Dual RG flows in 4D

IPPP/18/35 CP3-Origins-2018-019 DNRF90 Dual RG flows in 4D Steven Abela,1 , Borut Bajcb,2 and Francesco Sanninoc,3 a arXiv:1805.07611v2 [hep-th] 6 Mar 2019 b IPPP, Durham University, South Road, Durham, DH1 3LE b J. Stefan Institute, 1000 Ljubljana, Slovenia 3 CP -Origins & the Danish IAS, University of Southern Denmark, Denmark Abstract We present a prescription for using the a central charge to determine the flow of a strongly coupled supersymmetric theory from its weakly coupled dual. The approach is based on the equivalence of the scale-dependent a-parameter derived from the fourdilaton amplitude with the a-parameter determined from the Lagrange multiplier method with scale-dependent R-charges. We explicitly demonstrate this equivalence for massive free N = 1 superfields and for weakly coupled SQCD. 1 [email protected] [email protected] 3 [email protected] 2 1 Contents 1 Introduction 2 2 Dilaton scattering a versus Lagrange multiplier a 4 3 A perturbative calculation of a non-perturbative 3.1 UV (0 < t < ∞): magnetic theory . . . . . . . . . 3.2 UV (0 < t < ∞): electric theory . . . . . . . . . . 3.3 IR (−∞ < t < 0): magnetic theory . . . . . . . . 3.4 IR (−∞ < t < 0): electric theory . . . . . . . . . flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10 11 11 13 4 Equality of critical exponents 15 5 Conclusion 17 A The Lagrange multipliers 17 1 Introduction Renormalization Group (RG) flow of Quantum Field Theories (QFTs) is thought to be irreversible. In two dimensions this irreversibility is encompassed by the Zamalodchikov c-theorem, which states that one can define a monotonically decreasing parameter that interpolates between the central charges c [1] of two conformal theories related by an RG flow. An equivalent parameter in 4 dimensions is Cardy’s proposal of the a anomaly, the coefficient of the Euler density in the trace of the energy momentum tensor [2]. In a remarkable paper [3], Komargodski and Schwimmer (KS) produced a general form for this coefficient to show that its value will inevitably decrease if a system goes from a UV to an IR fixed point. The method that was used in [3] is a cousin of ’t Hooft anomaly matching, in the sense that a spectator dilaton field is introduced that compensates the anomaly and restores exact Weyl symmetry at all scales, which is spontaneously broken by a dilaton VEV. Using such a set-up the a parameter can be deduced from the 4 dilaton amplitude. The change in the a parameter between fixed points, aIR − aU V , is then found to be always negative by relating it via the optical theorem to the cross-section. Thus the weak form of the a-theorem, that its value will decrease if a system flows from a UV fixed point to an IR one, can be considered proven. However the strong version, namely that there exists a monotonically decreasing a function with unambiguous physical meaning all along the flow, appears to be still open because of the presence of scheme dependent β 2 terms in the four dilaton amplitude, as discussed in [4–6]. Indeed Jack and Osborn [7–9] showed the existence of a function â related to a through the beta functions, that coincides with it at fixed points and that flows with energy scale µ as dâ µ = χIJ β J β I , (1.1) dµ where β I are the beta functions of couplings λI , and χIJ is a metric on the space of couplings. The problem of proving the monotonicity of the function â (and hence the irreversibility of RG flow) is then reduced to one of proving the positive-definiteness of the metric χ on the space of functions. This problem remains to be solved (for a review see for example [10]). 2 Our purpose here is to point out that the parametric closeness of a and â suggests a method of tracking the approximate flow of a strongly coupled theory. Indeed generally, for flow between nearby fixed points, it seems natural to attempt a perturbative expansion in terms of the beta functions rather than in terms of any couplings [11]. In this letter we explore the a-parameter as the basis for such an approach, showing how one can use it to follow the flow of arbitrarily strongly coupled SQCD theories between fixed points. Central to this approach is of course the fact that it is already known how to map the particle content of strongly coupled “electric” SQCD theories to weakly coupled “magnetic” ones via Seiberg duality [12,13]. Thus one can already determine all the discrete parameters of strongly coupled theories, as well as much of their holomorphic data, even when they are away from fixed points. The question we will address here is how one can also determine the flow of the coupling in the strong theory, up to the aforementioned corrections of order β 2 , by mapping from the weak theory. The approach continues in the spirit of ’t Hooft anomaly matching, by considering the flow of the a parameter. In order to define such a flow we will use the KS determination of a which involves a certain integration of the 4-dilaton amplitude over the Mandelstam variable [14]: Z ImA(s) f4 aU V − aKS (µ) = ds , (1.2) 4π s>µ2 s3 where f is the dilaton decay constant and s is the Mandelstam variable, and where we impose an IR cut-off on the integral, s > µ2 , in order to generate a running a-parameter, which we denote aKS (µ). The cut-off induced scale dependence in the a parameter interpolates its value smoothly and monotonically between its fixed point values. If one supposes that there exist dual descriptions of the entire flow between the UV and IR fixed points, then the flow induced in the a-parameter in the dual theories is identical by the above prescription. The route from the a parameters to the couplings is via R-charges and hence anomalous dimensions. Indeed at the fixed points there already exist well known relations between the anomalous dimensions of fields, their R-charges (via the superconformal field theory), and the a and c parameters. The latter relations for example take the form4 ã = 3TrR3 − TrR ; c̃ = 9TrR3 − 5TrR , (1.4) where here R denotes the charges of states contributing to the ’t Hooft anomalies (i.e. it would be R − 1 if the superfield has charge R). Thus one prescription for defining a set of R-charges along the flow is to continue to solve (1.4) for R(µ) away from the fixed points, using aKS (µ) as defined in (1.2). We should stress that such a prescription (and the anomalous dimensions it gives rise to) corresponds to a choice of renormalisation scheme. However as the right-hand-side of (1.2) is the integral of a physical quantity (namely, by the optical theorem, the 4-dilaton cross-section) this particular choice has a physical meaning which is similar to that of the “sliding scale” scheme [15, 16]. Moreover it is independent of perturbation theory, so it has the same interpretation irrespective of whether one is using the electric or magnetic formulation. 4 Here and in the following we will interchangeably use a and ã related by a= 3 ã . 32(4π)2 3 (1.3) A second reason to favour “flowing” R-charges defined in such a way is that they appear to coincide with those of the Lagrange-multiplier method suggested by Kutasov [17, 18] 5 . The starting point of our discussion will be to demonstrate this unexpected equivalence, for flows near fixed points in the Banks-Zaks limit. This gives some physical meaning to the Lagrange-multiplier method when the theory is strongly coupled. Remarkably the RGscheme implicit in applying (1.4) to (1.2), appears to correspond to that implicit in the Lagrange-multiplier technique6 . Consequently one can determine the R-charges of the strongly coupled theory from those of the weakly coupled theory, by way of the matched a-parameters, which have a welldefined physical meaning in terms of the 4-dilaton amplitude, independent of whether the description is strongly or weakly coupled. From there it is straightforward to determine the anomalous dimensions, hence the NSVZ beta function, and ultimately the gauge coupling in the strongly coupled description. 2 Dilaton scattering a versus Lagrange multiplier a Let us begin by showing (in the Banks-Zaks limit ) that the a(µ) parameter one extracts for SQCD at scale µ along the flow between two fixed points using the KS definition [14], coincides with the Lagrange multiplier a-parameter of [17]. Figure 1: Contour for aKS (µ). First consider aKS (µ) in more detail. The prescription of (1.2) can be understood in terms of the contour integral of A/s3 around the loop shown in figure 1, where the radius of the inner contour is µ2 . The amplitude in this integral is treated as holomorphic in the upper half-plane of complex s, with branch-cuts arranged along the real axis. The integral in (1.2) corresponds to going along the I2 portion of the contour above the branch-cuts of the amplitude which run along the real axis to plus infinity in the s channel (and minus infinity in the u-channel). In the IR the amplitude behaves as  2(4−∆IR ) ∆IR −2  m s s2 , (2.1) A(s) = 8(aU V − aIR ) 4 + O f f4 where ∆IR > 4 is the lowest dimension of the irrelevant operators (of the dilaton) in the IR theory, and hence m is the scale of the relevant operators that we added into the UV theory that generated them upon integrating out degrees of freedom. In the limit that µ → 0 we 5 This method relies on there being a Lagrangian description of the theory, which will be assumed in the following. 6 If along the flow a gauge invariant operator becomes free, a new accidental symmetry arises and one should properly define aK (µ) along the lines of [19]. 4 may simply neglect the terms with inverse powers of m (along with I3 which tends to zero) and performing the integral find by Cauchy’s theorem [4], Z ∞ ImA(s) 8π(aU V − aIR ) ds = I2 = 2 , (2.2) − I1 = 4 f s3 0 where we also require Schwartz reflectivity of the Amplitude (namely A(s) = A(s̄)). This (by way of the optical theorem) is enough to establish the weak a-theorem. By contrast at finite µ the answer for I1 is of course µ dependent. To demonstrate what happens let us first revisit the simple example of free scalar fields of mass m discussed in [3]. Using standard perturbation theory (with the conventions of [3]) their contribution to the 4 dilaton amplitude is found to be Z
m4 1 A = − 240 (aU V − aIR ) 4 dx log(m2 − sx(1 − x)) + log(m2 + sx(1 − x)) + const. , f 0 (2.3) where we assume only these fields contribute to aU V − aIR . The constant term is independent of s and contains counter-terms to remove infinities, but it is not important for the discussion. Expanding the logarithms in s/m2 and performing the x integral gives the leading contribution in (2.1) (which can be used to check the pre-factor). Alternatively we note that the new absorptive contribution to A comes from the region of the integral where the argument of the first logarithm is negative, sx(1 − x) > m2 . Taking s → s + iǫ in order to be above the branch-cuts, we find ImA = 240π(aU V − aIR ) m4 p 1 − 4m2 /s . f4 Inserting this into the integral I2 with a cut-off then gives a running a parameter Z ImA(s) f4 aU V − aKS (µ) = ds, 4π s>µ2 s3 = (aU V − aIR ) (1 − ρ(µ/2m)) , where ρ(x) = ( (1 − x−2 ) 0 3/2 1 + 23 x−2
; x≥1 ; x≤1. (2.4) (2.5) (2.6) For later comparison it is useful to rearrange the expression as aKS = aIR + (aU V − aIR ) ρ(µ/2m) , (2.7) making it clear that ρ(µ/2m) correctly scales the contributions of the scalars to the aKS parameter continuously and monotonically, with ρ = 0 at µ = 2m to ρ = 1 at µ → ∞. Thus we may interpret the KS integral of (1.2) as simply counting the imaginary (absorptive) contributions to the amplitude from states that are able to go on shell when s > µ2 (a useful reference in this context is [20]). In order to compare the running aKS derived above with the continuously varying aK function devised for SUSY theories in [17], we need to extend the simple case above to N = 1 SUSY. Consider the free field theory, consisting of Nf pairs of superfields Φa and Φ̃a , a = 1 . . . Nf . The Lagrangian of [3] can be made supersymmetric in the obvious way, 5 by coupling the fields in a superpotential mass-term W ⊃ mΩΦ̃I∆f ×∆f Φ, where Ωf is the canonically normalised dilaton superfield with hΩi = 1. This gives a supersymmetry preserving mass m to ∆f pairs of superfields. (As the superpartner of the dilaton does not appear in any loops of interest we can ignore it.) The amplitude is of course augmented by superpartner diagrams, but now supersymmetry guarantees that the coefficient of terms such as (2.3) vanish, because otherwise (as these terms are not zero in the limit of vanishing external momenta and finite f ) they would signal a renormalisation of the superpotential. The non-vanishing terms of interest are, in the standard Passarino-Veltman notation, of the form sB0 (s, m2 , m2 ) and friends. Thus the contributions of interest are of the form Z
m2 s 1 dx log(m2 − sx(1 − x)) − log(m2 + sx(1 − x)) + const. , A = − 24 (aU V − aIR ) 4 f 0 (2.8) where as before the second term is really the u Mandelstam variable with t → 0. Following the above treatment of the massive scalar, we deduce a running aKS from the absorptive part which is ImA = 24π(aU V − aIR ) m2 p s 1 − 4m2 /s . f4 Inserting this into (1.2) then gives (2.7), but with a modified scaling function, ( 3/2 (1 − x−2 ) ; x≥1 ρ(x) = 0 ; x≤1. (2.9) (2.10) Let us now compare this expression with the aK function of [17]. The Lagrange multiplier method for this simple case goes as follows. In general the running a-parameter is defined by adding a Lagrange multiplier for each relevant operator. In this case there is only one of them, which imposes the constraint from the mass term. The a-function is therefore given simply by     ãK = (Nf − ∆f ) 3(R − 1)3 − (R − 1) + ∆f 3(r − 1)3 − (r − 1) + λ∆f [r − 1] , (2.11) where R is the R−charge of the Nf −∆f chiral superfields that remain massless and r is the R-charge of the last ∆f flavours, which is considered to be a function of the energy scale. Thus the R-symmetry we are following along the flow is a linear combination of the superconformal R-symmetry of the deep U V and the SU(Nf )×SU(Nf ) flavour symmetry with  which it mixes because of the mass-term (specifically the diag ∆f INf −∆f , (∆f − Nf )I∆f component)7 . ∂a = ∂R = One first solves to maximise the a-function with respect to unfixed R-charges, ∂a ∂r 0. In the absence of the mass-term constraint this simply chooses the free-field value of 2/3 for both R and r. However at arbitrary Lagrange multiplier values one finds R = 2/3 , √ 1−λ r = 1− . 3 7 This is true for ∆f < Nf : when ∆f = Nf there is of course no relevant R-symmetry left. 6 (2.12) The case where λ = 0 corresponds to R = r = 2/3 in the deep UV, while λ = 1 corresponds to r = 1, which is the value forced upon it by the mass-term in the deep IR. Substituting these values into ãK we have ãU V = 29 Nf and ãIR = 92 (Nf −∆f ), and a running a−parameter given by 3 (2.13) ãK = ãIR + (ãU V − ãIR ) (1 − λ) 2 . Comparison with (2.10) shows that the two a-functions precisely coincide if one makes the 2 . Note that the a-functions in the supersymmetric case match identification λ ≡ 4m µ2 essentially because of the non-renormalisation theorem, and that as usual the Lagrange multiplier is essentially the “coupling” that induces the flow. For the SUSY gauge theories of interest the situation is more complicated but the interpretation is always the same; namely aKS counts the physical states that are able to contribute to the absorptive part of the 4-dilaton amplitude. Meanwhile aK tracks the mixing of the UV R-symmetry with flavour symmetry along the flow [18]. We will now show that at weak coupling, close to the Caswell-Bank-Zaks fixed point, they are equivalent in this case as well8 . Consider SQCD with Nf flavours of quarks Q and Q̃ flowing from the asymptotically free theory to the fixed point. The aKS parameter was derived in terms of the gauge coupling in [14]; Z ∞ Nc2 dλ βλ , (2.14) aKS (µ) = aU V − 2 128π g−2 (µ) λ2 where λ = 1/g 2 . In the limit µ → 0 this expression reduces to eq. (3.12) of [14]. Using b2 = Nc N f , (8π 2 )2 where g∗2 = 8π 2 ǫ, Nf 3Nc − Nf ≪ 1, Nc ǫ = (2.15) (2.16) the integral gives aKS (µ) = aU V − Nc3 Nf 2 g (µ)(2g∗2 − g 2 (µ)) , 32(8π 2 )3 (2.17) with g 2 (µ) being a solution of the 2-loop RGE, dg 2 = b2 g 4 (g 2 − g∗2 ) . d log µ (2.18) Again we can compare this parameter to the continuously varying aK -function of [17]. In an SU (N ) gauge theory it can be written in generality as X ãK = 2(Nc2 − 1) + |ri | (a1 (Ri ) − (Ri − RiIR ) a′1 (Ri )) , (2.19) i where |ri | is the dimension of the representation ri , the prime means derivative with respect to R and where a1 (r) ≡ 3(r − 1)3 − (r − 1) . (2.20) 8 It would be interesting to look for reasons behind this equivalence that are valid beyond weak coupling, along the lines of [21, 22]. For the present work the equivalence in the weakly coupled theories is sufficient. 7 In the case of electric SQCD this gives
∗ ãK (µ) = 2(Nc2 − 1) + 2Nc Nf a1 (RQ ) − (RQ − RQ )a′1 (RQ ) . (2.21) In order to compare with aKS we relate the R-charges to the anomalous dimensions through γQ  2 1+ . (2.22) RQ = 3 2 This equation holds along the flow, but only at the endpoints of the flow does RQ coincide with the respective super-conformal R-charges of the fixed points. The anomalous dimension can be perturbatively calculated at 1-loop as γQ = − and then using UV RQ = 2 , 3 Nc 2 g , 8π 2 ∗ RQ =1− (2.23) Nc , Nf (2.24) we easily find the same leading contribution as that in (2.17), and hence aK ≡ aKS as we wished to prove. At this point one could ask, what is the meaning of equating a scheme independent quantity such as aKS (µ) with a scheme dependent one such as aK (µ). This is of course what we always do when we calculate a cross section (in which we are bound to choose a scheme), and compare it to its (scheme independent) measured value. The theoretical result becomes scheme independent only when all terms in perturbation theory are taken into account, but never at finite order. Therefore the equivalence is only a perturbative one. Nevertheless as we shall now show, what it does do is allow us to develop a perturbative description of the flow in a strongly coupled theory. 3 A perturbative calculation of a non-perturbative flow We now wish to explore how this equivalence can be used to determine the gauge coupling flow in a strongly coupled description. To do so we will consider a strongly coupled SQCD (in the conformal window) when one invokes a flow by adding a mass term for one flavour, and will make use of the well-known duality between this theory and Higgsing in a weakly coupled magnetic description, described in [12]9 . The original electric SQCD theory is an N = 1 SU(Nc ) theory with Nf + 1 flavours of Q and Q̃ quarks and anti-quarks. We add a mass-term of the form We = m QNf +1 Q̃Nf +1 (3.1) in its superpotential. In the IR, i.e. at energy below m, it flows to a new theory with Nf flavours, hence effectively there is a UV fixed point with Nf + 1 flavours at energy above m, and an IR fixed point with Nf flavours. If we take 2Nf = 3Nc + 1 then the theory is expected to be strongly coupled for large Nc all along the flow. 9 For a pedagogical description of such a set-up see for example [23]. 8 Meanwhile the magnetic description is an SU(Ñc + 1) theory with Ñc = Nf − Nc and, as well as Nf +1 flavours of quarks q and q̃, it contains an elementary (Nf +1)×(Nf +1) meson Φ formed from a composite of the electric quarks, which we will take to be Φ ≡ Λ1 Q · Q̃ where Λ is the dynamical scale of the theory, and a superpotential N +1 Wm = m ΛΦNff +1 + ỹ Φq̃ · q , (3.2) whose first term derives from the mass-term, and where the Yukawa coupling is ỹ = Λ/Λ̂ with Λ̂ ∼ Λ. The magnetic theory which has Nf = 3Ñc − 1 is arbitrarily weakly coupled, so its flow can be followed perturbatively. In particular the linear meson term in the superpotential causes a Higgsing down to SU(Ñc ). For completeness we summarise the flows as seen in the two dual theories in Table 1, where the RG scale is defined with respect to m, that is t ≡ log (µ/m) . (3.3) We can easily determine the difference between the UV and IR a-central charges ãU V − ãIR = = 2Nc (Nf + 1)a1 (1 − Nc /(Nf + 1)) − 2Nc Nf a1 (1 − Nc /Nf ) 6Nc2 (2Nf + 1) , Nf2 (Nf + 1)2 (3.4) which is positive for all Nf > 0, and thus the weak a-theorem is satisfied. magnetic theory electric theory IR (t < 0) UV (t > 0) Nf flavours Ñc colours Nf flavours Nc colours Nf + 1 flavours Ñc + 1 colours Nf + 1 flavours Nc colours Table 1: The dual theories considered in the text with Nc = Nf − Ñc . We consider throughout the case of Nf = 32 Nc + 12 = 3Ñc − 1. As discussed, our aim is to determine the gauge coupling for the original strongly coupled electric theory. In order to do this we first consider in detail the dual of the UV theory, and the dual of the IR theory, both of which are known. By choosing Nf ≈ 3Ñc and large Ñc ≡ Nf − Nc , the magnetic theory10 is made perturbative both in the UV and the IR so we can calculate its flow with good accuracy along the whole RG trajectory. As we also know the (in principle non-perturbative) interacting electric theories in both the UV and the IR, we assume that the flow of the magnetic theory is dual to that of the strongly coupled electric theory along the whole trajectory. Before entering into the explicit computation, let us clarify the idea and the procedure we will follow. The magnetic theory is perturbative and is thus under control in the whole region between the deep UV (µ = +∞, t = +∞) and the deep IR (µ = 0, t = −∞). The gauge and Yukawa couplings are continuous along the whole flow, while the two beta functions and the two anomalous dimensions are, due to the mass independent character of the NSVZ scheme, discontinuous at the explicit quark mass scale (µ = m, t = 0). 10 It is convenient to take the magnetic theory to be perturbative as then only one coupling - the electric gauge coupling - is non-perturbative and thus the matching of a-parameters determines it uniquely. 9 However in one half of the flow (µ < m) we are able to use the a parameters to track (perturbatively and in the NSVZ scheme) the evolution of the strongly coupled theory from that in the weakly coupled one. In particular the electric theory is non-perturbative and what we know from Seiberg duality is that it is equivalent to the magnetic theory at µ = m (t = 0), which becomes the “new” UV, and in the deep IR at µ = 0 (t = −∞). By continuity11 we assume that the perturbative magnetic and non-perturbative electric theories are dual between these two endpoints. In this region the physical quantity aKS is by definition the same in the magnetic and electric theory, and as motivated in the previous section we equate aK and aKS , allowing us to explicitly calculate the central charge in the whole energy range 0 < µ < m in the magnetic theory, as a perturbative function of the gauge and Yukawa coupling constants. This (via the equivalence of the a parameters) yields the non-perturbative R-charges and hence anomalous dimensions of the strongly coupled electric theory, which in turn yields the explicit numerical solution of the RGE for the electric gauge coupling constant. 3.1 UV (0 < t < ∞): magnetic theory We start in the UV with the magnetic theory, which is an SU(Ñc + 1) gauge theory with Nf + 1 quarks q + q̃ and (Nf + 1)2 singlet meson fields with the superpotential of (3.2). We will work in terms of (Ñc + 1)g̃ 2 (Ñc + 1)ỹ 2 α̃g ≡ , α̃ ≡ . (3.5) y (4π)2 (4π)2 The theory is asymptotically free when Nf + 1 = 3Ñc since b1 = 3(Ñc + 1) − (Nf + 1) = 3 , (3.6) and at t → ∞ all couplings go to zero. For µ ≫ Λ the 1-loop approximation is sufficient,  1 and one has the usual evolution with dynamical scale given by Λ ≡ µ exp − 2b1 α̃g (µ) . Towards the IR the flow approaches a Banks-Zaks fixed point that for larger Ñc becomes increasingly perturbative. Indeed the two-loop RGEs (see for example [11, 24]) give the fixed points to be at    Nf +1 N +1 − 3 2 Ñf +1 + 1 Nf →3Ñc −1 7 43 Ñc +1 c −−−−−−−→ + + O(1/Ñc3 ) , α̃g (0+ ) = 2  2 N +1 N +1 N +1 2Ñc 2Ñc 6 + 8 Ñf +1 − 4 Ñf +1 + 2 (Ñ f+1)3 c c c    Nf +1 1 2 1 − (Ñ +1)2 −3 Nf →3Ñc −1 7 1 Ñc +1 c + + O(1/Ñc3 ) . (3.7) −−−−−−−→ α̃y (0+ ) = 2  2 N +1 N +1 N +1 Ñc Ñc +2 f 6+8 f −4 f Ñc +1 Ñc +1 (Ñc +1)3 Defining (+) + + ∆(+) g̃ (t) = α̃g (t) − α̃g (0 ) , ∆ỹ (t) = α̃y (t) − α̃y (0 ) , the 2-loop RGEs can be rephrased as dα̃g (t) dt dα̃y (t) dt 11 = −2α̃g2 (t) × " Nf + 1 Nf + 1   Nf + 1 (3.8) 2 ∆(+) ∆(+) ỹ (t) g̃ (t) + 2 (Ñc + 1)3 Ñc + 1       Nf + 1 1 (+) ∆(+) (t) + 2 + 1 ∆ (t) . = 2α̃y (t) × −2 1 − g̃ ỹ (Ñc + 1)2 Ñc + 1 6−4 Ñc + 1 +2 For a discussion on this point see for example [23]. 10 # , (3.9) The terms proportional to α̃g (0+ ) and α̃y (0+ ) in these expressions are the one-loop terms, while the remaining terms are two-loop. Finally we can calculate the R-charges at the t = 0+ fixed point: Rq (0+ ) = 1 − Ñc + 1 , Nf + 1 RΦ (0+ ) = 2 − 2Rq (0+ ) , (3.10) which are perturbative (free) Rq = RΦ = 2/3 in the large Ñc limit with Nf → 3Ñc − 1, in accord with the magnetic theory being parametrically perturbative for all positive t. 3.2 UV (0 < t < ∞): electric theory Apart from the far UV (as it is also asymptotically free) the form of the electric dual theory is known only in the t → 0+ limit, where it is an SU(Nc ) gauge theory with Nf + 1 quarks Q + Q̃ and vanishing superpotential. In the same limit the fixed point determines the value of the R-charge: Nc . (3.11) RQ (0+ ) = 1 − Nf + 1 In the large Nc limit (as we have Nf = 23 Nc + 1) this value is clearly interacting, RQ → 1/3, in accord with the composite object Q · Q̃ becoming free. We do not know the value of the electric gauge coupling at other values of t. 3.3 IR (−∞ < t < 0): magnetic theory We now turn to the flow of interest, towards the IR, for t < 0. Here the magnetic theory is an SU(Ñc ) gauge theory with Nf quarks q + q̃ and Nf × Nf gauge singlet mesons, which for convenience we continue to call Φ. At t = 0 the boundary conditions of the couplings, α̃g ≡ Ñc g̃ 2 , (4π)2 α̃y ≡ Ñc ỹ 2 , (4π)2 (3.12) are determined by continuity12 , Ñc α̃g (0+ ) , Ñc + 1 Ñc α̃y (0+ ) . α̃y (0− ) = Ñc + 1 α̃g (0− ) = (3.13) The flow of the magnetic theory can be determined perturbatively from the RGEs. Defining ∆g̃(−) (t) = α̃g (t) − α̃g (−∞) , ∆ỹ(−) (t) = α̃y (t) − α̃y (−∞) , 12 (3.14) Note that we are using the NSVZ scheme which is a mass independent scheme: this means that the perturbative gauge couplings are continuous passing the mass scale, while the beta functions or anomalous dimensions are not. The apparent discontinuity in (3.13) is clear from the way the coupling constants are defined in (3.12), i.e. with the discontinuity being in the number of colours. 11 where the new fixed point is at    Nf N − 3 2 Ñf + 1 Nf →3Ñc −1 7 25 Ñc c −−−−−−−→ α̃g (−∞) = + + O(1/Ñc3 ) ,  2 2 Nf Nf Nf 6 Ñ 9 Ñ c c + 2 Ñ 3 6 + 8 Ñ − 4 Ñ c c c    Nf − 3 2 1 − Ñ12 Nf →3Ñc −1 8 1 Ñc c + + O(1/Ñc3 ) , −−−−−−−→ α̃y (−∞) =  2 2 Nf Nf Nf 3Ñc 9Ñc 6 + 8 Ñ − 4 Ñ + 2 Ñ 3 c c (3.15) c they are dα̃g (t) dt dα̃y (t) dt = −2α̃g2 (t) × " 6−4 Nf +2 Nf  (−) ∆g̃ (t) + 2  Nf 2 (−) ∆ỹ (t) Ñc       Nf 1 (−) (−) ∆g̃ (t) + 2 = 2α̃y (t) × −2 1 − + 1 ∆ỹ (t) . Ñc2 Ñc Ñc Ñc3 # , (3.16) The evolution is shown in Fig. 2 for Ñc = 100 and Nf = 3Ñc − 1. α̃g α̃y 0.04 0.010 0.03 0.008 0.006 0.02 0.004 0.01 0.002 t -30 000 -25 000 -20 000 -15 000 -10 000 -5000 t -30 000 0 -25 000 -20 000 -15 000 -10 000 -5000 0 Figure 2: The perturbative running of the gauge (left) and Yukawa (right) coupling constants of the magnetic theory for Ñc = 100 and Nf = 3Ñc − 1, from the UV fixed point (t = 0 with Nf + 1 quarks and Ñc + 1 colours), to the IR fixed point (t = −∞ with Nf quarks and Ñc colours). The lower and upper lines denote the UV and IR values of the couplings. It is useful to explicitly express the flowing R-charges in terms of the couplings. This we can do because the theory is perturbative (approximately, order by order in perturbation theory). From the usual definition of the NSVZ beta function and the relation in (2.22), we have   Nf 2 (Rq − 1) , β(α̃g ) = −6α̃g f (α̃g ) 1 + Ñc β(α̃y ) = 3α̃y (2Rq + RΦ − 2) , (3.17) with f (x) ≡ 1 . 1 − 2x 12 (3.18) Comparison with the r.h.s. of (3.16) gives  
2 Nf (−) 2 1 ∆g̃(−) + ∆ỹ + O ∆2 , Rq (t) − Rq (−∞) = − 1− 3 3 Ñc Ñc2
2 RΦ (t) − RΦ (−∞) = ∆ỹ(−) + O ∆2 . (3.19) 3 Their evolution for t < 0 is shown in Fig. 3. Note that although we do not display them explicitly, the order ∆2 terms as derived from Eq.(3.17), are actually required later in order to get consistent convergence to the IR fixed point in the strongly coupled description. RΦ Rq 0.6655 0.673 0.6650 0.672 0.6645 0.671 0.6640 0.670 0.6635 0.669 t -30 000 -25 000 -20 000 -15 000 -10 000 -5000 t -30 000 -25 000 -20 000 -15 000 -10 000 -5000 Figure 3: The “flowing” R-charges (green) of the quark (left and meson (right) in the magnetic SQCD with gauge SU(Ñc ) and Nf quarks q + q̃ and Nf2 mesons Φ, with Nf = 3Ñc − 1. The flow has been found using the perturabative relations (3.17) and (3.17) and using Ñc = 100. The blue straight lines are the values Rq (0+ ) and RΦ (0+ ) obtained in the fixed point above the mass m. Notice that the values Rq (0− ) and RΦ (0− ) do not coincide with them: although the gauge couplings g̃ and ỹ are continuous, the R-charges are not: they are in some sense proportional to the non-continuous beta-functions. Finally the orange straight lines are the limiting values Rq (−∞) and RΦ (−∞) obtained from the IR fixed point couplings. 3.4 IR (−∞ < t < 0): electric theory Up to this point, for t < 0, everything has been perturbative. Now let us now consider the original electric theory in the range −∞ < t < 0. In the limit t → −∞ the theory is SU(Nc ) SQCD with Nf quarks Q + Q̃ and no superpotential. Let us assume that the same pair of dual theories describe the physics along the whole RGE running. As the parameter aKS is a function of the amplitude its definition is independent of which description is being used and hence its value in the electric and magnetic theories is the same all along the flow. We will adopt the assumption, motivated in the Introduction, that in regions where the beta functions are small the aKS -function is the same as the function aK derived using the Lagrange multiplier definition [17]. Hence using (2.21) and equating aK ’s in the two descriptions as in the Appendix, one finds 2(Nc2 − 1) + 2Nf Nc (a1 (RQ (t)) − (RQ (t) − RQ (−∞))a′1 (RQ (t))) = 2(Ñc2 − 1) + 2Nf Ñc (a1 (Rq (t)) − (Rq (t) − Rq (−∞))a′1 (Rq (t)) + Nf2 (a1 (RΦ (t)) − (RΦ (t) − RΦ (−∞))a′1 (RΦ (t))) , (3.20) 13 which can be used to determine RQ (t). Its behaviour is shown in Fig. 4. RQ 0.3370 0.3365 0.3360 0.3355 0.3350 0.3345 t -30 000 -25 000 -20 000 -15 000 -10 000 -5000 0 Figure 4: The R-charge (green) of the quark in the electric SQCD with gauge group SU(Nc ) and Nf quarks Q + Q̃, with Nf = 3Ñc − 1, using (3.20). As before, the blue straight line is the value at t = 0+ , while the orange line is the asymptotic value in the IR. From there it is straightforward to determine the gauge coupling from the NSVZ beta function. Defining the electric gauge coupling as Nc g 2 , (4π)2 αg ≡ (3.21) and using β(αg ) = − 6αg2 f (αg ) one can now integrate, to find F (αg (t)) − F (αg (0 )) = Z 1 F (x) ≡ 6  − where  Nf 1+ (RQ − 1) Nc t dt 0 ′   , Nf (RQ (t′ ) − 1) 1+ Nc 1 + 2 log x x  . (3.22)  , (3.23) (3.24) This can then be solved for αg . Note that as mentioned the O(∆2 ) terms in Eq.(3.19) are required here. If they are omitted then there are order 1/Nc2 errors in the integrand, which over the order −t ∼ Nc2 running required to get to the fixed point, translates into errors of order unity: in other words there would not be proper convergence to a fixed point. Of course we do not know the numerical value of the boundary condition, αg (0− ), in the electric theory, but since the r.h.s. of (3.23) is negative, and since the gauge coupling must obey αg < 1/2 in order that f (αg ) defined in (3.18) does not change sign, there is a maximum allowed value of αg (0− ) given by  Z −∞  Nf ′ ′ max − (RQ (t ) − 1) = F (1/2) . (3.25) dt 1 + F (αg (0 )) + Nc 0 For our inputs this is given by αgmax (0− ) = 0.0216164 . 14 (3.26) As an illustrative example we take three different inputs (0.99, 0.96, 0.9) for the ratio αg (0− )/αgmax (0− ) and obtain numerically the flows shown in Fig. 5 for the non-perturbative coupling αg (t). There is of course only one correct numerical boundary condition at t = 0− corresponding to the electric theory dual to the perturbative magnetic one, but unfortunately it cannot be determined13 . All we know is that it must be non-perturbative, so too small ratios αg (0− )/αgmax (0− ) are unacceptable because they would not reproduce the known anomalous dimensions in the deep IR. The entire flow including the R-charges can of course be expressed in terms of the Lagrange multipliers of [17], in the manner described in the introduction and in Section 2. We included them for completeness in the Appendix. α� ●●●●●●●●● ●●●● ●●● ●● ���� ●● ●● ● ���� ●● ●● ● ■ ���� ���� ● ● ● ● ◆ ��� ● ■■■■■■■■■■■■■■■■■ ● ■■■■■ ● ■■■■ ���� ● ■■■ ● ■■ ● ■■ ■■ ● ■ ■ ●● ���� ■■ ● ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆ ◆◆◆◆ ■ ■ ● ● ◆◆◆ ■ ■ ● ◆◆◆■ ● ■● ■● ◆◆ ���� ■● ◆◆ ■● ■● ◆ ◆ ■● ◆ ■ ◆ -�� ��� -�� ��� -�� ��� -�� ��� -�� ��� -���� � Figure 5: The non-perturbative running of the gauge coupling constant of the electric theory SU(Nc ) with Nf quarks Q + Q̃, for three different values of the ratio αg (0− )/αgmax (0− ), and Nc g 2 for Ñc = 100, Nf = 3Ñc − 1 and Nc = Nf − Ñc . Note that αg ≡ (4π) 2 . 4 Equality of critical exponents The critical exponent provides a mild but nevertheless important check on the consistency of this picture. It is defined as the minimal eigenvalue of the matrix of coupling derivatives of the beta functions around the fixed point14 :     ∂βαa ′ . (4.1) β ≡ min positive eigenvalues ∂αb F.P. It is a renormalization scheme independent quantity and therefore should be equal for dual theories [25, 26]. Usually of course this equivalence cannot be checked because one cannot 13 It would be interesting to attempt to extend the approach to include the mass term explicitly with another Lagrange multiplier, as for the free-field theory in the introduction. However as one would have to describe a Higgsing in the magnetic theory, this would be significantly more complicated, and it is not clear how one could fix several Lagrange multipliers with only a single a-parameter. 14 Here one cannot compare the full matrices or even all their eigenvalues, because for example the dimensions of the matrices do not agree. However the minimal eigenvalue has a physical scheme-independent meaning in both descriptions. 15 (mag) compute in the strongly coupled theory. However our prescription (a(el) ) allows it KS = aKS to be checked explicitly, as we now show. In the magnetic description, the theory is perturbative and so we can simply use (3.16) to evaluate the critical exponent:        2 Nf Nf Nf   2 2 −2α̃ (−∞) 2 + 2 −2α̃ (−∞) 6 − 4 ∂βαa 3 g g   Ñc Ñc Ñc (4.2) =       . N ∂αb F.P. f 1 2α̃y (−∞) 1 − Ñ 2 (−2) 2α̃y (−∞) 2 Ñ + 1 c c For Nf = 3Ñc − 1 one obtains the leading order in 1/Ñc approximation, using (3.15): ′ βmag = 7 , 3Ñc2 (4.3) while the second, larger, eigenvalue is found to be equal to 14/(3Ñc ). In the strongly coupled electric description, there is a single gauge coupling, so that βel′ = ∂βαg (t) ∂αg (t) = t→−∞ d β (t) dt αg d α (t) dt g = t→−∞ d log (RQ (t) − RQ (−∞)) dt . (4.4) t→−∞ Usually in the non-perturbative theory the relation between RQ (t) and αg (t) is not known. Here however we have a relation between RQ (t) and the known Rq (t) and RΦ (t) of the perturbative magnetic theory through (3.20). We may therefore expand a around t = −∞, X 1 ∂ 2a a(t) = a(−∞) + (−∞) (Ri (t) − Ri (−∞)) (Rj (t) − Rj (−∞)) + . . . , (4.5) 2 ∂Ri ∂Rj i,j and from there must find ∂ 2 ael (−∞) (RQ (t) − RQ (−∞))2 2 ∂RQ ≈ ∂ 2 amag (−∞) (Rq (t) − Rq (−∞))2 2 ∂Rq + ∂ 2 amag (−∞) (RΦ (t) − RΦ (−∞))2 . (4.6) 2 ∂RM Since the second derivative of a over the R-charges is proportional to the b-central charge of a conserved current (in this case it is the baryon current) and thus strictly non-zero, we must have the same scaling, ′ RQ (t) − RQ (−∞) ∼ Rq (t) − Rq (−∞) ∼ RΦ (t) − RΦ (−∞) ∼ exp (βmag t) , (4.7) in the asymptotic region t → −∞. But then from (4.4) we consistently find ′ βel′ = βmag . (4.8) We conclude that ael = amag along the flow is compatible with the equality of the electric and magnetic critical exponents. Of course this is not a particularly restrictive condition, and many other relations would have given equality. For example amag = A(ael ) (4.9) A(ael (−∞)) = ael (−∞) , A′ (ael (−∞)) 6= 0 . (4.10) for an arbitrary function A(x) with would suffice. 16 5 Conclusion In this paper we discussed the use of the a central charge as a method of determining the flow in a strongly coupled supersymmetric theory from its weakly coupled dual. Although there are other examples of exact duality in field theory along an entire flow (e.g. [27]) this method seems particularly general and well suited to N = 1 supersymmetry. Crucial to the approach is the equivalence of the scale-dependent a-parameter determined from the four-dilaton amplitude with an IR cut-off, and the a-parameter determined in the Lagrange multiplier method of Ref. [17, 18] with “flowing” R-charges. We showed that this equivalence holds directly for massive free N = 1 superfields, as well as weakly coupled SQCD. Assuming it to hold generally amounts to a particularly physical choice of RG scheme, in which the running R-charges are always determined precisely from the four-dilaton amplitude. In this scheme, which is clearly well defined regardless of which formulation is being used, one can map the flow of a weakly coupled magnetic dual to the original strongly coupled electric theory. The specific system we considered was the well-known pair of original SQCD Seiberg duals, with the magnetic description (with weak gauge and Yukawa coupling constants) running perturbatively from a fixed point in the UV to a different fixed point in the IR due to a mass-deformation, and the electric SQCD dual running between strongly coupled fixed points due to a meson-induced Higgsing. We should add that the mapping only seems to work straightforwardly in the direction of magnetic to electric, as in that case there is only one R-charge to determine (namely that of the electric quarks), and there is only one parameter (namely the a-parameter) with which to do it. Mapping in the converse direction may be possible in conjunction with a-maximisation [28], but is less obvious. Acknowledgments We are extremely grateful to Colin Poole for interesting discussions. BB acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0035). The work of FS is partially supported by the Danish National Research Foundation under the grant DNRF:90. BB thanks CP3 Origins Odense for hospitality. A The Lagrange multipliers Here we explicitly show how the Lagrange multipliers of [17,18] flow in the model discussed in Section 3. We start with the original magnetic a-function,   (A.1) ãmag = 2 Ñc2 − 1 + 2Ñc Nf a1 (Rq ) + Nf2 a1 (RΦ ) − λ̃g Nf (Rq − Rq (−∞)) + λ̃y Nf (2(Rq − Rq (−∞)) + (RΦ − RΦ (−∞))) . By a-maximisation we have [28] ∂ãmag ∂ãmag = = 0, ∂Rq ∂RΦ (A.2) which gives the following for the Lagrange multipliers: λ̃g = 2Ñc a′1 (Rq ) − 2Nf a′1 (RΦ ) , λ̃y = −Nf a′1 (RΦ ) . 17 (A.3) (A.4) Similarly for the electric theory one gets
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