dimensions, Journal of High Energy Physics, 2016, 118, DOI: 10.1007/JHEP07(2016)118