IIPPP/18/35CP3-Origins-2018-019 DNRF90
Dual RG flows in 4D
Steven Abel a, , Borut Bajc b, and Francesco Sannino c, a IPPP, Durham University, South Road, Durham, DH1 3LE b J. Stefan Institute, 1000 Ljubljana, Slovenia b 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 astrongly coupled supersymmetric theory from its weakly coupled dual. The approachis based on the equivalence of the scale-dependent a -parameter derived from the four-dilaton amplitude with the a -parameter determined from the Lagrange multipliermethod with scale-dependent R -charges. We explicitly demonstrate this equivalencefor massive free N = 1 superfields and for weakly coupled SQCD. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] M a r ontents a versus Lagrange multiplier a
43 A perturbative calculation of a non-perturbative flow 8 < t < ∞ ): magnetic theory . . . . . . . . . . . . . . . . . . . . . . . 103.2 UV (0 < t < ∞ ): electric theory . . . . . . . . . . . . . . . . . . . . . . . . 113.3 IR ( −∞ < t < −∞ < t < Renormalization Group (RG) flow of Quantum Field Theories (QFTs) is thought to beirreversible. In two dimensions this irreversibility is encompassed by the Zamalodchikov c -theorem, which states that one can define a monotonically decreasing parameter thatinterpolates between the central charges c [1] of two conformal theories related by an RGflow. An equivalent parameter in 4 dimensions is Cardy’s proposal of the a anomaly, thecoefficient 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 formfor this coefficient to show that its value will inevitably decrease if a system goes from aUV to an IR fixed point. The method that was used in [3] is a cousin of ’t Hooft anomalymatching, in the sense that a spectator dilaton field is introduced that compensates theanomaly and restores exact Weyl symmetry at all scales, which is spontaneously broken bya dilaton VEV. Using such a set-up the a parameter can be deduced from the 4 dilatonamplitude. The change in the a parameter between fixed points, a IR − a UV , is then foundto be always negative by relating it via the optical theorem to the cross-section. Thus theweak form of the a -theorem, that its value will decrease if a system flows from a UV fixedpoint to an IR one, can be considered proven. However the strong version, namely thatthere exists a monotonically decreasing a function with unambiguous physical meaning allalong the flow, appears to be still open because of the presence of scheme dependent β terms in the four dilaton amplitude, as discussed in [4–6].Indeed Jack and Osborn [7–9] showed the existence of a function ˆ a related to a throughthe beta functions, that coincides with it at fixed points and that flows with energy scale µ as µ d ˆ adµ = χ IJ β J β I , (1.1)where β I are the beta functions of couplings λ I , and χ IJ is a metric on the space of couplings.The problem of proving the monotonicity of the function ˆ a (and hence the irreversibility ofRG flow) is then reduced to one of proving the positive-definiteness of the metric χ on thespace of functions. This problem remains to be solved (for a review see for example [10]).2ur purpose here is to point out that the parametric closeness of a and ˆ a suggests amethod 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 expansionin terms of the beta functions rather than in terms of any couplings [11]. In this letter weexplore the a -parameter as the basis for such an approach, showing how one can use it tofollow 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 theparticle 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 parametersof strongly coupled theories, as well as much of their holomorphic data, even when they areaway from fixed points. The question we will address here is how one can also determinethe flow of the coupling in the strong theory, up to the aforementioned corrections of order β , by mapping from the weak theory.The approach continues in the spirit of ’t Hooft anomaly matching, by considering theflow of the a parameter. In order to define such a flow we will use the KS determinationof a which involves a certain integration of the 4-dilaton amplitude over the Mandelstamvariable [14]: a UV − a KS ( µ ) = f π (cid:90) s>µ Im A ( s ) s ds , (1.2)where f is the dilaton decay constant and s is the Mandelstam variable, and where weimpose an IR cut-off on the integral, s > µ , in order to generate a running a -parameter,which we denote a KS ( µ ). The cut-off induced scale dependence in the a parameter interpo-lates its value smoothly and monotonically between its fixed point values. If one supposesthat 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 aboveprescription.The route from the a parameters to the couplings is via R -charges and hence anomalousdimensions. Indeed at the fixed points there already exist well known relations betweenthe 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 form ˜ a = 3Tr R − Tr R ; ˜ c = 9Tr R − R , (1.4)where here R denotes the charges of states contributing to the ’t Hooft anomalies (i.e. itwould be R − 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 a KS ( µ ) as defined in (1.2). We should stressthat such a prescription (and the anomalous dimensions it gives rise to) corresponds to achoice of renormalisation scheme. However as the right-hand-side of (1.2) is the integralof a physical quantity (namely, by the optical theorem, the 4-dilaton cross-section) thisparticular 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 sameinterpretation irrespective of whether one is using the electric or magnetic formulation. Here and in the following we will interchangeably use a and ˜ a related by a = 332(4 π ) ˜ a . (1.3) R -charges defined in such a way is that they appearto coincide with those of the Lagrange-multiplier method suggested by Kutasov [17, 18] .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 tothe Lagrange-multiplier method when the theory is strongly coupled. Remarkably the RG-scheme implicit in applying (1.4) to (1.2), appears to correspond to that implicit in theLagrange-multiplier technique .Consequently one can determine the R -charges of the strongly coupled theory fromthose of the weakly coupled theory, by way of the matched a -parameters, which have a well-defined physical meaning in terms of the 4-dilaton amplitude, independent of whether thedescription is strongly or weakly coupled. From there it is straightforward to determine theanomalous dimensions, hence the NSVZ beta function, and ultimately the gauge couplingin the strongly coupled description. a versus Lagrange multiplier a Let us begin by showing (in the Banks-Zaks limit ) that the a ( µ ) parameter one extractsfor 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 a KS ( µ ) . First consider a KS ( µ ) in more detail. The prescription of (1.2) can be understood interms of the contour integral of A /s around the loop shown in figure 1, where the radiusof the inner contour is µ . The amplitude in this integral is treated as holomorphic in theupper half-plane of complex s , with branch-cuts arranged along the real axis. The integralin (1.2) corresponds to going along the I portion of the contour above the branch-cuts ofthe amplitude which run along the real axis to plus infinity in the s channel (and minusinfinity in the u -channel). In the IR the amplitude behaves as A ( s ) = 8( a UV − a IR ) s f + O (cid:18) m − ∆ IR ) s ∆ IR − f (cid:19) , (2.1)where ∆ IR > m is the scale of the relevant operators that we added into the UV theorythat generated them upon integrating out degrees of freedom. In the limit that µ → This method relies on there being a Lagrangian description of the theory, which will be assumed inthe following. If along the flow a gauge invariant operator becomes free, a new accidental symmetry arises and oneshould properly define a K ( µ ) along the lines of [19]. m (along with I which tends to zero)and performing the integral find by Cauchy’s theorem [4], − I = 8 π ( a UV − a IR ) f = I = 2 (cid:90) ∞ ds Im A ( s ) s , (2.2)where we also require Schwartz reflectivity of the Amplitude (namely A ( s ) = A (¯ s )). This(by way of the optical theorem) is enough to establish the weak a -theorem.By contrast at finite µ the answer for I is of course µ dependent. To demonstrate whathappens let us first revisit the simple example of free scalar fields of mass m discussedin [3]. Using standard perturbation theory (with the conventions of [3]) their contributionto the 4 dilaton amplitude is found to be A = −
240 ( a UV − a IR ) m f (cid:90) dx (cid:0) log( m − sx (1 − x )) + log( m + sx (1 − x )) (cid:1) + const . , (2.3)where we assume only these fields contribute to a UV − a IR . The constant term is inde-pendent of s and contains counter-terms to remove infinities, but it is not important forthe discussion. Expanding the logarithms in s/m and performing the x integral gives theleading contribution in (2.1) (which can be used to check the pre-factor). Alternatively wenote that the new absorptive contribution to A comes from the region of the integral wherethe argument of the first logarithm is negative, sx (1 − x ) > m . Taking s → s + i(cid:15) in orderto be above the branch-cuts, we findIm A = 240 π ( a UV − a IR ) m f (cid:112) − m /s . (2.4)Inserting this into the integral I with a cut-off then gives a running a parameter a UV − a KS ( µ ) = f π (cid:90) s>µ Im A ( s ) s ds, = ( a UV − a IR ) (1 − ρ ( µ/ m ) ) , (2.5)where ρ ( x ) = (cid:40) (1 − x − ) / (cid:0) x − (cid:1) ; x ≥
10 ; x ≤ . (2.6)For later comparison it is useful to rearrange the expression as a KS = a IR + ( a UV − a IR ) ρ ( µ/ m ) , (2.7)making it clear that ρ ( µ/ m ) correctly scales the contributions of the scalars to the a KS parameter continuously and monotonically, with ρ = 0 at µ = 2 m to ρ = 1 at µ → ∞ . Thuswe 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 > µ (auseful reference in this context is [20]).In order to compare the running a KS derived above with the continuously varying a K 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 N f pairs of superfields Φ a and˜Φ a , a = 1 . . . N f . The Lagrangian of [3] can be made supersymmetric in the obvious way,5y coupling the fields in a superpotential mass-term W ⊃ m Ω ˜Φ I ∆ f × ∆ f Φ, where Ω f isthe canonically normalised dilaton superfield with (cid:104) Ω (cid:105) = 1. This gives a supersymmetrypreserving mass m to ∆ f pairs of superfields. (As the superpartner of the dilaton does notappear in any loops of interest we can ignore it.)The amplitude is of course augmented by superpartner diagrams, but now supersym-metry guarantees that the coefficient of terms such as (2.3) vanish, because otherwise (asthese terms are not zero in the limit of vanishing external momenta and finite f ) theywould signal a renormalisation of the superpotential. The non-vanishing terms of interestare, in the standard Passarino-Veltman notation, of the form s B ( s, m , m ) and friends.Thus the contributions of interest are of the form A = −
24 ( a UV − a IR ) m sf (cid:90) dx (cid:0) log( m − sx (1 − x )) − log( m + sx (1 − x )) (cid:1) + const . , (2.8)where as before the second term is really the u Mandelstam variable with t → a KS from theabsorptive part which isIm A = 24 π ( a UV − a IR ) m f s (cid:112) − m /s . (2.9)Inserting this into (1.2) then gives (2.7), but with a modified scaling function, ρ ( x ) = (cid:40) (1 − x − ) / ; x ≥
10 ; x ≤ . (2.10)Let us now compare this expression with the a K function of [17]. The Lagrange mul-tiplier method for this simple case goes as follows. In general the running a -parameter isdefined by adding a Lagrange multiplier for each relevant operator. In this case there isonly one of them, which imposes the constraint from the mass term. The a -function istherefore given simply by˜ a K = ( N f − ∆ f ) (cid:2) R − − ( R − (cid:3) + ∆ f (cid:2) r − − ( r − (cid:3) + λ ∆ f [ r − , (2.11)where R is the R − charge of the N f − ∆ 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 super-conformal R -symmetry of the deep U V and the SU( N f ) × SU( N f ) flavour symmetry withwhich it mixes because of the mass-term (specifically the diag (cid:16) ∆ f I Nf − ∆ f , (∆ f − N f ) I ∆ f (cid:17) component) .One first solves to maximise the a -function with respect to unfixed R -charges, ∂a∂r = ∂a∂R =0. In the absence of the mass-term constraint this simply chooses the free-field value of 2 / R and r . However at arbitrary Lagrange multiplier values one finds R = 2 / ,r = 1 − √ − λ . (2.12) This is true for ∆ f < N f : when ∆ f = N f there is of course no relevant R -symmetry left. λ = 0 corresponds to R = r = 2 / λ = 1 correspondsto r = 1, which is the value forced upon it by the mass-term in the deep IR. Substitutingthese values into ˜ a K we have ˜ a UV = N f and ˜ a IR = ( N f − ∆ f ), and a running a − parametergiven by ˜ a K = ˜ a IR + (˜ a UV − ˜ a IR ) (1 − λ ) . (2.13)Comparison with (2.10) shows that the two a -functions precisely coincide if one makes theidentification λ ≡ m µ . Note that the a -functions in the supersymmetric case matchessentially because of the non-renormalisation theorem, and that as usual the Lagrangemultiplier is essentially the “coupling” that induces the flow.For the SUSY gauge theories of interest the situation is more complicated but theinterpretation is always the same; namely a KS counts the physical states that are able tocontribute to the absorptive part of the 4-dilaton amplitude. Meanwhile a K tracks themixing of the UV R -symmetry with flavour symmetry along the flow [18]. We will nowshow that at weak coupling, close to the Caswell-Bank-Zaks fixed point, they are equivalentin this case as well .Consider SQCD with N f flavours of quarks Q and ˜ Q flowing from the asymptoticallyfree theory to the fixed point. The a KS parameter was derived in terms of the gauge couplingin [14]; a KS ( µ ) = a UV − N c π (cid:90) ∞ g − ( µ ) dλλ β λ , (2.14)where λ = 1 /g . In the limit µ → b = N c N f (8 π ) , g ∗ = 8 π N f (cid:15) , (2.15)where (cid:15) = 3 N c − N f N c (cid:28) , (2.16)the integral gives a KS ( µ ) = a UV − N c N f π ) g ( µ )(2 g ∗ − g ( µ )) , (2.17)with g ( µ ) being a solution of the 2-loop RGE, dg d log µ = b g ( g − g ∗ ) . (2.18)Again we can compare this parameter to the continuously varying a K -function of [17].In an SU ( N ) gauge theory it can be written in generality as˜ a K = 2( N c −
1) + (cid:88) i | r i | ( a ( R i ) − ( R i − R IR i ) a (cid:48) ( R i )) , (2.19)where | r i | is the dimension of the representation r i , the prime means derivative with respectto R and where a ( r ) ≡ r − − ( r − . (2.20) 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.
7n the case of electric SQCD this gives˜ a K ( µ ) = 2( N c −
1) + 2 N c N f (cid:0) a ( R Q ) − ( R Q − R ∗ Q ) a (cid:48) ( R Q ) (cid:1) . (2.21)In order to compare with a KS we relate the R -charges to the anomalous dimensions through R Q = 23 (cid:16) γ Q (cid:17) . (2.22)This equation holds along the flow, but only at the endpoints of the flow does R Q coin-cide with the respective super-conformal R -charges of the fixed points. The anomalousdimension can be perturbatively calculated at 1-loop as γ Q = − N c π g , (2.23)and then using R UV Q = 23 , R ∗ Q = 1 − N c N f , (2.24)we easily find the same leading contribution as that in (2.17), and hence a K ≡ a KS as wewished to prove.At this point one could ask, what is the meaning of equating a scheme independentquantity such as a KS ( µ ) with a scheme dependent one such as a K ( µ ). This is of coursewhat we always do when we calculate a cross section (in which we are bound to choosea scheme), and compare it to its (scheme independent) measured value. The theoreticalresult becomes scheme independent only when all terms in perturbation theory are takeninto 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 perturbativedescription of the flow in a strongly coupled theory. We now wish to explore how this equivalence can be used to determine the gauge couplingflow 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 weaklycoupled magnetic description, described in [12] .The original electric SQCD theory is an N = 1 SU( N c ) theory with N f + 1 flavours of Q and ˜ Q quarks and anti-quarks. We add a mass-term of the form W e = m Q N f +1 ˜ Q N f +1 (3.1)in its superpotential. In the IR, i.e. at energy below m , it flows to a new theory with N f flavours, hence effectively there is a UV fixed point with N f + 1 flavours at energy above m , and an IR fixed point with N f flavours. If we take 2 N f = 3 N c + 1 then the theory isexpected to be strongly coupled for large N c all along the flow. For a pedagogical description of such a set-up see for example [23]. N c + 1) theory with ˜ N c = N f − N c and, aswell as N f +1 flavours of quarks q and ˜ q , it contains an elementary ( N f +1) × ( N f +1) mesonΦ formed from a composite of the electric quarks, which we will take to be Φ ≡ Q · ˜ Q where Λ is the dynamical scale of the theory, and a superpotential W m = m ΛΦ N f +1 N f +1 + ˜ y Φ˜ q · q , (3.2)whose first term derives from the mass-term, and where the Yukawa coupling is ˜ y = Λ / ˆΛwith ˆΛ ∼ Λ. The magnetic theory which has N f = 3 ˜ N c − N c ). For completeness we summarise theflows as seen in the two dual theories in Table 1, where the RG scale is defined with respectto m , that is t ≡ log ( µ/m ) . (3.3)We can easily determine the difference between the UV and IR a -central charges˜ a UV − ˜ a IR = 2 N c ( N f + 1) a (1 − N c / ( N f + 1)) − N c N f a (1 − N c /N f )= 6 N c (2 N f + 1) N f ( N f + 1) , (3.4)which is positive for all N f >
0, and thus the weak a -theorem is satisfied.IR ( t <
0) UV ( t > N f flavours N f + 1 flavours˜ N c colours ˜ N c + 1 colourselectrictheory N f flavours N f + 1 flavours N c colours N c coloursTable 1: The dual theories considered in the text with N c = N f − ˜ N c . We considerthroughout the case of N f = N c + = 3 ˜ N c − . As discussed, our aim is to determine the gauge coupling for the original strongly coupledelectric 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 N f ≈ N c and large˜ N c ≡ N f − N c , the magnetic theory is made perturbative both in the UV and the IR sowe can calculate its flow with good accuracy along the whole RG trajectory. As we alsoknow the (in principle non-perturbative) interacting electric theories in both the UV andthe IR, we assume that the flow of the magnetic theory is dual to that of the stronglycoupled electric theory along the whole trajectory.Before entering into the explicit computation, let us clarify the idea and the procedurewe will follow. The magnetic theory is perturbative and is thus under control in the wholeregion 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 betafunctions and the two anomalous dimensions are, due to the mass independent characterof the NSVZ scheme, discontinuous at the explicit quark mass scale ( µ = m , t = 0). It is convenient to take the magnetic theory to be perturbative as then only one coupling - the electricgauge coupling - is non-perturbative and thus the matching of a -parameters determines it uniquely. µ < m ) we are able to use the a parameters to track(perturbatively and in the NSVZ scheme) the evolution of the strongly coupled theoryfrom that in the weakly coupled one. In particular the electric theory is non-perturbativeand 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 continuity we assume that the perturbative magnetic and non-perturbative electrictheories are dual between these two endpoints. In this region the physical quantity a KS isby definition the same in the magnetic and electric theory, and as motivated in the previoussection we equate a K and a KS , allowing us to explicitly calculate the central charge in thewhole energy range 0 < µ < m in the magnetic theory, as a perturbative function of thegauge and Yukawa coupling constants. This (via the equivalence of the a parameters) yieldsthe non-perturbative R -charges and hence anomalous dimensions of the strongly coupledelectric theory, which in turn yields the explicit numerical solution of the RGE for theelectric gauge coupling constant. < t < ∞ ): magnetic theory We start in the UV with the magnetic theory, which is an SU( ˜ N c + 1) gauge theory with N f + 1 quarks q + ˜ q and ( N f + 1) singlet meson fields with the superpotential of (3.2). Wewill work in terms of ˜ α g ≡ ( ˜ N c + 1)˜ g (4 π ) , ˜ α y ≡ ( ˜ N c + 1)˜ y (4 π ) . (3.5)The theory is asymptotically free when N f + 1 = 3 ˜ N c since b = 3( ˜ N c + 1) − ( N f + 1) = 3 , (3.6)and at t → ∞ all couplings go to zero. For µ (cid:29) Λ the 1-loop approximation is sufficient,and one has the usual evolution with dynamical scale given by Λ ≡ µ exp (cid:16) − b ˜ α g ( µ ) (cid:17) .Towards the IR the flow approaches a Banks-Zaks fixed point that for larger ˜ N c becomesincreasingly perturbative. Indeed the two-loop RGEs (see for example [11, 24]) give thefixed points to be at˜ α g (0 + ) = (cid:16) N f +1˜ N c +1 + 1 (cid:17) (cid:16) N f +1˜ N c +1 − (cid:17) N f +1˜ N c +1 − (cid:16) N f +1˜ N c +1 (cid:17) + 2 N f +1( ˜ N c +1) N f → N c − −−−−−−−→
72 ˜ N c + 432 ˜ N c + O (1 / ˜ N c ) , ˜ α y (0 + ) = 2 (cid:16) −
1( ˜ N c +1) (cid:17) (cid:16) N f +1˜ N c +1 − (cid:17) N f +1˜ N c +1 − (cid:16) N f +1˜ N c +1 (cid:17) + 2 N f +1( ˜ N c +1) N f → N c − −−−−−−−→ N c + 7˜ N c + O (1 / ˜ N c ) . (3.7)Defining ∆ (+) ˜ g ( t ) = ˜ α g ( t ) − ˜ α g (0 + ) , ∆ (+) ˜ y ( t ) = ˜ α y ( t ) − ˜ α y (0 + ) , (3.8)the 2-loop RGEs can be rephrased as d ˜ α g ( t ) dt = − α g ( t ) × (cid:34)(cid:18) − N f + 1˜ N c + 1 + 2 N f + 1( ˜ N c + 1) (cid:19) ∆ (+) ˜ g ( t ) + 2 (cid:18) N f + 1˜ N c + 1 (cid:19) ∆ (+) ˜ y ( t ) (cid:35) ,d ˜ α y ( t ) dt = 2 ˜ α y ( t ) × (cid:20) − (cid:18) −
1( ˜ N c + 1) (cid:19) ∆ (+) ˜ g ( t ) + (cid:18) N f + 1˜ N c + 1 + 1 (cid:19) ∆ (+) ˜ y ( t ) (cid:21) . (3.9) For a discussion on this point see for example [23]. α g (0 + ) and ˜ α y (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: R q (0 + ) = 1 − ˜ N c + 1 N f + 1 , R Φ (0 + ) = 2 − R q (0 + ) , (3.10)which are perturbative (free) R q = R Φ = 2 / N c limit with N f → N c −
1, inaccord with the magnetic theory being parametrically perturbative for all positive t . < t < ∞ ): electric theory Apart from the far UV (as it is also asymptotically free) the form of the electric dual theoryis known only in the t → + limit, where it is an SU( N c ) gauge theory with N f + 1 quarks Q + ˜ Q and vanishing superpotential. In the same limit the fixed point determines the valueof the R-charge: R Q (0 + ) = 1 − N c N f + 1 . (3.11)In the large N c limit (as we have N f = N c + 1) this value is clearly interacting, R Q → / Q · ˜ Q becoming free. We do not know the value of theelectric gauge coupling at other values of t . −∞ < t < ): magnetic theory We now turn to the flow of interest, towards the IR, for t <
0. Here the magnetic theoryis an SU( ˜ N c ) gauge theory with N f quarks q + ˜ q and N f × N f gauge singlet mesons, whichfor convenience we continue to call Φ. At t = 0 the boundary conditions of the couplings,˜ α g ≡ ˜ N c ˜ g (4 π ) , ˜ α y ≡ ˜ N c ˜ y (4 π ) , (3.12)are determined by continuity ,˜ α g (0 − ) = ˜ N c ˜ N c + 1 ˜ α g (0 + ) , ˜ α y (0 − ) = ˜ N c ˜ N c + 1 ˜ α y (0 + ) . (3.13)The flow of the magnetic theory can be determined perturbatively from the RGEs. Defining∆ ( − ) ˜ g ( t ) = ˜ α g ( t ) − ˜ α g ( −∞ ) , ∆ ( − ) ˜ y ( t ) = ˜ α y ( t ) − ˜ α y ( −∞ ) , (3.14) Note that we are using the NSVZ scheme which is a mass independent scheme: this means that theperturbative gauge couplings are continuous passing the mass scale, while the beta functions or anomalousdimensions are not. The apparent discontinuity in (3.13) is clear from the way the coupling constants aredefined in (3.12), i.e. with the discontinuity being in the number of colours. α g ( −∞ ) = (cid:16) N f ˜ N c + 1 (cid:17) (cid:16) N f ˜ N c − (cid:17) N f ˜ N c − (cid:16) N f ˜ N c (cid:17) + 2 N f ˜ N c N f → N c − −−−−−−−→
76 ˜ N c + 259 ˜ N c + O (1 / ˜ N c ) , ˜ α y ( −∞ ) = 2 (cid:16) − N c (cid:17) (cid:16) N f ˜ N c − (cid:17) N f ˜ N c − (cid:16) N f ˜ N c (cid:17) + 2 N f ˜ N c N f → N c − −−−−−−−→
13 ˜ N c + 89 ˜ N c + O (1 / ˜ N c ) , (3.15)they are d ˜ α g ( t ) dt = − α g ( t ) × (cid:34)(cid:18) − N f ˜ N c + 2 N f ˜ N c (cid:19) ∆ ( − ) ˜ g ( t ) + 2 (cid:18) N f ˜ N c (cid:19) ∆ ( − ) ˜ y ( t ) (cid:35) ,d ˜ α y ( t ) dt = 2 ˜ α y ( t ) × (cid:20) − (cid:18) − N c (cid:19) ∆ ( − ) ˜ g ( t ) + (cid:18) N f ˜ N c + 1 (cid:19) ∆ ( − ) ˜ y ( t ) (cid:21) . (3.16) The evolution is shown in Fig. 2 for ˜ N c = 100 and N f = 3 ˜ N c − - - - - - - α ˜ g - - - - - - α ˜ y Figure 2:
The perturbative running of the gauge (left) and Yukawa (right) coupling constantsof the magnetic theory for ˜ N c = 100 and N f = 3 ˜ N c − , from the UV fixed point ( t = 0 with N f + 1 quarks and ˜ N c + 1 colours), to the IR fixed point ( t = −∞ with N f quarks and ˜ N c 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 wecan do because the theory is perturbative (approximately, order by order in perturbationtheory). From the usual definition of the NSVZ beta function and the relation in (2.22),we have β ( ˜ α g ) = − α g f ( ˜ α g ) (cid:18) N f ˜ N c ( R q − (cid:19) ,β ( ˜ α y ) = 3 ˜ α y (2 R q + R Φ − , (3.17)with f ( x ) ≡ − x . (3.18)12omparison with the r.h.s. of (3.16) gives R q ( t ) − R q ( −∞ ) = − (cid:18) − N c (cid:19) ∆ ( − ) ˜ g + 23 N f ˜ N c ∆ ( − ) ˜ y + O (cid:0) ∆ (cid:1) ,R Φ ( t ) − R Φ ( −∞ ) = 23 ∆ ( − ) ˜ y + O (cid:0) ∆ (cid:1) . (3.19)Their evolution for t < terms as derived from Eq.(3.17), are actually required later in orderto get consistent convergence to the IR fixed point in the strongly coupled description. - - - - - - R q - - - - - - R Φ Figure 3:
The “flowing” R-charges (green) of the quark (left and meson (right) in themagnetic SQCD with gauge SU( ˜ N c ) and N f quarks q + ˜ q and N f mesons Φ , with N f =3 ˜ N c − . The flow has been found using the perturabative relations (3.17) and (3.17) andusing ˜ N c = 100 . The blue straight lines are the values R q (0 + ) and R Φ (0 + ) obtained in thefixed point above the mass m . Notice that the values R q (0 − ) and R Φ (0 − ) do not coincidewith them: although the gauge couplings ˜ g and ˜ y are continuous, the R -charges are not: theyare in some sense proportional to the non-continuous beta-functions. Finally the orangestraight lines are the limiting values R q ( −∞ ) and R Φ ( −∞ ) obtained from the IR fixed pointcouplings. −∞ < t < ): electric theory Up to this point, for t <
0, everything has been perturbative. Now let us now considerthe original electric theory in the range −∞ < t <
0. In the limit t → −∞ the theory isSU( N c ) SQCD with N f quarks Q + ˜ Q and no superpotential.Let us assume that the same pair of dual theories describe the physics along the wholeRGE running. As the parameter a KS is a function of the amplitude its definition is inde-pendent of which description is being used and hence its value in the electric and magnetictheories is the same all along the flow.We will adopt the assumption, motivated in the Introduction, that in regions wherethe beta functions are small the a KS -function is the same as the function a K derived usingthe Lagrange multiplier definition [17]. Hence using (2.21) and equating a K ’s in the twodescriptions as in the Appendix, one finds2( N c −
1) + 2 N f N c ( a ( R Q ( t )) − ( R Q ( t ) − R Q ( −∞ )) a (cid:48) ( R Q ( t )))= 2( ˜ N c −
1) + 2 N f ˜ N c ( a ( R q ( t )) − ( R q ( t ) − R q ( −∞ )) a (cid:48) ( R q ( t ))+ N f ( a ( R Φ ( t )) − ( R Φ ( t ) − R Φ ( −∞ )) a (cid:48) ( R Φ ( t ))) , (3.20)13hich can be used to determine R Q ( t ). Its behaviour is shown in Fig. 4. - - - - - - R Q Figure 4:
The R-charge (green) of the quark in the electric SQCD with gauge group SU( N c )and N f quarks Q + ˜ Q , with N f = 3 ˜ N c − , using (3.20). As before, the blue straight line isthe 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 betafunction. Defining the electric gauge coupling as α g ≡ N c g (4 π ) , (3.21)and using β ( α g ) = − α g f ( α g ) (cid:18) N f N c ( R Q − (cid:19) , (3.22)one can now integrate, to find F ( α g ( t )) − F ( α g (0 − )) = (cid:90) t dt (cid:48) (cid:18) N f N c ( R Q ( t (cid:48) ) − (cid:19) , (3.23)where F ( x ) ≡ (cid:18) x + 2 log x (cid:19) . (3.24)This can then be solved for α g . Note that as mentioned the O (∆ ) terms in Eq.(3.19) arerequired here. If they are omitted then there are order 1 /N c errors in the integrand, whichover the order − t ∼ N c running required to get to the fixed point, translates into errors oforder 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, α g (0 − ), inthe electric theory, but since the r.h.s. of (3.23) is negative, and since the gauge couplingmust obey α g < / f ( α g ) defined in (3.18) does not change sign, there is amaximum allowed value of α g (0 − ) given by F ( α maxg (0 − )) + (cid:90) −∞ dt (cid:48) (cid:18) N f N c ( R Q ( t (cid:48) ) − (cid:19) = F (1 / . (3.25)For our inputs this is given by α maxg (0 − ) = 0 . . (3.26)14s an illustrative example we take three different inputs (0 . , . , .
9) for the ratio α g (0 − ) /α maxg (0 − ) and obtain numerically the flows shown in Fig. 5 for the non-perturbativecoupling α g ( 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 unfortu-nately it cannot be determined . All we know is that it must be non-perturbative, sotoo small ratios α g (0 − ) /α maxg (0 − ) are unacceptable because they would not reproduce theknown anomalous dimensions in the deep IR.The entire flow including the R -charges can of course be expressed in terms of theLagrange 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 theorySU( N c ) with N f quarks Q + ˜ Q , for three different values of the ratio α g (0 − ) /α maxg (0 − ) , andfor ˜ N c = 100 , N f = 3 ˜ N c − and N c = N f − ˜ N c . Note that α g ≡ N c g (4 π ) . The critical exponent provides a mild but nevertheless important check on the consistencyof this picture. It is defined as the minimal eigenvalue of the matrix of coupling derivativesof the beta functions around the fixed point : β (cid:48) ≡ min (cid:26) positive eigenvalues (cid:18) ∂β α a ∂α b (cid:19) F . P . (cid:27) . (4.1)It is a renormalization scheme independent quantity and therefore should be equal for dualtheories [25, 26]. Usually of course this equivalence cannot be checked because one cannot It would be interesting to attempt to extend the approach to include the mass term explicitly withanother Lagrange multiplier, as for the free-field theory in the introduction. However as one would haveto describe a Higgsing in the magnetic theory, this would be significantly more complicated, and it is notclear how one could fix several Lagrange multipliers with only a single a -parameter. Here one cannot compare the full matrices or even all their eigenvalues, because for example the di-mensions of the matrices do not agree. However the minimal eigenvalue has a physical scheme-independentmeaning in both descriptions. a ( el ) KS = a ( mag ) KS ) allows itto 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: (cid:18) ∂β α a ∂α b (cid:19) F . P . = − α g ( −∞ ) (cid:16) − N f ˜ N c + 2 N f ˜ N c (cid:17) − α g ( −∞ ) (cid:18) (cid:16) N f ˜ N c (cid:17) (cid:19) α y ( −∞ ) (cid:16) − N c (cid:17) ( −
2) 2 ˜ α y ( −∞ ) (cid:16) N f ˜ N c + 1 (cid:17) . (4.2)For N f = 3 ˜ N c − / ˜ N c approximation, using (3.15): β (cid:48) mag = 73 ˜ N c , (4.3)while the second, larger, eigenvalue is found to be equal to 14 / (3 ˜ N c ).In the strongly coupled electric description, there is a single gauge coupling, so that β (cid:48) el = ∂β α g ( t ) ∂α g ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t →−∞ = ddt β α g ( t ) ddt α g ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t →−∞ = ddt log ( R Q ( t ) − R Q ( −∞ )) (cid:12)(cid:12)(cid:12)(cid:12) t →−∞ . (4.4)Usually in the non-perturbative theory the relation between R Q ( t ) and α g ( t ) is not known.Here however we have a relation between R Q ( t ) and the known R q ( t ) and R Φ ( t ) of theperturbative magnetic theory through (3.20). We may therefore expand a around t = −∞ , a ( t ) = a ( −∞ ) + (cid:88) i,j ∂ a∂R i ∂R j ( −∞ ) ( R i ( t ) − R i ( −∞ )) ( R j ( t ) − R j ( −∞ )) + . . . , (4.5)and from there must find ∂ a el ∂R Q ( −∞ ) ( R Q ( t ) − R Q ( −∞ )) ≈ ∂ a mag ∂R q ( −∞ ) ( R q ( t ) − R q ( −∞ )) + ∂ a mag ∂R M ( −∞ ) ( R Φ ( t ) − R Φ ( −∞ )) . (4.6)Since the second derivative of a over the R -charges is proportional to the b -central chargeof a conserved current (in this case it is the baryon current) and thus strictly non-zero, wemust have the same scaling, R Q ( t ) − R Q ( −∞ ) ∼ R q ( t ) − R q ( −∞ ) ∼ R Φ ( t ) − R Φ ( −∞ ) ∼ exp ( β (cid:48) mag t ) , (4.7)in the asymptotic region t → −∞ . But then from (4.4) we consistently find β (cid:48) el = β (cid:48) mag . (4.8)We conclude that a el = a mag along the flow is compatible with the equality of the electricand magnetic critical exponents. Of course this is not a particularly restrictive condition,and many other relations would have given equality. For example a mag = A ( a el ) (4.9)for an arbitrary function A ( x ) with A ( a el ( −∞ )) = a el ( −∞ ) ,A (cid:48) ( a el ( −∞ )) (cid:54) = 0 . (4.10)would suffice. 16 Conclusion
In this paper we discussed the use of the a central charge as a method of determining theflow in a strongly coupled supersymmetric theory from its weakly coupled dual. Althoughthere are other examples of exact duality in field theory along an entire flow (e.g. [27]) thismethod seems particularly general and well suited to N = 1 supersymmetry. Crucial tothe approach is the equivalence of the scale-dependent a -parameter determined from thefour-dilaton amplitude with an IR cut-off, and the a -parameter determined in the Lagrangemultiplier method of Ref. [17, 18] with “flowing” R -charges. We showed that this equiva-lence 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, inwhich the running R -charges are always determined precisely from the four-dilaton ampli-tude. In this scheme, which is clearly well defined regardless of which formulation is beingused, one can map the flow of a weakly coupled magnetic dual to the original strongly cou-pled electric theory. The specific system we considered was the well-known pair of originalSQCD Seiberg duals, with the magnetic description (with weak gauge and Yukawa couplingconstants) running perturbatively from a fixed point in the UV to a different fixed point inthe IR due to a mass-deformation, and the electric SQCD dual running between stronglycoupled fixed points due to a meson-induced Higgsing.We should add that the mapping only seems to work straightforwardly in the directionof magnetic to electric, as in that case there is only one R -charge to determine (namelythat 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 conjunctionwith a -maximisation [28], but is less obvious. Acknowledgments
We are extremely grateful to Colin Poole for interesting discussions. BB acknowledges thefinancial 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 underthe 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 discussedin Section 3. We start with the original magnetic a -function,˜ a mag = 2 (cid:16) ˜ N c − (cid:17) + 2 ˜ N c N f a ( R q ) + N f a ( R Φ ) (A.1) − ˜ λ g N f ( R q − R q ( −∞ )) + ˜ λ y N f (2( R q − R q ( −∞ )) + ( R Φ − R Φ ( −∞ ))) . By a -maximisation we have [28] ∂ ˜ a mag ∂R q = ∂ ˜ a mag ∂R Φ = 0 , (A.2)which gives the following for the Lagrange multipliers:˜ λ g = 2 ˜ N c a (cid:48) ( R q ) − N f a (cid:48) ( R Φ ) , (A.3)˜ λ y = − N f a (cid:48) ( R Φ ) . (A.4)17imilarly for the electric theory one gets˜ a el = 2 (cid:0) N c − (cid:1) + 2 N c N f a ( R Q ) − λ g N f ( R Q − R Q ( −∞ )) , (A.5)giving λ g = 2 N c a (cid:48) ( R Q ) (A.6)Plugging (A.3) and (A.4) into (A.1), (A.6) into (A.5), and equating the two a -centralcharges, we obtain (3.20).From the perturbative knowledge of R q ( t ) and R Φ ( t ) we can thus draw ˜ λ g ( t ) in themagnetic theory discussed in the main text, while from the non-perturbative knowledge of R Q ( t ) using (3.20) we get λ g ( t ) for the strongly coupled electric theory. The graphs areshown in figs. 6 and 7. - - - - - - λ ˜ g - - - - - - λ ˜ y Figure 6:
The Lagrange multipliers of the magnetic theory. - - - - - - λ g Figure 7:
The Lagrange multiplier of the electric theory.
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