Critical exponent η at O(1/N^3) in the chiral XY model using the large N conformal bootstrap
CCritical exponent η at O (1 /N ) in the chiral XY model using thelarge N conformal bootstrap J.A. Gracey,Theoretical Physics Division,Department of Mathematical Sciences,University of Liverpool,P.O. Box 147,Liverpool,L69 3BX,United Kingdom.
Abstract.
We compute the O (1 /N ) correction to the critical exponent η in the chiral XY orchiral Gross-Neveu model in d -dimensions. As the leading order vertex anomalous dimensionvanishes, the direct application of the large N conformal bootstrap formalism is not immediatelypossible. To circumvent this we consider the more general Nambu-Jona-Lasinio model for ageneral non-abelian Lie group. Taking the abelian limit of the exponents of this model producesthose of the chiral XY model. Subsequently we provide improved estimates for η in the threedimensional chiral XY model for various values of N . LTH 1251 a r X i v : . [ h e p - t h ] J a n Introduction.
The Gross-Neveu model, [1, 2], is one of the core quantum field theories that is on a similar levelto scalar φ theory since it too has a quartic self-interaction but is purely fermionic. By contrastalthough it is renormalizable in two space-time dimensions rather than four, more interestinglyit is asymptotically free, [1]. It shares this feature with four dimensional non-abelian gaugetheories such as Quantum Chromodynamics (QCD) [3, 4] and, moreover, has the property ofdynamical symmetry breaking leading to non-perturbative mass generation, [1]. In more recentyears the original Gross-Neveu model of [1] and its extensions have been studied in the context ofultraviolet completion, [5, 6]. This is where the two dimensional theory is extended to becomea renormalizable one in four dimensions which is achieved in several stages. The first is tointroduce a Hubbard-Stratonovich transformation which produces an auxiliary scalar field. Intwo dimensions this field becomes dynamical and corresponds to a bound state of two fermions,[1]. In the ultraviolet completion to four dimensions, however, this auxiliary field develops afundamental propagator, [5]. So to ensure four dimensional renormalizability it is necessary forthe scalar field to have a quartic self-interaction. This in essence describes the structure of aGross-Neveu-Yukawa theory which is the ultimate point of the ultraviolet completion exercise,[5]. Indeed this particular theory has been of interest for many years due to its potential tomodel a composite Higgs field in extensions of the Standard Model, [7], for example.Our description of this connection between two and four dimensions is primarily for theoriginal Gross-Neveu model which has a discrete chiral symmetry. In this context it is sometimesreferred to as the Ising Gross-Neveu model. It is not the only version of a Gross-Neveu model inthat alternatively one can endow the basic quartic fermion interaction with a continuous chiralsymmetry. In two dimensions this is known as the chiral Gross-Neveu model, [1], and that has anultraviolet completion to what is termed the chiral XY model, [6]. This model as well as the Isingcase have been the subject of considerable interest in the last decade or so due to its underlyingconnection with models of phase transitions in graphene. Stretching a sheet of carbon atoms canchange the electrical properties of the material from a semi-conductor to a Mott insulator, [8, 9].Indeed the problem of evaluating the critical exponents describing the phase transition in thedimension of interest, which is three, has generated an immense amount of activity. The maintheoretical approaches include the application of the functional renormalization group, [10, 11],conformal bootstrap, [12, 13], Monte Carlo methods, [14], the (cid:15) expansion, [15, 16], and thelarge N expansion, [17]. We note that this is a representative set of recent articles rather thana definitive one. The connection between the (cid:15) expansion and the large N approach is quiteclose. For the former the renormalization group functions are calculated to high loop order inthe ultraviolet completed four dimensional model. Then the critical exponents at the d = 4 − (cid:15) dimensional Wilson-Fisher fixed point are computed before the (cid:15) expansion of an exponent issummed up prior to setting (cid:15) = . In some Gross-Neveu universality classes the same exercise hasbeen additionally performed in the original two dimensional theory. In that case the exponentsare determined at the Wilson-Fisher fixed point in an ¯ (cid:15) expansion near d = 2 + 2¯ (cid:15) dimensionseven though the theory is only perturbatively renormalizable in strictly two dimensions. Despitethese respective theories not being in the neighbourhood of three dimensions, accurate estimatesfor exponents have been extracted in certain cases. For instance, the current situation in theIsing Gross-Neveu model has been summarized in [16].By contrast in the large N critical point approach pioneered in scalar O ( N ) models in[18, 19, 20], the first three terms of critical exponents have been computed in d -dimensionalspace-time as a function of d . The large N expansion is renormalizable, [21, 22], primarily dueto 1 /N being a dimensionless parameter. Using the first three terms in the series for relativelylow N , estimates of exponents in three dimensions have been shown to be competitive with2hose of the other techniques mentioned. It is these large N exponents depending on d thatdrive the close connection with the exponents determined from the (cid:15) expansion of the explicitperturbative renormalization group functions. This is due to the general observation that theexponents depend on both variables N and (cid:15) . So expanding those in powers of 1 /N and (cid:15) theexpressions derived using either technique separately should be in agreement in the area whereterms in the double expansion overlap. Indeed this is the case in many models. Moreover thelarge N exponents can be expanded to all orders in (cid:15) at each order in 1 /N thereby providingnon-trivial checks on future perturbative results.While the main focus so far has been on the Ising and chiral Gross-Neveu models and theirconnections with graphene phase transitions, the chiral XY model itself has been explored inrelation to certain phases in matter. For instance, the correlated symmetry protected phaseswere investigated in depth in [23]. As this model clearly is of importance to these areas it isworth summarizing the current status of the analytic evaluation of the exponents. For instance,the renormalization group functions of the chiral XY model are available to four loops, [6, 15].Equally the (cid:15) expansion of the large N critical exponents are known to be in agreement withthose high order results. More specifically the exponents η , η φ and 1 /ν corresponding to thefermion and scalar field dimensions as well as the correlation length exponent respectively areeach known to O (1 /N ), [17, 24]. However, for η this means only the first two terms in 1 /N are available since the fermion has zero canonical dimension. However the large N conformalbootstrap formalism of [20] allows one to determine the O (1 /N ) term of η . This has beenachieved for the Ising Gross-Neveu model in [25, 26, 27], as well as a variant known as thechiral Heisenberg Gross-Neveu model in [28] but not yet for the chiral XY case. This is dueto a technical reason. In the large N conformal bootstrap technique, that is founded on theearly work of [29, 30, 31, 32], the expansion is in powers of the vertex anomalous dimensions.In particular in one of the amplitude variables the vertex anomalous dimension appears in thedenominator. However in the chiral XY Gross-Neveu case the leading order large N vertexexponent is zero. So one cannot immediately effect the formalism at the outset. In reality onehas a mathematically ill-defined quantity akin to 0 /
0. However since the application of theoriginal formalism of [28, 29] to the chiral XY model can indeed produce non-singularly derivedexpressions for η to O (1 /N ), [24], there ought to be a way of applying the general large N conformal bootstrap technology of [20] as well.Given the need for having as accurate as possible data on the phase transition exponentsfor the chiral XY model we have found an indirect method to determine η at O (1 /N ) in d -dimensions. This is the main topic of the article and it is worth outlining our strategy at theoutset. Instead of considering the chiral XY model itself we will carry out a computation ina generalized theory which possesses a non-abelian symmetry. In essence this is the Nambu-Jona-Lasinio model, [33]. Critical exponents have been computed in the large N expansion tohigh order in several cases previously. For instance, the exponents of the fields for the group SU ( N c ) have been determined at O (1 /N ) as well as that for the correlation length, [24, 34]. Acalculation of η at O (1 /N ) for the SU (2) Nambu-Jona-Lasinio model was given in [35]. Howeverwe will consider the case of a general Lie group rather than specific unitary ones. The reasonfor this is that then the exponents will depend on the general colour Casimir group invariantssuch as the standard ones of C F and C A as well as higher rank ones. In this way previousresults should be accessible in various limits. En route though we will identify several errors inearlier results. Unlike the present situation, when the earlier exponents were determined no highorder (cid:15) -expansion perturbative results were available to check against. That is not the case nowsince four loop chiral XY model results are available, [15]. Large N exponents for that modelare subsequently deduced from the general non-abelian Nambu-Jona-Lasinio model by simplytaking the abelian limit. It will be evident from the expression for η at O (1 /N ) that there3re no singularities meaning that a smooth abelian limit can be taken. Indeed we will be ableto show the origin of the problem in the direct large N conformal bootstrap application to thechiral XY model. As a corollary we will provide an improved estimate for η in three dimensionsto the same number of terms as β φ and ν .The article is organized as follows. We discuss the Lagrangians of the various theories we willconsider for the large N analysis in Section 2 where the large N critical point formalism is brieflyreviewed. This is applied in the following section to determine the fermion critical exponent at O (1 /N ) in the Nambu-Jona-Lasinio model. Subsequently we complete the evaluation of bothscalar field anomalous dimensions at O (1 /N ) in Section 4 by computing the vertex anomalousdimensions. These and the other exponents are necessary for the O (1 /N ) evaluation of thecorrelation length exponent 1 /ν which is carried out in Section 5. The focus then alters to thelarge N conformal bootstrap formalism with the derivation of the formal underlying equationsprovided in Section 6. The details are necessary due to the presence of more than one vertexunlike previous large N applications of the technique. The actual evaluation of η at O (1 /N ) isgiven in Section 7 prior to extracting the corresponding expression for the chiral XY model inthe abelian limit in Section 8. Estimates of the exponents in three dimensions are also providedthere. Finally, we provide concluding remarks in Section 9. To begin with we introduce the theories we will focus on. First the four dimensional chiral XYmodel has the renormalizable Lagrangian L XY = i ¯ ψ i ∂/ψ i + 12 ( ∂ µ ˜ σ ) + 12 ( ∂ µ ˜ π ) + g ¯ ψ i (cid:16) ˜ σ + iπγ (cid:17) ψ i + 124 g (cid:16) ˜ σ + ˜ π (cid:17) (2.1)where 1 ≤ i ≤ N introduces our expansion parameter. The renormalization group functionsof (2.1) are available in [15] and the (cid:15) expansion of the associated critical exponents at theWilson-Fisher fixed point can be summed up to obtain estimates of the corresponding theory inthree dimensions. Here our convention for (cid:15) is defined by d = 4 − (cid:15) . Underlying the Wilson-Fisher fixed point is a universal theory which is driven by a core interaction that is relevant atcriticality in all dimensions. It is this universal theory that one applies the large N formalismof [18, 19, 20] to as the expansion parameter 1 /N is dimensionless in all space-time dimensions.In the case of (2.1) the universal Lagrangian is L XYU = i ¯ ψ i ∂/ψ i + σ ¯ ψ i ψ i + iπ ¯ ψ i γ ψ i − g (cid:16) σ + π (cid:17) . (2.2)The scalar fields σ and π are related to ˜ σ and ˜ π of (2.1) respectively by a coupling constantrescaling. This is because at criticality the coupling constant is in effect fixed and plays no roledue to the scaling and conformal symmetry. In (2.2) each field has a dimension that comprisesa canonical part and an anomalous piece. The former is determined from ensuring the action isdimensionless in d -dimensions and we define the full dimensions of the three fields by α ψ = µ + η , β σ = 1 − η − χ σ , β π = 1 − η − χ π (2.3)where η is the fermion anomalous dimension with χ σ and χ π corresponding to the anomalousdimensions of the respective σ and π a vertex operators. At this stage we assume that the lattertwo are unequal prior to their determination.The problem, however, is that at leading order in 1 /N both χ σ and χ π are zero, [24]. Asthe large N conformal bootstrap programme is based on an expansion in the vertex anomalous4imension, for (2.2) the expansion cannot begin. This is because the factor associated withthe Polyakov conformal triangle, that is at the core of each vertex in the large N bootstrapformalism, depends on 1 /χ σ and 1 /χ π which we will demonstrate later. While previous workproduced various large N exponents to O (1 /N ), [24, 27], and showed that the vertex exponentsare non-zero at O (1 /N ) for (2.2), a strategy needs to be devised to avoid the problem of thissingularity in the bootstrap construction at leading order. The direction we have chosen to go inis to consider a more general theory which contains the universality class (2.2) in a specific limitand this is the non-abelian Nambu-Jona-Lasinio model which has the universal Lagrangian, [34], L NJLU = i ¯ ψ iI ∂/ψ iI + σ ¯ ψ iI ψ iI + iπ a ¯ ψ iI γ T aIJ ψ iJ − g (cid:16) σ + π a (cid:17) . (2.4)Here the fermion field has a non-abelian symmetry which we take to be any Lie group, classicalor exceptional, with generators T a and structure constants f abc while the index ranges are1 ≤ a ≤ N A and 1 ≤ I ≤ N c where N c and N A are the respective dimensions of the fundamentaland adjoint representations and π a is a vector in group space. The four dimensional equivalentto (2.1) is L NJL = i ¯ ψ iI ∂/ψ iI + 12 ( ∂ µ ˜ σ ) + 12 ( ∂ µ ˜ π a ) + g ¯ ψ iI (cid:16) ˜ σδ IJ + i ˜ π a γ T aIJ (cid:17) ψ iJ + 124 g (cid:16) ˜ σ + ˜ π a (cid:17) (2.5)which has a similar coupling structure to (2.1).Comparing (2.2) and (2.4) the main difference is in the decoration of fields with indicesleading to Feynman rules with Lie group dependence. Structurally in terms of Feynman graphsthat will arise the only difference is the appearance of group factors. These will depend on theusual group Casimirs which are defined byTr (cid:16) T a T b (cid:17) = T F δ ab , T a T a = C F I N c , f acd f bcd = C A δ ab (2.6)where I N c is the N c × N c unit matrix. From the equations defining C F and C A we have therelation C F N c = T F N A . (2.7)However at high loop order it is known that new higher rank Lie group Casimirs can appear.For instance, these first occur in a four dimensional gauge theory in the four loop β -functionof QCD, [36]. For (2.4) it transpires that the same feature is present in higher order large N exponents and in particular the square of the fully symmetric tensor d abcdF arises where d abcdF = 16 Tr (cid:16) T a T ( b T c T d ) (cid:17) . (2.8)As will be evident from our final expressions the large N exponents of (2.4) will depend onthe group entities T F , C F , C A , N c , N A and d abcdF d abcdF . In light of this and observing that it ispossible to extract (2.2) from (2.4) by formally setting T a → C F → , T F → , N c → , C A → , d abcdF d abcdF → . (2.9)This then completes our computational strategy. We will focus in the main throughout ondetermining the large N critical exponents of (2.4) before finding those of (2.2) as a corollaryvia (2.9). 5aving justified why our main focus will be on (2.4) we recall the basic formalism for thelarge N critical point computations, [18, 19]. It is centred on the asymptotic behaviour of thepropagators in the limit to the Wilson-Fisher fixed point in d -dimensions where d does not playthe role of a regulator. Representing the asymptotic scaling form of each propagator by thename of its field, the coordinate space forms are, [37], ψ ( x ) ∼ A ψ x/ ( x ) α ψ (cid:104) A (cid:48) ψ ( x ) λ (cid:105) , σ ( x ) ∼ B σ ( x ) β σ (cid:104) B (cid:48) σ ( x ) λ (cid:105) π ( x ) ∼ C ( x ) β π (cid:104) B (cid:48) π ( x ) λ (cid:105) (2.10)where A ψ , B σ and B π are x -independent but d -dependent amplitudes. These will appear in twocombinations which we define by z = A ψ B σ , y = A ψ B π (2.11)and together with η and the other exponents can be expanded in powers of 1 /N through η ( µ ) = ∞ (cid:88) n =1 η n ( µ ) N n , z ( µ ) = ∞ (cid:88) n =1 z n ( µ ) N n , y ( µ ) = ∞ (cid:88) n =1 y n ( µ ) N n (2.12)for example. In addition to the leading terms of (2.10) we have included corrections to scalingcorresponding to the terms with ( x ) λ and associated amplitudes A (cid:48) ψ , B (cid:48) σ and B (cid:48) π . The exponent λ is usually used to determine the exponent 1 /ν which is related to the correlation length andwe will consider it here too. In that case the canonical value of λ is ( µ − x ) α = a ( α )2 α (cid:90) k e ikx ( k ) µ − α (2.13)in general to set notation where a ( α ) = Γ( µ − α )Γ( α ) . (2.14)Consequently we have ψ − ( x ) ∼ r ( α ψ − x/A ψ ( x ) µ − α ψ +1 (cid:104) − A (cid:48) ψ s ( α ψ − x ) λ (cid:105) σ − ( x ) ∼ p ( β σ ) B σ ( x ) µ − β σ (cid:104) − B (cid:48) σ q ( β σ )( x ) λ (cid:105) π − ( x ) ∼ p ( β π ) B π ( x ) µ − β π (cid:104) − B (cid:48) π q ( β π )( x ) λ (cid:105) (2.15)where the various functions are defined by p ( β ) = a ( β − µ ) a ( β ) , r ( α ) = αp ( α )( µ − α ) q ( β ) = a ( β − µ + λ ) a ( β − λ ) a ( β − µ ) a ( β ) , s ( α ) = α ( α − µ ) q ( α )( α − µ + λ )( α − λ ) (2.16)for arbitrary α , β and λ . 6 Evaluation of η . The asymptotic scaling forms of the propagators are necessary in order to algebraically repre-sent the behaviour of the Schwinger-Dyson equations in the critical region. To determine η at O (1 /N ) we consider the 2-point functions of the three fields of (2.2) and to the order we areinterested the relevant graphs that contribute are provided in Figures 1, 2 and 3. In each thereare no self-energy corrections on any of the propagators. This is because the propagator powersinclude the non-zero anomalous dimensions η , χ σ and χ π that account for such effects. In termsof counting with respect to the ordering parameter 1 /N , each closed fermion loop contributesa power of N whereas a σ or π a field has a factor of 1 /N associated with it. This is accountedfor through the N dependence of the amplitude combination variables z and y defined in (2.12).So, for example, all the two loop graphs differ from the respective one loop graphs of theirSchwinger-Dyson equations by a power of 1 /N . For the scalar O ( N ) theory of [18, 19] variousthree loop graphs also contribute at O (1 /N ) but the corresponding graphs are absent here.This is because the extra diagrams have closed fermion loops with three scalars external to thatloop. Such graphs are zero because of the trace over an odd number of γ -matrices. Substitutingthe propagators into the skeleton Schwinger-Dyson equation for ψ equates algebraically to0 = r ( α −
1) + zZ V σ ( x ) χ σ +∆ − C F yZ V π ( x ) χ π +∆ + (cid:104) z ( x ) χ σ +2∆ − C F zy ( x ) χ σ + χ π +2∆ + C F ( C F − C A ) y ( x ) χ π +2∆ (cid:105) Σ + O (cid:18) N (cid:19) (3.1)where we have included the various group theory factors associated with the non-abelian sym-metry, as well as the vertex renormalization constants Z V σ and Z V π , and effected the γ -algebra.These will have poles in the analytic regularization ∆ that is introduced through the replace-ment, [18, 19], χ σ → χ σ + ∆ , χ π → χ π + ∆ . (3.2)Such a regulator is required since the actual two loop corrections themselves represented by Σ for the ψ equation are divergent. ψ − + ++ ++ + Figure 1: Skeleton Schwinger-Dyson 2-point function for ψ at O (1 /N ).Once the group and amplitude dependence are factored off the remaining contribution is theFeynman integral itself. Its value is the same irrespective of which of the two interactions are7nvolved. Expressions for both Σ and Π are provided in [37] where the latter arises in Figures2 and 3. The representation of the respective skeleton Schwinger-Dyson equations analogous to(3.1) are 0 = p ( β σ ) + N N c zZ V σ ( x ) χ σ +∆ − N (cid:104) N c z ( x ) χ σ +2∆ − C F N c yz ( x ) χ σ + χ π +2∆ (cid:105) Π + O (cid:18) N (cid:19) (3.3)and 0 = p ( β π ) − N T F yZ V π ( x ) χ π +∆ − N T F (cid:104) y ( x ) χ π +2∆ − ( C F − C A ) yz ( x ) χ σ + χ π +2∆ (cid:105) Π + O (cid:18) N (cid:19) . (3.4)In all three cases we have not included vertex renormalization constants in the higher order termssince those counterterms would contribute to the next order in the respective expansions. Alsowe have cancelled off a common factor of x whose power is related to the canonical dimension.What remains are factors of x whose exponent involves a linear combination of the regulatorand the vertex anomalous dimensions. These cannot be neglected despite being small. This isbecause in the approach to criticality there is no overall scale. For instance as x → x ) χ σ +∆ = 1 + χ σ N ln( x ) + ∆ ln( x ) + O (cid:18) ∆; 1 N (cid:19) (3.5)to the order of interest. If the ln( x ) terms remain then they would give singularities in thelimit as x → ∞ . However the simple pole in each of Σ and Π is removed by the vertexcounterterm. Once this is fixed the remaining terms involve χ σ or χ π as well as a termdependent on the counterterm but where each is multiplied by ln( x ). These terms are removedby defining the respective vertex anomalous dimensions. A consistency check on this is that thesame solution for each of χ σ and χ π emerges from all three equations. σ − ++ + Figure 2: Skeleton Schwinger-Dyson 2-point function for σ at O (1 /N ).Eliminating z and y at successive orders in 1 /N from (3.1), (3.3) and (3.4) and expandingthe scaling functions for the 2-point functions of (2.16), we find η = − T F + C F N c ]Γ(2 µ − µ Γ( µ − − µ )Γ ( µ ) N c T F (3.6)at leading order. From the O (1 /N ) parts of the equation, after renormalization we deduce χ σ = − µ [ C F N c − T F ] η [ T F + C F N c ][ µ − , χ π = − µ [2 T F − C F N c + C A N c ] η µ − T F + C F N c ] (3.7)8o ensure no ln( x ) terms remain resulting in η = (cid:20) [2(2 µ − C F N c + T F ) − C F N c T F − µC F C A N c ] Ψ( µ )2[ µ − µ − µ − µ [ µ − [ C F N c + T F ] + (2 µ − µ + 1) µ [ µ − C F N c T F − µ µ − C F C A N c (cid:21) η [ T F + C F N c ] (3.8)from the finite part of the consistency equations. Here we use the shorthand notationΨ( µ ) = ψ (2 µ − − ψ (1) + ψ (2 − µ ) − ψ ( µ ) (3.9)where ψ ( z ) is the Euler ψ function. O (1 /N ) . One of the key ingredients to proceeding to higher order in the large N expansion is the provisionof the vertex anomalous dimensions. The general approach is similar to conventional perturba-tion theory in that at successive loop orders the wave function renormalization constants areneeded first prior to renormalizing the mass and coupling constants. While the fermion anoma-lous dimension is associated with the critical exponent η those for the scalar fields require χ σ and χ π and we summarize their determination in this section. While χ σ and χ π were deducedas a corollary to finding η by excluding ln( x ) terms thereby ensuring the scaling limit couldbe taken smoothly, the vertex dimensions can also be computed directly from the two vertex3-point functions. The leading order 1 /N corrections for the σ ¯ ψψ vertex are shown in Figure 4.Those for the π a ¯ ψγ T a ψ are virtually the same but with the modification that the external σ line is replaced by one representing π a . It is for this reason that we have not illustrated them.To extract expressions for χ σ and χ π from Figure 4 we follow the method of [18, 19, 38, 39]which is the large N critical point renormalization formalism. One uses the asymptotic scalingforms of the propagators (2.10) but ∆ regularized. This produces an expansion in powers of1 /N where at each order the terms are a Laurent series in ∆. The poles corresponding tothe divergences are absorbed into the vertex renormalization constants Z V σ and Z V π in generalalthough the O (1 /N ) terms of each are already available. Once the divergences are removedthe remaining finite part does not have a non-singular limit to criticality. By this we meanthat in coordinate space ln( x ) parts are present. Equally carrying out the computation inmomentum space the partner ln( p ) dependence remains. In each case there are terms whichinvolve χ σ ln( x ) and χ π ln( x ). Thus to ensure a scale free finite Green’s function these π − ++ + Figure 3: Skeleton Schwinger-Dyson 2-point function for π at O (1 /N ).9 Figure 4: Leading order corrections to σ ¯ ψψ x or p dependence. In the way we have presentedthe larger formalism in the previous section, the values that we extract for χ σ and χ π fromthe graphs of Figure 4 fully agree with (3.7). + + ++ + + ++ + + + Figure 5: Vertex corrections to σ ¯ ψψ O (1 /N ).At O (1 /N ) there are a considerably larger number of diagrams to evaluate. For instance,each of the three vertices in both graphs of Figure 4 will gain vertex corrections. These aredisplayed in Figure 5 where again we will only illustrate the graphs for the σ ¯ ψψ π a ¯ ψγ T a ψ are obtained by swapping the external scalar in each graph. In additionto these there will be contributions from expanding the graphs of Figure 4 out to O (1 /N ) dueto the N dependence in the anomalous part of the propagator exponents in each graph. Moresignificantly the set of graphs in Figure 6 need to be included. These are shown separately,partly for a reason that will become evident later, but mainly because they can be regarded asprimitive in the sense that the highest order of pole in ∆ is simple. One aspect of the evaluationof the graphs of Figure 6 requiring care is that of determining the group theory factor associatedwith the three loop light-by-light graphs. This is particularly more involved for the π a ¯ ψγ T a ψ vertex function when all the scalar fields are π a ones. In this instance one has to rationalizetraces of the form Tr (cid:16) T a T b T c T d (cid:17) for a general Lie group. To handle this we have used the color.h module that is written in the symbolic manipulation language Form , [40, 41]. Indeed10e have used
Form to handle the algebra associated with solving all the various Schwinger-Dyson self-consistency equations. The color.h routine is an encoding of the Lie group Casimiranalysis of [42] that goes beyond the rank 2 ones of (2.6). In particular the earlier trace can berewritten in terms of d abcdF d abcdF and d abc d abc where d abc = 12 Tr (cid:16) T a T ( b T c ) (cid:17) (4.1)is fully symmetric. Though for all the light-by-light graphs of Figure 6 the combination d abc d abc cancels in the final expressions for not only χ σ and χ π but also all the O (1 /N ) and higherorder exponents where d abcdF d abcdF occurs. + + ++ + + ++ + + + Figure 6: Additional corrections to σ ¯ ψψ O (1 /N ).Assembling the values of the various diagrams with their associated group theory factor andextracting the piece associated with the separate vertex exponents we find χ σ = (cid:34) µ [ µ −
1] [ T F − C F N c T F ]Θ( µ ) − µ [ µ − C F N c − µ (6 µ − µ + 14)[ µ − C F N c T F − µ (2 µ − µ − µ − µ − T F − µ Ψ( µ )2[ µ − (cid:104) µ − C F N c − T F ] − µC F C A N c (cid:105)(cid:35) × η [ T F + C F N c ] (4.2)from the graphs of Figure 5 and 6 whereΘ( µ ) = ψ (cid:48) ( µ ) − ψ (cid:48) (1) . (4.3)Similarly we find χ π = (cid:20) µ Ψ( µ )4[ µ − (cid:104) µ − C F N c − T F ) − µ − C F N c − T F ) C A N c + µC A N c (cid:105) µ Θ( µ )8[ µ − (cid:34) d abcdF d abcdF N c C F T F − T F − C A N c (cid:35) + µ (2 µ − µ − C F N c + 3 µ [ µ − C F N c T F − µ (2 µ − µ − (cid:34) d abcdF d abcdF N c C F T F − T F (cid:35) − µ (7 µ − µ − C F C A N c − µ (2 µ − µ + 1)12[ µ − C A N c − µ (8 µ − µ + 19 µ − µ − T F − µ (5 µ − µ − C A N c T F (cid:35) η [ T F + C F N c ] . (4.4)Both (4.2) and (4.4) are more involved than the expression for η primarily due to the light-by-light contributions with the term involving Θ( µ ) arising from the non-planar ones. We notethough that similar to [24] the vertex anomalous dimensions are equal at O (1 /N ) for the SU (2)group. + ++ ++ ++ Figure 7: Graphs for O (1 /N ) correction to the σ skeleton Schwinger-Dyson 2-point function todetermine λ using shorthand representation for the internal scalar fields. λ . Having (4.2) and (4.4) now means the scalar wave function anomalous dimensions are known at O (1 /N ). This was achieved by using the leading terms of the asymptotic scaling forms of thepropagators (2.10). To deduce other exponents one considers the correction to scaling terms andto find ν in particular at O (1 /N ) we take λ = 1 / (2 ν ) where the canonical value of λ is ( µ − O ( N )12odel of [18, 19] for the Gross-Neveu universality class there is a reordering of the diagramswith corrections on the σ and π a lines in both their self-consistency equations, [25, 26]. Thisstems from the correction term of the 2-point functions of (2.16) and in particular q ( β σ ) and q ( β π ). Specifically the leading order large N term of a ( β − µ + λ ) of (2.16) is O ( N ) for both β σ and β π rather than O (1) for the analogous function in the scalar theory of [19]. Consequentlyto find ν at O (1 /N ) the two graphs of Figures 2 and 3 with corrections on the σ and π a internalpropagators have to be included.This is evident when the self-consistency equation for λ is constructed. Clearly the correctionto scaling terms in (2.10) are of different length dimension to the leading terms. So the leadingand correction terms with regard to x in the algebraic representation of the Schwinger-Dysonequations decouple. The leading term that allowed us to deduce η has been dealt with butthree equations are left each involving the correction amplitudes A (cid:48) ψ , B (cid:48) σ and B (cid:48) π . The threeequations are combined into one equation involving a 3 × λ as the onlyunknown at leading order. However examining the N dependence of the terms in the matrix itis evident that it is different in different rows. Omitting the higher order graphs for the σ and π a Schwinger-Dyson equations would have left an inconsistent set of equations. The final stepto find λ is to set the determinant of this matrix to zero whence we find λ = − (2 µ − η . (5.1)The consequence of the reordering due to the singular nature of q ( β σ ) and q ( β π ) is that atnext order the higher order graphs of Figure 7 have to be included where the dot on a scalar lineindicates the propagator with the correction to scaling term of (2.10). Here we have introduceda shorthand notation to compress the number of graphs that actually contribute. In Figure7 the zigzag line represents both σ and π a fields. So, for instance, this means that in realitythere are four graphs from each possible choice zigzag line in the three loop graphs and eightfor the four loop ones. In total that is 48 graphs for each scalar field Schwinger-Dyson equation.The values of each graph independent of the group theory factor are available in [43]. Howeverappending the group values the solution to the determinant of the self-consistency equation tonext order in 1 /N gives λ = (cid:34)(cid:34) µ − µ − C F T F + 4 d abcdF d abcdF N c T F + 4 T F N c + (3 µ − µ + 2) C F C A N c T F (cid:35) µ [ T F + C F N c ][ µ − µ − η − (cid:34) C F N c T F + 2 T F + 124 C A C F N c + 2 d abcdF d abcdF N c T F (cid:35) µ (2 µ − µ − µ − (cid:104) Φ( µ ) + Ψ ( µ ) (cid:105) + (cid:104) µ ( µ − µ − µ − C F C A N c T F − ( µ + 1)( µ − ( µ − µ − C F N c − (2 µ − µ + 1)(2 µ − µ + 4 µ − T F − ( µ − µ − (2 µ − µ − µ − C F N c T F − (12 µ − µ + 251 µ − µ + 111 µ + 28 µ − C F N c T F + 124 µ ( µ − µ − µ + 20) C A C F N c + µ ( µ − µ − (2 µ − C A C F N c − µ (3 µ − µ − d abcdF d abcdF N c T F (cid:35) Ψ( µ )[ µ − [ µ − + (cid:20) µ (2 µ − µ − C F C A N c T F − µ (2 µ + 1)( µ − C F N c µ (2 µ − µ + 12) T F + 32 µ (2 µ − µ − C F N c T F + 32 µ (2 µ + 5 µ − C F N c T F − µ ( µ − C A C F N c + 34 µ (2 µ + 5)( µ − C A C F N c + 3 µ (5 µ − d abcdF d abcdF N c T F (cid:35) Θ( µ )[ µ − µ − µ − (2 µ − µ − µ + 1)2 µ [ µ − C F N c + µ (3 µ − µ − µ − C F C A N c T F + (8 µ − µ + 212 µ − µ + 145 µ + 48 µ − µ + 32 µ − µ [ µ − [ µ − T F + (24 µ − µ + 126 µ − µ − µ + 21 µ − µ [ µ − C F N c T F + [24 µ − µ + 730 µ − µ + 868 µ − µ − µ + 96 µ − µ [ µ − [ µ − C F N c T F − µ (8 µ − µ + 85 µ − µ + 20)48[ µ − [ µ − C A C F N c + 3 µ (2 µ − µ − C A C F N c − µ (4 µ − µ + 26 µ − µ + 7)2[ µ − [ µ − d abcdF d abcdF N c T F (cid:35) η [ T F + C F N c ] . (5.2)The contributions from the light-by-light graphs of Figure 7 are evident.Finally we can note that the expressions for the exponents to this stage are in agreementwith previously determined exponents in the Gross-Neveu model, [25, 26, 37, 43], in the limit C F → , T F → , N c → , C A → , d abcdF d abcdF → π a ¯ ψγ T a ψ vertex. For the Gross-Neveu XY modeltaking the limit of (2.9) we also find agreement with [17, 24]. These provide important checkson our exponents for (2.4). α α α = f ( α i , a i ) α α α a a a Figure 8: Definition of internal indices in the Polyakov conformal triangle of a scalar Yukawainteraction. ψ − + + Figure 9: Schwinger-Dyson equation for ψ .14 = σ − +0 = π − + Figure 10: Schwinger-Dyson equations for σ and π a . N conformal bootstrap equations. Before concentrating on the O (1 /N ) evaluation of η using the conformal bootstrap we devotethis section to recalling one of the key features of the formalism. This is the Polyakov conformaltriangle, [29, 44], which represents the full vertex function in the theory at criticality. Ordinarilya vertex operator has a non-zero anomalous dimension. In other words in coordinate space thesum of the exponents of the propagators joining to a 3-point vertex has a non-zero part thatmeans the overall dimension does not equal the canonical dimension. This is in fact the reasonwhy the conformal integration rule referred to as uniqueness, [19, 31], cannot directly be appliedto the two loop graphs of Figures 1, 2 and 3. The canonical dimension of each vertex in thetheory matches the value of (2 µ + 1) satisfied by the uniqueness rule of [37] but χ σ and χ π have small but non-zero values. The Polyakov conformal triangle construction allows one toexploit uniqueness as a computational tool by replacing the full vertex with a one loop graphwhere the exponents of the internal lines are arbitrary but are chosen to make the new internalvertices unique. For a Yukawa vertex the most general conformal triangle is shown in Figure 8,[20, 25, 26]. Specifically a + a + α = 2 µ + 1 a + a + α = 2 µ + 1 a + a + α = 2 µ + 1 (6.1)where α i are the exponents of the original bare vertex of the underlying Lagrangian and a i are thearbitrary internal exponents. By way of orientation, for example, one could have α = α = α and α = β σ for the σ ¯ ψψ vertex of (2.4). The overall function f ( α i , a i ) represents the normalization.While it is straightforward to solve the simultaneous equations (6.1) to find the a i in terms of α i in general we will leave this to its explicit use of the construction for (2.4).One of the reasons for determining the various exponents for (2.4) at O (1 /N ) using theskeleton Schwinger-Dyson equations was in part to establish η , χ σ and χ π at this order for thecomputation of η at O (1 /N ). Since this will use a different formalism knowledge of η will serveas an independent check while χ σ and χ π are necessary to extract η . The main differencein the two formalisms rests in how the class of Feynman diagrams are analysed in the criticalregion. In the conformal bootstrap the key graphs are the primitive ones in the sense that notonly are there no self-energy corrections but there are no vertex corrections. In the former casethis was because the propagators are dressed but now in the bootstrap approach the verticesare also dressed. So the focus is not on 2-point functions per se but instead on the vertex or3-point functions which we denote by V σ and V π for the respective vertices of (2.4). The methodto construct the consistency equations for each here has been discussed previously, [25, 26, 37],15 δ (cid:48) V σ ( δ (cid:48) ) = + 2 δ (cid:48) + 2 δ (cid:48) δ (cid:48) V π ( δ (cid:48) ) = + 2 δ (cid:48) + 2 δ (cid:48) Figure 11: δ (cid:48) regularized vertex functions.based on the early work of [29, 30, 44, 45, 46]. As that work involved Lagrangians with asingle coupling constant we have repeated those derivations for (2.4) which has two independentvertices and will now summarize.The starting point is the Schwinger-Dyson equations for the three 2-point functions of Fig-ures 9 and 10 where the blob represents the full vertex which will correspond to the Polyakovconformal triangle. Clearly the graphs in each figure contain the explicit ones of Figures 1, 2and 3. The next step in the derivation is to use the Schwinger-Dyson equation defining eachvertex function to replace the bare vertex in each 2-point function of Figures 9 and 10, [45, 46].These are shown in Figures 11 and 12 where the shaded box indicates all possible contributionsto the respective 4-point functions. The two relations illustrated in Figure 11, for example, areused to substitute for the bare vertices in the ψ Schwinger-Dyson equation of Figure 9. In bothequations for V σ and V π one has to take account of the two interactions in (2.2) rather thanone as in previous Gross-Neveu bootstrap computations. The appearance of the regularizingparameter δ (cid:48) is due to the fact that in the formalism of [45, 46] there is at least one divergentgraph in the final result for the conformal bootstrap consistency equation for each field. Theregularization is introduced in the exponent of the external line where the designation of 2 δ and2 δ (cid:48) appears. Unlike [20] we will denote the regularizing parameters by δ and δ (cid:48) instead of (cid:15) and (cid:15) (cid:48) in order to avoid confusion with the parameter that is usually associated with the regulatorof dimensional regularization. By contrast we recall that here the regularization is analytic.The situation after following the procedure of [20, 45, 46] is illustrated in Figure 13. In thatequation the Lorentz index indicates that the original full Schwinger-Dyson equation of Figure9 has been multiplied by the vector x µ , [45, 46], where x is the location of one of the externalvertices with the other at the origin. In following the manipulations to arrive at this equationin Figure 13 the contribution from the graphs involving the 4-point boxes have been eliminated.In terms of notation the directed solid line represents a coordinate space vector joining theendpoints of the respective full 3-point vertices. At this point we divide the graphs into thosewith the directed line joining the external points of one full vertex subgraph and those wherethere is a directed line between two different full vertices. The former set are divergent whilethe latter are finite. To see this explicitly one replaces the full vertex by the equivalent Polyakovtriangle defined by (6.1). As each of the vertices of the resulting three loop graphs are unique16 δV σ ( δ ) = + 2 δ δV π ( δ ) = + 2 δ Figure 12: δ regularized vertex functions.it is a simple exercise to apply the Yukawa uniqueness relation to find explicit expressions forall the graphs. For the graphs where the evaluation produces a finite expression then there isa cancellation due to the relative minus sign. This is because the regularized vertex functions V σ ( δ (cid:48) ) and V π ( δ (cid:48) ) are unity in the limit δ (cid:48) →
0, [20, 45, 46]. By contrast for the divergent graphsone has to retain the next term of the Taylor series in δ (cid:48) before taking the δ (cid:48) → δ rather than δ , [20],we are merely avoiding the appearance of δ internally in the conformal triangle. There isno ambiguity as ultimately the regularization will be lifted. We note that in Figure 11 andother figures we only include the relevant argument of the vertex functions for brevity. The fulldependence on the various exponents and parameters in fact is V σ = V σ (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) , V π = V π (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) . (6.2)The parameters ¯ z and ¯ y are similar in origin to those of the skeleton Schwinger-Dyson equationsof the earlier approach in that they depend on the amplitude combinations A ψ B σ and A ψ B π .They differ in value, however, due to the normalization factor in the definition of the conformaltriangle shown in Figure 8. The result of the exercise is the conformal bootstrap equation for ψ which is r ( α ψ −
1) = ¯ zt σ ∂∂δ (cid:48) V σ (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) δ =0 ,δ (cid:48) =0 − C F ¯ yt π ∂∂δ (cid:48) V π (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) δ =0 ,δ (cid:48) =0 (6.3)where we have included the group factors associated with (2.4) and t σ = a ( a ψ | σ − a ( α ψ − a ( b σ ) a ( β σ )( α ψ − ( a ψ | σ − a ( β σ − b σ ) t π = a ( a ψ | π − a ( α ψ − a ( b π ) a ( β π )( α ψ − ( a ψ | π − a ( β π − b π ) . (6.4)17 = ψ − µ + 2 δ (cid:48) + 2 δ (cid:48) + 2 δ (cid:48) + 2 δ (cid:48) − V σ ( δ (cid:48) ) 2 δ (cid:48) − V π ( δ (cid:48) ) 2 δ (cid:48) − V σ ( δ (cid:48) ) 2 δ (cid:48) − V π ( δ (cid:48) ) 2 δ (cid:48) Figure 13: δ (cid:48) regularized Schwinger-Dyson equation for ψ consistency equation.These are related to the residue with respect to δ (cid:48) of the earlier divergent graphs of Figure 9.In the expression for t σ the notation is a ψ | σ = µ + − β σ , b σ = µ + − α ψ + β σ (6.5)while a ψ | π = µ + − β π , b π = µ + − α ψ + β π (6.6)for t π which are the solutions to (6.1) for the internal exponents. In arriving at (6.3) we havealso made use of another result of [20, 45, 46] in the derivation of the bootstrap equations forthe vertex functions which is1 = V σ (¯ z, ¯ y ; α ψ , β σ , β π ; 0 , , V π (¯ z, ¯ y ; α ψ , β σ , β π ; 0 , . (6.7)These reflect the sum of all the graphs with dressed propagators and vertices contributing tothe two 3-point vertices. The 1 /N leading order graphs for both V σ and V π are shown in Figures15 and 16 which contain the graphs of Figures 4 and 5. The regularization is not necessary forthese bootstrap equations since none of the contributing graphs are divergent.The final part of the exercise in constructing the bootstrap equations is to repeat the proce-dure that gave (6.3) for the remaining two equations of Figure 10. The process is similar exceptthat now in these graphs there is a δ regularization on the external σ and π a fields. So theinitial part of the derivation of the equations uses the regularized results of Figure 12 where the4-point boxes will contain graphs with both σ and π a fields. This is the reason for the simplerforms compared with Figure 11. The result of implementing the regularized vertex of Figure 12is shown in Figure 17 for σ where the Lorentz index indicates that the scaling function has beenmultiplied by x µ too. The corresponding graphs for π a are similar to Figure 17 with the σ exter-nal legs replaced with π a fields in addition to the replacement of σ − µ with π − µ . While the fourgraphs of Figure 17 differ from the corresponding ones of Figure 13 it is still the case that thosewith the directed line connecting two conformal triangles are finite. Evaluating the remaining18 δ −→ δ − δ − δδ δ (cid:48) −→ δ (cid:48) − δ (cid:48) − δ (cid:48) δ (cid:48) Figure 14: Distribution of regularizing parameters δ and δ (cid:48) in a conformal triangle. = + Figure 15: Respective O (1 /N ) graphs V σσσ and V σππ contributing to V σ .divergent graphs one arrives at the respective conformal bootstrap consistency equation for σ and π a which are p ( β σ ) = N N c ¯ zt σ ∂∂δ V σ (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) δ =0 ,δ (cid:48) =0 p ( β π ) = − N T F ¯ yt π ∂∂δ V π (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) δ =0 ,δ (cid:48) =0 (6.8)where the same factors t σ and t π occur. Eliminating these and the amplitudes ¯ z and ¯ y finallyproduces the consistency equation for η which is r ( α ψ −
1) = p ( β σ ) N N c (cid:20) ∂∂δ (cid:48) V σ (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) (cid:20) ∂∂δ V σ (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) − + C F p ( β π ) N T F (cid:20) ∂∂δ (cid:48) V π (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) (cid:20) ∂∂δ V π (¯ z, ¯ y ; α ψ , β σ , β π ; δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) − (6.9) = + Figure 16: Respective O (1 /N ) graphs V πσσ and V πππ contributing to V π .19here the restriction indicates that both regularizations are lifted at the end of the evaluationof the underlying graphs. At this stage (6.9) is exact and its solution would determine η toall orders. However to achieve this would require the explicit values from all the contributinggraphs. σ − µ + 2 δ + 2 δ − V σ ( δ ) 2 δ − V σ ( δ ) 2 δ Figure 17: δ regularized Schwinger-Dyson equation for σ consistency equation.The consistency equation (6.9) can be refined though by recalling the observation of in[20]. For the moment we consider a general situation and denote a regularized n -point functionby Γ ( n ) (∆ i , δ, δ (cid:48) ) where each vertex is represented by a conformal triangle, the j th vertex hasanomalous dimension χ j ≡ j and two of the external legs have δ and δ (cid:48) regularizations. Thenthe Green’s function will have a simple structure. In the unregularized case it will formally beΓ(∆ i , , / ( (cid:81) nj =1 ∆ j ) but in the regularized caseΓ ( n ) (∆ i , δ, δ (cid:48) ) = Γ(∆ i , δ, δ (cid:48) )(∆ − δ )(∆ − δ (cid:48) )( (cid:81) nj =3 ∆ j ) (6.10)where legs 1 and 2 are chosen to have the respective regularizations δ and δ (cid:48) and Γ ( n ) (∆ i , δ, δ (cid:48) )is analytic in all the ∆ i and both regulators. Then ∂∂δ Γ ( n ) (∆ i , δ, δ (cid:48) ) = 1∆ + 1( (cid:81) nj =1 ∆ j ) ∂∂δ Γ(∆ i , δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) δ =0 ,δ (cid:48) =0 ∂∂δ (cid:48) Γ ( n ) (∆ i , δ, δ (cid:48) ) = 1∆ + 1( (cid:81) nj =1 ∆ j ) ∂∂δ (cid:48) Γ(∆ i , δ, δ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) δ =0 ,δ (cid:48) =0 (6.11)for instance. The first term depends only on ∆ and ∆ respectively since the equivalent of(6.7) has been employed and the relation can be written more compactly as, [20], ∂∂δ Γ ( n ) (∆ i , δ, δ (cid:48) ) = 1∆ (cid:34) ∂ Γ ( n ) ∂δ (cid:12)(cid:12)(cid:12)(cid:12) res ; δ =0 ,δ (cid:48) =0 (cid:35) ∂∂δ (cid:48) Γ ( n ) (∆ i , δ, δ (cid:48) ) = 1∆ (cid:34) ∂ Γ ( n ) ∂δ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) res ; δ =0 ,δ (cid:48) =0 (cid:35) (6.12)where res indicates the contribution from the derivative of the residue of Γ ( n ) (∆ i , δ, δ (cid:48) ) only.This was for a general scenario but returning to (2.4) and applying this procedure to (6.9) wehave r ( α ψ −
1) = p ( β σ ) N N c (cid:34) σ ∂V σ ∂δ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) res ; δ =0 ,δ (cid:48) =0 (cid:35) (cid:34) σ ∂V σ ∂δ (cid:12)(cid:12)(cid:12)(cid:12) res ; δ =0 ,δ (cid:48) =0 (cid:35) − + C F p ( β π ) N T F (cid:34) π ∂V π ∂δ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) res ; δ =0 ,δ (cid:48) =0 (cid:35) (cid:34) π ∂V π ∂δ (cid:12)(cid:12)(cid:12)(cid:12) res ; δ =0 ,δ (cid:48) =0 (cid:35) − (6.13)20here we have now set χ σ = 2∆ σ , χ π = 2∆ π (6.14)and the arguments of the two vertex functions have been suppressed for brevity but are thesame as the corresponding terms in (6.13). + + ++ + + ++ + + + Figure 18: O (1 /N ) graphs contributing to V σ .For practical purposes it is best to expand the two vertex functions by setting V σ = ∞ (cid:88) i =1 V σ i N i , V π = ∞ (cid:88) i =1 V π i N i (6.15)although we will only need V σ , V σ , V π and V π to determine η . If one focuses on the leadingorder large N piece of (6.13) it is equivalent to that which produced η earlier. So, for instance,a contribution to the O (1 /N ) part from the first term on the right hand side of (6.13) can besimplified to ∆ σ (cid:20) ∂V σ ∂δ (cid:48) − ∂V σ ∂δ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) res ; δ =0 ,δ (cid:48) =0 (6.16)for the V σ vertex while the contribution from the second term involving V π is similar. η . Having derived the key equation (6.13) satisfied by η we now turn to details of its determina-tion. This entails computing the corrections to the vertex functions of the conformal bootstrapequations (6.7) and (6.13). The leading order contributions are shown in Figures 15 and 16 andcorrespond to V σ and V π respectively where V σ = V σσσ + V σππ , V π = V πσσ + V πππ . (7.1)21t next order the graphs that comprise V σ and V π are illustrated in Figures 18 and 19 and areclearly primitive. While the leading order 1 /N terms of both V σ and V π are needed to verify η the next term in the series is required for η . This N dependence arises through the exponents α ψ , β σ and β π . Therefore what are sometimes termed 3-gamma graphs, which are illustratedin Figures 15 and 16, have to be computed. This is achieved via the method developed in [20]using the properties of the Polyakov conformal triangle. + + ++ + + ++ + + + Figure 19: O (1 /N ) graphs contributing to V π .For instance, the explicit designation of the exponents on the lines of both V σσσ and V σππ are shown in Figures 20 and 21 where γ = 1 − η (7.2)is set for shorthand due to β σ and β π differing only in the piece involving the respective vertexanomalous dimensions. The factor of 2 associated with ∆ σ and ∆ π is a consequence of (6.5)and (6.6). The graph of Figure 20 was evaluated in [27]. If we set V σσσ = − Q ∆ σ (∆ σ − δ )(∆ σ − δ (cid:48) ) exp[ F σσσ ( δ, δ (cid:48) , ∆ σ )] (7.3)which is consistent with the general form of Γ ( n ) then we recall that F σσσ ( δ, δ (cid:48) , ∆ σ ) = (cid:34) B γ − B α ψ − − B − α ψ − (cid:35) ∆ σ − [ B γ − B ] δ + (cid:34) B − B α ψ − − α ψ − (cid:35) δ (cid:48) + (cid:34) C α ψ − − α ψ − (cid:35) δδ (cid:48) + (cid:34) C γ + C − C α ψ − + 2( α ψ − (cid:35) ∆ σ δ
22 12 (cid:34) α ψ − − C α ψ − − C (cid:35) δ (cid:48) + (cid:34) C − C γ − C α ψ − + 2( α ψ − (cid:35) ∆ σ δ (cid:48) −
12 [ C γ + C ] δ + (cid:34) C α ψ − − C γ − C − α ψ − (cid:35) ∆ σ (7.4)with the shorthand notation, [20], B z = ψ ( µ − z ) + ψ ( z ) , B = ψ (1) + ψ ( µ ) C z = ψ (cid:48) ( z ) − ψ (cid:48) ( µ − z ) , C = ψ (cid:48) ( µ ) − ψ (cid:48) (1) (7.5)and Q = − π µ a ( α ψ − a ( γ )( α ψ − Γ( µ ) . (7.6)We note that in [27] not all the terms quadratic in combinations of the parameters δ or δ (cid:48) wererecorded. We have included them here for completeness and as an aid to an interested readeralthough they do not contribute to the consistency equation. α + ∆ σ − δ (cid:48) γ − σ α + ∆ σ γ − ∆ σ − δ (cid:48) α + ∆ σ + δ (cid:48) α + ∆ σ γ − ∆ σ α αγ − ∆ σ + δα + ∆ σ − δ α + ∆ σ − δ Figure 20: Regularized one loop 3-gamma graph denoted by V σσσ contributing to V σ containinga σ propagator and including the conformal triangles.We have rederived (7.4) that appeared in [27] for V σσσ in order to extend it for V σππ . Thelatter is more general than the former since it will depend on χ π as well as χ σ . This leads toa check in that formally taking the limit χ π → χ σ in V σππ should produce the expression for V σσσ . If we defineΓ σππ = − Q ∆ π (∆ σ − δ )(∆ π − δ (cid:48) ) exp[ F σππ ( δ, δ (cid:48) , ∆ σ , ∆ π )] (7.7)and repeat the procedure that resulted in (7.4) we find F σππ ( δ, δ (cid:48) , ∆ σ , ∆ π ) = [ B − B γ ] δ + (cid:34) B − B α ψ − − α ψ − (cid:35) δ (cid:48) + [ B γ − B ] ∆ σ (cid:34) B γ − B − B α ψ − − α ψ − (cid:35) ∆ π −
12 [ C γ + C ] ∆ σ + (cid:34) α ψ − − C α ψ − − C γ − C (cid:35) ∆ π + [ C γ + C ] δ ∆ σ + 2 (cid:34) C α ψ − − α ψ − (cid:35) ∆ σ ∆ π + (cid:34) α ψ − − C α ψ − (cid:35) δ (cid:48) ∆ σ + (cid:34) C − C α ψ − − C γ + 1( α ψ − (cid:35) δ (cid:48) ∆ π −
12 [ C γ + C ] δ + 12 (cid:34) α ψ − − C α ψ − − C (cid:35) δ (cid:48) + 2 (cid:34) α ψ − − C α ψ − (cid:35) δ ∆ π + (cid:34) C α ψ − − α ψ − (cid:35) δδ (cid:48) . (7.8)Clearly this is consistent with (7.4) since F σππ ( δ, δ (cid:48) , ∆ σ , ∆ σ ) = F σσσ ( δ, δ (cid:48) , ∆ σ ). These two ex-pressions are sufficient to calculate the contribution to (6.13) from V σ . For the second termof (6.13) involving V π the expressions for the corresponding 3-gamma graphs can be deducedfrom those of V σ by formally swapping ∆ σ and ∆ π in (7.4) and (7.8). α + ∆ π − δ (cid:48) γ − π α + ∆ π γ − ∆ π − δ (cid:48) α + ∆ π + δ (cid:48) α + ∆ π γ − ∆ π α αγ − ∆ σ + δα + ∆ σ − δ α + ∆ σ − δ Figure 21: Regularized one loop 3-gamma graph denoted by V σππ contributing to V σ containinga π propagator and including the conformal triangles.To complete the evaluation of η requires the contribution to (6.13) from V σ and V π . Thevalues of the four topologies underlying the graphs in Figures 18 and 19 have been computedpreviously, [25, 26, 27]. Strictly though it is the combination of (6.16) and the π a counterpart thathave been calculated. The values of the individual terms cannot be extracted as the underlyingthree loop integral does not have unique vertices to allow the integration to proceed. While itwould appear from the figures that there are three distinct topologies this would be the case ifthere was no regularization. With non-zero δ and δ (cid:48) through specified external legs there aretwo independent contributions from the three loop non-planar graphs. Indeed the contributionto (6.7) from these non-planar graphs is the same but not for (6.13). For each of the graphs ofFigures 20 and 21 we have computed the associated group factor by again using the color.h π a fields connect to the fermion loop.The next stage is to assemble all the components contributing to (6.7) and (6.13). Thepurpose of the two equations of (6.7) is to fix the values of ¯ z and ¯ y at O (1 /N ). While thesewere eliminated to arrive at (6.13) combinations of them are present as factors in each of thegraphs of Figures 15, 16, 18 and 19. However we can now illuminate the difficulties with a directevaluation in the U (1) case. The formal solution for ¯ z in (2.4) at leading order is¯ z = (cid:104) [2 C F − C A ]∆ σ + 2 C F ∆ σ ∆ π (cid:105) C A Q . (7.9)While C A vanishes in the abelian limit producing a singularity, when the leading order valuesfor the vertex dimensions are included these are proportional to C A . So that overall the leadingorder variable ¯ z is finite in the abelian limit. For a direct evaluation one could not determine ¯ z since it would equate to the mathematically ill-defined quantity of 0 /
0. The situation for ¯ y issimilar. It was this that forced us to consider the more general Lagrangian. For (2.4) we find¯ z = − ( C F N c − T F ) Γ (2 µ − ( µ − (1 − µ ) N c T F ¯ y = − (2 C F N c − T F − C A N c ) Γ (2 µ − ( µ − (1 − µ ) N c T F (7.10)for instance which is finite in the abelian limit. With these values and those for the next orderwe ultimately arrive at our main result for (2.2) which is η = (cid:34)(cid:34) µ ( µ + 8) d abcdF d abcdF N c T F + 12 µ (3 µ + 1) C A C F T F N c − µ (5 µ − C F C A N c − µ ( µ − C F C A N c + ( µ + 10 µ + µ − T F − (2 µ + 22 µ − µ + 3) C F N c T F − (6 µ − µ + 3) C F N c T F + (2 µ − µ + 1) C F N c (cid:105) Θ( µ )4[ µ − − (cid:104) µ (2 µ − C F C A N c − µC A C F T F N c − µ C F C A N c − µ − T F + 4(4 µ − C F N c T F + 4(4 µ − C F N c T F − µ − C F N c (cid:105) Φ( µ )8[ µ − + (cid:34) T F − C F N c T F − C A C F N c + 32 d abcdF d abcdF N c T F (cid:35) (cid:20) Θ( µ ) + 1[ µ − (cid:21) µ Ξ( µ )[ µ − (cid:104) d abcdF d abcdF N c + 24 T F − C F N c T F − C A C F N c T F (cid:105) µ Θ( µ )Ψ( µ )8[ µ − T F − (cid:104) µ − C F N c T F − µC A C F T F N c − µ C F C A N c − µ − T F + 4 µ (2 µ − C F C A N c + 4(4 µ − C F N c T F − µ − C F N c (cid:105) ( µ )8[ µ − + (cid:34) (24 µ − µ + 9)2 µ [ µ − C F N c T F − (12 µ − µ + 2)4[ µ − C A C F T F N c − (19 µ − µ − µ − C F C A N c − (2 µ − µ − µ + 56 µ − µ + 3)2 µ [ µ − T F − (12 µ − µ + 20 µ + 76 µ − µ + 9)2 µ [ µ − C F N c T F − µ ( µ + 2)( µ − µ − C F C A N c
25 (11 µ − µ − µ [ µ − C F N c − µ (2 µ + 1)( µ − µ − d abcdF d abcdF N c T F (cid:35) Ψ( µ ) − µ − µ + 2)8[ µ − C A C F T F N c − µ (8 µ − µ − µ − µ − C F C A N c − µ − µ − µ − C F C A N c − (2 µ − µ − µ − µ + 21 µ − µ + 1)2 µ [ µ − T F − (12 µ − µ − µ + 11 µ + 108 µ − µ + 30 µ − µ [ µ − C F N c T F + 3(15 µ − µ + 33 µ − µ + 1)2 µ [ µ − C F N c T F + (3 µ − (2 µ − µ [ µ − C F N c − µ ( µ + 1)(2 µ − µ − d abcdF d abcdF N c T F (cid:35) η [ T F + C F N c ] (7.11)which is considerably more involved than the other exponents. We note that electronic versionsof (7.11) and all the large N exponents of (2.4) are available in an attached data file.Like the η result of [20] (7.11) contains the function Ξ( µ ) which is related to the function I ( µ ) of that paper via I ( µ ) = − µ −
1) + Ξ( µ ) . (7.12)The (cid:15) expansion of I ( µ ) around d = D − (cid:15) is known for integer D . In the case of D = 2 and4 this was derived in [47] and built on the earlier expansion provided in [20]. When D = 3 theexpansion was provided in [48] where the leading order term is I ( ) = 2 ln 2 − π ζ (3) (7.13)which was derived in [20] using uniqueness methods. So, for instance, we can record the firstthree terms of the large N expansion of each exponent in three dimensions for (2.4) which are η | d =3 = 8[ C F N c + T F ]3 π N c T F N + (cid:104) C F N c − C A C F N c − C F N c T F + 28 T F (cid:105) π N c T F N + (cid:104) π ln(2) d abcdF d abcdF N c − π ln(2) C A C F N c T F − π ln(2) C F N c T F + 2592 π ln(2) T F + 1134 ζ (3) C A C F N c T F + 54432 ζ (3) C F N c T F − ζ (3) d abcdF d abcdF N c − ζ (3) T F − π C A C F N c T F + 2916 C A C F N c T F + 1368 π C A C F N c T F − C A C F N c T F − π C A C F N c T F − C A C F N c T F − π C F N c T F + 33536 C F N c T F + 1152 π C F N c T F + 94848 C F N c T F − π C F N c T F − C F N c T F + 3240 π d abcdF d abcdF N c − d abcdF d abcdF N c + 1128 π T F − T F (cid:105) π N c T F N + O (cid:18) N (cid:19) β σ | d =3 = 1 + 32[ C F N c − T F ]3 π N c T F N + 64[45 C A C F N c − C F N c + 27 π C F N c T F − C F N c T F − π T F + 304 T F ]27 π N c T F N + O (cid:18) N (cid:19) β π | d =3 = 1 + 8[3 C A N c − C F N c + 4 T F ]3 π N c T F N (cid:104) π C A C F N c T F − C A C F N c T F + 864 C A C F N c T F + 936 C A C F N c T F − C F N c T F − C F N c T F + 216 π C F T F + 992 C F T F − π d abcdF d abcdF N c + 3456 d abcdF d abcdF N c (cid:105) π C F N c T F N + O (cid:18) N (cid:19) ν (cid:12)(cid:12)(cid:12)(cid:12) d =3 = 1 − C F N c + T F ]3 π N c T F N + (cid:104) C A C F N c T F − π C A C F N c T F + 108 π C A C F N c T F − C A C F N c T F − C A C F N c T F − π C F N c T F + 1264 C F N c T F + 624 C F N c T F + 108 π C F N c T F − C F N c T F + 162 π d abcdF d abcdF N c + 54 π T F + 1264 T F (cid:105) π N c T F [ C F N c + T F ] N + O (cid:18) N (cid:19) (7.14)from which we will be able to deduce the analogous expressions for the chiral XY model. Finallywe note that (7.11) agrees with known expressions in the limit of (5.3).Exponent Pad´e N t = 2 N t = 4 N t = 6 N t = 8 η XY [1,1] 0.087176 0.038058 0.024343 0.017894[1,2] 0.078821 0.037198 0.024104 0.017797 η XY φ [0,1] 0.880984 0.936726 0.956909 0.967330[1,1] 0.902301 0.943303 0.960064 0.969176[0,2] 0.907741 0.944124 0.960325 0.969290 ν XY [0,1] 0.787284 0.880984 0.917378 0.936726[1,1] 0.902616 0.928416 0.943409 0.953210Table 1. Pad´e approximants for exponents in chiral XY model for N t = 2, 4, 6 and 8.Exponent Reference Method N t = 2 η XY [15] [2 ,
2] Pad´e 0.117[15] [3 ,
1] Pad´e 0.108[11] functional RG 0.062 η XY φ [15] [2 ,
2] Pad´e 0.810[15] [3 ,
1] Pad´e 0.788[11] functional RG 0.88 ν XY [15] [2 ,
2] Pad´e 0.840[15] [3 ,
1] Pad´e 0.841[11] functional RG 0.862Table 2. Exponent estimates in chiral XY model for N t = 2 from [2 ,
2] and [3 ,
1] Pad´eapproximants of four dimensional (cid:15) expansion [15] and the functional renormalization group[11].
Having established the critical exponents for the non-abelian Nambu-Jona-Lasinio model, (2.4),we can now derive those for the chiral XY model in the abelian limit. It is clear that there are27o singular colour group factors for the results of (3.6), (3.7), (3.8), (4.2), (4.4), (5.1), (5.2) and(7.11). Therefore it is safe to take the abelian limit of (2.9) giving η XY = − µ − µ Γ(1 − µ )Γ( µ − ( µ ) χ XY σ = χ XY π = 0 λ XY = − (2 µ − η XY (8.1)at leading order which clearly illustrates the issue surrounding the vanishing leading order vertexanomalous dimension for both Yukawa interactions. At next order we have η XY = (cid:20) (5 µ − µ ( µ −
1) + Ψ( µ ) (cid:21) η XY χ XY σ = χ XY π = − µ (4 µ − µ + 7)2( µ − η XY λ XY = (cid:34) µ ( µ − µ − η XY + (8 µ − µ + 108 µ − µ − µ + 24 µ − µ ( µ − µ − − µ − µ + 31 µ − µ + 9 µ + 6 µ − µ − ( µ − Ψ( µ ) − µ (2 µ − µ − µ − (cid:104) Φ( µ ) + Ψ ( µ ) (cid:105) + 3 µ (3 µ − µ − µ −
2) Θ( µ ) (cid:35) η XY . (8.2)We have checked that these expressions are in agreement with earlier work, [17, 24]. Finally weobtain the main result for this model which is η XY = (cid:34) (cid:104) Φ( µ ) + 3Ψ ( µ ) (cid:105) − (2 µ − µ − µ + 18 µ − µ ( µ − Ψ( µ ) −
14 Θ( µ ) − (4 µ − µ − µ − µ + 84 µ − µ + 20 µ − µ ( µ − (cid:35) η XY . (8.3)The expression is considerably simpler than its non-abelian counterpart similar to the O (1 /N )exponents. One aspect of this is that the function Ξ( µ ) is absent which is not unrelated to thevanishing of χ σ and χ π at leading order.While the exponents to O (1 /N ) agree with previous derivations, it is possible to check themagainst perturbative computations in four dimensions. In [15] the β -function and anomalousdimensions of the chiral XY model were determined to four loops in the MS scheme. Moreoverthe (cid:15) expansion of the corresponding critical exponents were computed to O ( (cid:15) ). Therefore ifwe expand our expressions in the same (cid:15) expansion they ought to be in agreement. Setting d = 4 − (cid:15) in (8.1), (8.2) and (8.3) we find η XY (cid:12)(cid:12)(cid:12) d =4 − (cid:15) = (cid:20) (cid:15) − (cid:15) − (cid:15) + (cid:20) ζ − (cid:21) (cid:15) + O ( (cid:15) ) (cid:21) N t + (cid:20) − (cid:15) + 234 (cid:15) − (cid:15) − (cid:20) ζ + 194 (cid:21) (cid:15) + O ( (cid:15) ) (cid:21) N t + (cid:20) (cid:15) − (cid:15) + 774 (cid:15) + (cid:20) ζ + 58116 (cid:21) (cid:15) + O ( (cid:15) ) (cid:21) N t + O (cid:18) N t (cid:19) η XY φ (cid:12)(cid:12)(cid:12) d =4 − (cid:15) = 2 (cid:15) + (cid:20) − (cid:15) + 3 (cid:15) + 32 (cid:15) + (cid:20) − ζ (cid:21) (cid:15) + O ( (cid:15) ) (cid:21) N t + (cid:20) (cid:15) + 12 (cid:15) − (cid:15) + (cid:20) ζ + 512 (cid:21) (cid:15) + O ( (cid:15) ) (cid:21) N t + O (cid:18) N t (cid:19) η XY for N t = 2, 4, 6 and 8.1 ν XY (cid:12)(cid:12)(cid:12)(cid:12) d =4 − (cid:15) = 2 − (cid:15) + (cid:20) − (cid:15) + 13 (cid:15) − (cid:15) − (cid:20)
34 + 12 ζ (cid:21) (cid:15) + O ( (cid:15) ) (cid:21) N t + (cid:20) (cid:15) − (cid:15) + (cid:20) − ζ (cid:21) (cid:15) + (cid:20) ζ − ζ + 744 ζ + 2052 (cid:21) (cid:15) + O ( (cid:15) ) (cid:21) N t + O (cid:18) N t (cid:19) (8.4)where the combination η φ = 4 − µ − η − χ σ (8.5)tallies with the definition used in [15, 16]. We have also set N = 4 N t to adjust for the differenttrace and group conventions used here in the underlying master integrals for the skeleton Dyson-Schwinger and conformal bootstrap equations. So N t corresponds to the N used in [15, 16]. Itis straightforward to verify that each expansion is in full agreement with [15] after allowingfor a different convention in the definition of d in terms of (cid:15) . In light of this agreement withhigh order perturbation theory we can obtain expressions for the exponents in three dimensions.29nalytically we have η XY (cid:12)(cid:12)(cid:12) d =3 = 43 π N t + 16027 π N t − π + 20]243 π N t + O (cid:18) N t (cid:19) η XY φ (cid:12)(cid:12)(cid:12) d =3 = 1 − π N t + 54427 π N t + O (cid:18) N t (cid:19) ν XY (cid:12)(cid:12)(cid:12)(cid:12) d =3 = 1 − π N t + [216 π + 2912]27 π N t + O (cid:18) N t (cid:19) (8.6)or η XY (cid:12)(cid:12)(cid:12) d =3 = 0 . N t + 0 . N t − . N t + O (cid:18) N t (cid:19) η XY φ (cid:12)(cid:12)(cid:12) d =3 = 1 − . N t + 0 . N t + O (cid:18) N t (cid:19) ν XY (cid:12)(cid:12)(cid:12)(cid:12) d =3 = 1 − . N t + 1 . N t + O (cid:18) N t (cid:19) (8.7)numerically.The next stage is to extract estimates for the three dimensional exponents for the case ofinterest in [15] which corresponds to N t = 2 here. To improve the convergence of the expansionwe have followed a similar approach to [28] and evaluated several Pad´e approximants for eachof the exponents. These have been plotted in Figures 22, 23 and 24 for N t = 2, 4, 6 and 8 for2 < d < η XY , η XY φ and 1 /ν XY respectively with numerical values for three dimensionsrecorded in Table 1. The latter three values of N t are selected to gauge the effect the higherorder 1 /N corrections have. In the case of η it is worth noting that we can plot the behaviourof η unlike [28]. This is because Ξ( µ ) is absent from η and it was not possible to write thefunction in an analytic form that could be passed to the plotting routine in that case. For η XY the [2 ,
1] Pad´e had a singularity at around d = 3 .
6. Equally for N t = 2 the [0 ,
2] Pad´e of1 /ν XY had a maximum near a similar value of d . We have omitted plots of both Pad´e’s for eachexponent.What is evident in the plots of η XY for various N t is that there is convergence of the O (1 /N )values down to N t = 4. For N t = 2 there is a difference between the O (1 /N ) and O (1 /N )curves but the latter is bounded by the leading two orders. This is similar to higher values of N t . Given this it might not be unreasonable to give 0 .
083 as a rough estimate of η XY for N t = 2being the average of the two Pad´e estimates. This is not inconsistent with the N t = 2 estimatesfrom other approaches which are summarized in Table 2. In the case of η XY φ it is clear fromFigure 23 that there is very little difference for the value of the exponent between successiveorders in 1 /N . Indeed comparing the values for N t = 2 with other estimates there is a degree ofconsistency. Finally the situation for 1 /ν XY does not appear to be as good especially at leadingorder for low values of N t due to the kink that is evident in Figure 24 in its behaviour across d -dimensions. Although it appears that this feature washes out for larger values of N t , it is notclear whether this would be the case for N t = 2 if an O (1 /N ) expression were available. Againone could assume that the two large N Pad´e approximants for 1 /ν XY were bounds of the trueresult. In this instance one would arrive at an estimate of around 0 .
845 which is not dissimilarto the perturbative Pad´e or functional renormalization group estimates of [11, 15].
We have provided the large N critical exponents for the Nambu-Jona-Lasinio universality classwith Lagrangian (2.4) for a general Lie group using the large N critical point formalism pioneered30igure 23: Pad´e approximants of η XY φ for N t = 2, 4, 6 and 8.in [18, 19, 20]. This included the large N conformal bootstrap formalism that allowed us toextract η . To achieve this we had to extend the formalism to the situation where there are twoindependent 3-point interactions. By independent we mean the vertex anomalous dimensions ofthe separate vertices were inequivalent. The derivation of the underlying bootstrap equationswas straightforward but was discussed at length. Indeed from that it is evident that they canbe readily generalized to the case of more independent 3-point interactions. Taking the moregeneral approach with a general Lie group means that results for exponents in various modelscan be extracted in certain limits. Previously η was calculated for a limited number of fermionicmodels including the SU (2) Nambu-Jona-Lasinio model, [35], as the anomalous dimensions ofeach vertex were the same. To produce the general Lie group results we were able to benefitfrom the Form encoded color.h package which was based on [42] that allowed us to expressour exponents in terms of the general group Casimir invariants. This was particularly importantfor the light-by-light graphs that contribute to several exponents at O (1 /N ) and η . One ofthe main consequences was that the abelian limit could be taken smoothly to deduce exponentsfor the chiral XY model. This circumvented the application of the bootstrap formalism directlyto (2.2) due to an ill-defined intermediate quantity that formally obstructs the direct derivation31igure 24: Pad´e approximants of ν XY for N t = 2, 4, 6 and 8.in the abelian case. The ultimate general expressions are analytic in the colour group invariantsallowing the chiral XY model exponents to emerge smoothly. Finally this more general groupapproach for the large N formalism offers a more compact way of extracting critical exponentsfor other universality classes. One benefit for instance is that it gives access to additionalinformation on the terms in the explicit perturbative series of the underlying renormalizationgroup functions. Acknowledgements.
This work was supported by a DFG Mercator Fellowship and in partthrough the STFC Consolidated ST/T000988/1. The graphs were drawn with the
Axodraw package [49]. Computations were carried out using the symbolic manipulation language
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