TThe Effective Bootstrap
Alejandro Castedo Echeverri a , Benedict von Harling b and Marco Serone a,ca SISSA and INFN, Via Bonomea 265, I-34136 Trieste, Italy b DESY, Notkestrasse 85, 22607 Hamburg, Germany c ICTP, Strada Costiera 11, I-34151 Trieste, Italy
Abstract
We study the numerical bounds obtained using a conformal-bootstrap method –advocated in ref. [1] but never implemented so far – where different points in theplane of conformal cross ratios z and ¯ z are sampled. In contrast to the most usedmethod based on derivatives evaluated at the symmetric point z = ¯ z = 1 /
2, we canconsistently “integrate out” higher-dimensional operators and get a reduced simpler,and faster to solve, set of bootstrap equations. We test this “effective” bootstrapby studying the 3D Ising and O ( n ) vector models and bounds on generic 4D CFTs,for which extensive results are already available in the literature. We also determinethe scaling dimensions of certain scalar operators in the O ( n ) vector models, with n = 2 , ,
4, which have not yet been computed using bootstrap techniques. a r X i v : . [ h e p - t h ] N ov ontents z → O ( n ) Symmetry . . . . . . . . . . . . . . . . . . . . . . 9 O ( n ) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 4D CFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 A Closer Look at the Spectrum of 3D O ( n ) Models . . . . . . . . . . . . . . . . . 21 There has recently been a great revival of interest in the conformal bootstrap program [2, 3]after ref. [4] observed that its applicability extends to Conformal Field Theories (CFTs) in d > d ≥ u = v = 1 /
4, or equivalently in z -coordinates at z = ¯ z = 1 / s and t channels. Taking higherand higher derivatives of the bootstrap equations evaluated at this point has proven to be veryeffective and successful in obtaining increasingly better bounds. We will denote this methodin the following as the “derivative method”. A drawback of the derivative method – both inits linear [4, 6, 9] or semi-definite [10, 11] programming incarnations – is the need to include alarge number of operators in the bootstrap equations. This makes any, even limited, analyticalunderstanding of the obtained results quite difficult.A possible approximation scheme is in fact available: ref. [12] has determined the rate ofconvergence of the Operator Product Expansion (OPE), on which the bootstrap equations arebased. This allows us to extract the maximal error from neglecting operators with dimensionslarger than some cutoff ∆ ∗ in the bootstrap equations and thus to consistently truncate them.2hese truncated bootstrap equations can then be evaluated at different points in the z -plane.This method, which we denote as the “multipoint method”, has been previously advocated byHogervorst and Rychkov in ref. [1] but has not yet been numerically implemented. The aim ofthis note is to provide such an implementation and study the resulting bounds. It is importantto emphasize that the method of ref. [1] combines what are in principle two independent ideas: i)multipoint bootstrap and ii) truncation of the bootstrap equations. One could study i) withoutii), or try to analyze ii) without i). We will not consider these other possibilities here.We begin in section 2 with a brief review of the results of refs. [1, 12, 13] on the convergenceof the OPE. We use generalized free theories as a toy laboratory to test some of the resultsobtained in ref. [12]. We then generalize the results of ref. [12] for CFTs with an O ( n ) globalsymmetry.We write the bootstrap equations and set the stage for our numerical computations in section3. Our results are then presented in section 4. For concreteness, we study bounds on operatordimensions and the central charge in 3D and 4D CFTs, with and without an O ( n ) globalsymmetry (with no supersymmetry). For these bounds, extensive results are already availablein the literature (see e.g. refs. [5–7, 10, 14–22]). In particular, we focus our attention on theregions where the 3D Ising and O ( n ) vector models have been identified. We show how theresults depend on the number N of points in the z -plane at which we evaluate the bootstrapequations and the cut-off ∆ ∗ on the dimension of operators in the bootstrap equations. Usingvalues for the dimension of the operator φ in O ( n ) vector models available in the literatureand a fit extrapolation procedure, we then determine the dimensions of the second-lowest O ( n )singlet and symmetric-traceless operators S (cid:48) and T (cid:48) for n = 2 , ,
4. To our knowledge, thesehave not been obtained before using bootstrap techniques. Our results are consistent with thosefrom analytical calculations using the (cid:15) -expansion [23, 24] with a mild tension with the result ofref. [24] for the dimension of T (cid:48) in the O (2) model. We notice from our results that the “kink”in the bound on the dimension of the lowest scalar (singlet) operator in 3D Ising and O ( n )vector models is already visible for relatively small ∆ ∗ , while the minimum in the central-chargebound is very sensitive to ∆ ∗ . For our numerical implementation, we discretize the spectrumand formulate the bootstrap equations as a linear program which we solve using the optimizer CPLEX by IBM . Since we focus on the truncated bootstrap equations with relatively low cutoff ∆ ∗ ,double precision as used by CPLEX is sufficient for our purposes. More refined implementationswith higher numerical precision, possibly adapting the method and optimizer of refs. [6, 9], arecertainly possible. More details on the numerical implementation are given in section 5. Weconclude in section 6. Convergence of the OPE
We begin with a brief review of the results of refs. [12,13] (see also ref. [1]) about the convergenceof the OPE in a euclidean, reflection positive, CFT in any number of dimensions. For moredetails see the original references. Consider the 4-point function of a scalar primary operator φ with scaling dimension ∆ φ : (cid:104) φ ( x ) φ ( x ) φ ( x ) φ ( x ) (cid:105) = g ( u, v ) x φ x φ , (2.1)where u ≡ x x x x and v ≡ x x x x (2.2)are the conformally-invariant cross-ratios ( x ij ≡ x i − x j ). Applying the OPE to the operatorpairs φ ( x ) φ ( x ) and φ ( x ) φ ( x ) in the 4-point function, one can write g ( u, v ) = 1 + (cid:88) ∆ ,l λ O g ∆ ,l ( z, ¯ z ) , (2.3)where u = z ¯ z , v = (1 − z )(1 − ¯ z ) and the sum runs over all primary operators O that appear inthe φ × φ OPE with ∆ and l being respectively their dimension and spin. For each primary, thesum over all its descendants is encoded in the conformal block function g ∆ ,l ( z, ¯ z ). In a euclideanCFT, ¯ z = z ∗ and the conformal blocks are regular everywhere in the complex z -plane, with theexception of a branch-cut along the real line [1 , + ∞ ). Thanks to reflection positivity, the OPEcoefficients λ O are real and thus λ O > (cid:88) (∆ ≥ ∆ ∗ ) ,l λ O g ∆ ,l ( z, ¯ z ) (2.4)of the sum in eq. (2.3) when it is truncated at some primary operator of dimension ∆ = ∆ ∗ . Todetermine this bound, one first uses that | g ∆ ,l ( z, ¯ z ) | ≤ g ∆ ,l ( | z | , | ¯ z | ) (2.5)as follows e.g. from a representation of the conformal blocks in terms of Gegenbauer polynomials[1]. It is therefore sufficient to estimate the remainder for real z = ¯ z . As was found in ref. [12],the most stringent bound is obtained by using the coordinate ρ ( z ) = z (1 + √ − z ) . (2.6) Bounds on the OPE convergence are obtained in an alternative way using crossing symmetry in ref. [25].Interestingly, ref. [25] sets bounds which are also valid for finite values of ∆ ∗ at z = ¯ z = 1 /
2, though they arerelative and not absolute bounds. It would be interesting to explore the approach followed in this paper further.We thank Slava Rychkov for having pointed out this reference to us. The branch-cut is best seen in Lorentzian signature, where z and ¯ z are two independent variables. At fixed ¯ z (z), g ∆ ,l ( z, ¯ z ) is a true analytic function in z (¯ z ) with a branch-cut along the line [1 , + ∞ ). z -plane is mapped to the unit disk in ρ and the branch-cut is mapped to the boundary ofthe disk. The conformal blocks in ρ are then defined for | ρ | <
1. In the manifestly reflectionpositive configuration with ¯ ρ = ρ = r , the function g ( u, v ) in eq. (2.3) can be written as g ( r ) = 1 + (cid:88) ∆ ,l λ O ∞ (cid:88) n =0 c n (∆ , l ) r ∆+ n , (2.7)where c n (∆ , l ) are positive coefficients whose explicit form is not important here and the sumover n takes into account the contributions from the descendants of each primary. It is convenientto rewrite g ( r ) as g ( β ) = (cid:90) ∞ d ∆ f (∆) e − β ∆ with f (∆) = (cid:88) k ρ k δ (∆ − ∆ k ) . (2.8)Here β ≡ − log r , k runs over all operators (primaries and their descendants) which are exchangedin the OPE and f (∆) is a spectral density with positive coefficients ρ k . Again, their explicit formis not relevant for our considerations.The behaviour of g ( β ) in the limit β → x → x , in whichcase z → r → − z → β / →
0) is dominated by the exchange of the identity operatorand one finds: g ( β ) ∼ β → φ β − φ . (2.9)Here a ∼ b means that a/b → ρ k are all positive, this asymptotic behaviour determines the leading,large-∆ behaviour of the integrated spectral density F (∆) = (cid:90) ∆0 f (∆ (cid:48) ) d ∆ (cid:48) (2.10)by means of the Hardy-Littlewood tauberian theorem (see e.g. [26]): F (∆) ∼ ∆ →∞ (2∆) φ Γ(4∆ φ + 1) . (2.11)The remainder (2.4) can then be bounded as follows: We first note that (cid:88) (∆ ≥ ∆ ∗ ) ,l λ O g ∆ ,l ( β ) ≤ (cid:90) ∞ ∆ ∗ f (∆) e − β ∆ d ∆ , (2.12) For simplicity, we use the same symbol to denote the functions g ( u, v ) and ˜ g ( r ) = g ( u ( r ) , v ( r )) etc. here andbelow. This is true in general only in d > d = 2, one has to be careful since scalar operators canhave arbitrarily small dimensions. See also the discussion after eq. (2.23). It is in fact sufficient that the coefficients are all positive for operators with dimension larger than some fixedvalue ∆ . ∗ , whereason the l.h.s. only primaries with dimension larger than ∆ ∗ and their descendents contribute.Using eq. (2.11), the r.h.s. can in turn be bounded as (cid:90) ∞ ∆ ∗ f (∆) e − β ∆ d ∆ = β (cid:90) ∞ ∆ ∗ e − β ∆ ( F (∆) − F (∆ ∗ )) d ∆ ≤ β (cid:90) ∞ ∆ ∗ e − β ∆ F (∆) d ∆ (cid:39) β (cid:90) ∞ ∆ ∗ e − β ∆ (2∆) φ Γ(4∆ φ + 1) d ∆ = β − φ φ Γ(4∆ φ + 1) Γ(4∆ φ + 1 , ∆ ∗ β ) , (2.13)where Γ( a, b ) is the incomplete Gamma function. Clearly, this bound applies for parametricallylarge values of ∆ ∗ , where eq. (2.11) holds. Using eq. (2.5), we finally get the bound on theremainder (cid:12)(cid:12)(cid:12) (cid:88) (∆ ≥ ∆ ∗ ) ,l λ O g ∆ ,l ( z, ¯ z ) (cid:12)(cid:12)(cid:12) ≤ ( − log | ρ ( z ) | ) − φ φ Γ(4∆ φ + 1) Γ(4∆ φ + 1 , − ∆ ∗ log | ρ ( z ) | ) . (2.14)This is valid in any number d > g ∆ ,l ( r ) ≡ (1 − r ) γ g ∆ ,l ( r ) (2.15)with γ = 1. Repeating the derivation reviewed above for a remainder involving the rescaledconformal blocks, it is straightforward to get the alternative bound (cid:12)(cid:12)(cid:12) (cid:88) (∆ ≥ ∆ ∗ ) ,l λ O g ∆ ,l ( z, ¯ z ) (cid:12)(cid:12)(cid:12) ≤ R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) (2.16)with R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) ≡ ( − log | ρ ( z ) | ) − φ + γ φ + γ Γ(4∆ φ + 1 − γ ) Γ(4∆ φ + 1 − γ, − ∆ ∗ log | ρ ( z ) | )(1 − | ρ ( z ) | ) γ . (2.17)For − ∆ ∗ log | ρ ( z ) | (cid:29)
1, eq. (2.17) can be approximated as R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) ≈ φ + γ ∆ φ − γ ∗ Γ(4∆ φ + 1 − γ ) | ρ ( z ) | ∆ ∗ (1 − | ρ ( z ) | ) γ . (2.18)We see that for | ρ ( z ) | not too close to 1 and ∆ ∗ (cid:38) φ , the bound is more stringent for γ = 1than for γ = 0. It was furthermore shown in ref. [13] that in d = 3 dimensions, γ = 1 is themaximal allowed value such that the Hardy-Littlewood tauberian theorem remains applicable,whereas it was conjectured without proof that the maximal allowed value in d = 4 dimensionsis γ = 3 /
2. Correspondingly we use eq. (2.17) with γ = 1 for the remainder both in 3 and 4dimensions in our numerical implementation. The fact that eq. (2.16) with γ = 0 is not optimal can be traced to using the inequality (2.12) in the derivation. g ( u, v ) turns into a positive definite function of a single variable. The remainder is then estimatedusing the Hardy-Littlewood tauberian theorem. One cannot naively apply these arguments toarbitrary derivatives of g ( u, v ) w.r.t. u and v , unless the resulting functions remain positivedefinite and derivatives can be brought inside the absolute value in the l.h.s. of eq. (2.16). Seethe appendix of ref. [27] for a recent discussion on how to estimate the remainder on derivativesof g ( u, v ). It would be interesting to verify if this allows us to also study truncated bootstrapequations with the derivative method. z → ∗ → ∞ . Of course, for any practical use, we need to know the value of ∆ ∗ beyond which we cantrust eq. (2.11) and thus the bound eq. (2.16). It is difficult to determine this value for a genericCFT. But we can get useful insights by considering exactly calculable CFTs, like generalizedfree theories (sometimes called mean field theories) for which the CFT data are known and thefunction g ( u, v ) in eq. (2.1) in any number of dimensions reads g ( u, v ) = 1 + u ∆ φ + (cid:16) uv (cid:17) ∆ φ = 1 + | z | φ + (cid:16) | z || − z | (cid:17) φ . (2.19)For values of ∆ ∗ such that eq. (2.11) is no good approximation, the r.h.s. of eq. (2.16) canclearly still overestimate the actual remainder, leading to no inconsistency. On the other hand,if it underestimates the actual remainder, eq. (2.16) is simply wrong. We define η ≡ R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) (cid:12)(cid:12)(cid:12) (cid:80) (∆ ≥ ∆ ∗ ) ,l λ O g ∆ ,l ( z, ¯ z ) (cid:12)(cid:12)(cid:12) (2.20)and check if and when η is smaller than 1, in which case eq. (2.16) is violated. The denominatorin eq. (2.20) is computed as (cid:88) (∆ ≥ ∆ ∗ ) ,l λ O g ∆ ,l ( z, ¯ z ) = g ( u, v ) − − (cid:88) (∆ < ∆ ∗ ) ,l λ O g ∆ ,l ( z, ¯ z ) . (2.21)In fig. 1, we show η as a function of ∆ ∗ evaluated at the symmetric point z = ¯ z = 1 /
2. Noticethat at the point of best convergence the actual remainder is always significantly smaller than R ,and that the ratio gets bigger and bigger as ∆ ∗ increases for large ∆ ∗ . In particular, η is greater In order to make the bound more stringent, one could then alternatively use the series representation in ref. [1]which includes contributions from primary operators and their descendants separately. Using this series truncatedat contributions corresponding to dimension ∆ ∗ instead of the full conformal blocks g ∆ ,l would make the r.h.s. ofthe inequality (2.12) the actual remainder to be bounded. This would thus make eq. (2.16) with γ = 0 morestringent. Here, however, we choose not to follow this approach. The reason is that the representations for thefull conformal blocks g ∆ ,l can be considerably faster calculated than (our implementation of) the truncated seriesrepresentation of ref. [1].
10 15 20 25 Δ * η Figure 1: η defined in eq. (2.20) as a function of ∆ ∗ in a generalized free theory in d = 4 dimensionsevaluated at the symmetric point z = ¯ z = 1 /
2. We have taken ∆ φ = 1 . γ = 1. than 1 for any value of ∆ ∗ . We have performed comparisons with GFTs in d = 3 dimensionswith γ = 0 , d = 4 dimensions with γ = 0 , / z and ∆ φ within theunitary bounds, finding analogous qualitative results. Somehow unexpectedly, we find that thebound (2.16) is never violated in GFTs, for any value of ∆ ∗ .When z →
1, both the numerator and the denominator of η in eq. (2.20) blow up, sincethe OPE is not convergent at z = ¯ z = 1. Operators with high scaling dimension are no longersuppressed and the remainder completely dominates the OPE. More precisely, we have R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) ∼ z, ¯ z → − φ ( − log | ρ ( z ) | ) − φ , (2.22)independently of γ . Notice that this limit is universal for any CFT that includes in its spectrum ascalar operator with dimension ∆ φ , because z = ¯ z → φ itself. In this casethe universal nature of the limit is trivially checked using eq. (2.19): g ( u, v ) ∼ z, ¯ z → − | − z | φ ∼ z, ¯ z → − φ ( − log | ρ ( z ) | ) − φ , (2.23)where in the last equality we have used that | − z | → (log | ρ ( z ) | ) / z → z fixed for d >
2, where large twistoperators are suppressed. The two-dimensional case is more subtle, because there is no longer agap between the identity (which has the minimum twist zero) and the other operators. Indeed,the results of refs. [28, 29] and those of ref. [12] in the euclidean do not straightforwardly applyfor d = 2. In this limit, the name remainder should actually be used for the finite sum of operators up to ∆ ∗ .
8n the euclidean, operators of any twist should be considered. However, given the resultsof refs. [28, 29], it is natural to expect that the leading behaviour (2.22) is expected to comefrom operators with parametrically high dimension and high spin for any CFT, asymptoticallyapproaching the GFT spectrum in this regime. It would be interesting to understand withineuclidean CFTs, where the twist does not play an obvious role, why this is so. O ( n ) Symmetry
The generalization of the OPE convergence estimate to CFTs with O ( n ) global symmetry isstraightforward. For concreteness, let us consider scalars φ i in the fundamental representation of O ( n ). The only non-trivial point is to identify a proper linear combination of 4-point functions (cid:104) φ i ( x ) φ j ( x ) φ k ( x ) φ l ( x ) (cid:105) (2.24)that leads to a positive definite series expansion, otherwise the Hardy-Littlewood tauberiantheorem does not apply. A possible choice is A η ≡ (cid:104) φ φ φ φ (cid:105) + | η | (cid:104) φ φ φ φ (cid:105) + η (cid:104) φ φ φ φ (cid:105) + η ∗ (cid:104) φ φ φ φ (cid:105) = a η ( u, v ) x φ x φ , (2.25)where for simplicity we have omitted the x -dependence of the fields. The parameter η can ingeneral take an arbitrary complex value, but it is enough for our purposes to consider η = ± ρ = ρ = r and any η , this correlator is manifestly positive definite, because it correspondsto the norm of the state φ | φ (cid:105) + ηφ | φ (cid:105) . (2.26)The leading term in a η ( u, v ) for x → x is given by the exchange of the identity operator in thefirst two correlators and hence is independent of η . On the other hand, expanding in conformalblocks in the (12)-(34) channel, we have [19] A η = 1 x φ x φ (cid:16) η ) (cid:16) (cid:88) S + λ S g ∆ ,l ( u, v ) (cid:17) + 4 (cid:16) − ηn (cid:17) (cid:88) T + λ T g ∆ ,l ( u, v ) (cid:17) , (2.27)where S and T denote operators in the singlet and rank-two symmetric representations of O ( n ),respectively. Both sums run over even spins. We can now repeat essentially verbatim the deriva-tion below eq. (2.6). For η = −
1, this gives rise to the bound (cid:12)(cid:12)(cid:12) (cid:88) (∆ ≥ ∆ ∗ ) ,l λ T g ∆ ,l ( z, ¯ z ) (cid:12)(cid:12)(cid:12) ≤ R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) , (2.28)where R is given in eq. (2.17). The factor 1/2 with respect to the non-symmetric case arisesbecause the identity operator is exchanged in two correlators but a factor 4 is present in thesecond term in the r.h.s. of eq. (2.27). For η = 1 we similarly get (cid:12)(cid:12)(cid:12) (cid:88) (∆ ≥ ∆ ∗ ) ,l (cid:18) λ S g ∆ ,l ( z, ¯ z ) + (cid:0) − n (cid:1) λ T g ∆ ,l ( z, ¯ z ) (cid:19)(cid:12)(cid:12)(cid:12) ≤ R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) . (2.29)9nother positive definite linear combination of correlators is B η ≡ (cid:104) φ φ φ φ (cid:105) + | η | (cid:104) φ φ φ φ (cid:105) + η (cid:104) φ φ φ φ (cid:105) + η ∗ (cid:104) φ φ φ φ (cid:105) = b η ( u, v ) x φ x φ , (2.30)corresponding to the norm of the state φ | φ (cid:105) + ηφ | φ (cid:105) . (2.31)Again, we consider η = ±
1. In the (12)-(34) channel the correlator B η can be written as B η = 1 x φ x φ (cid:16) η ) (cid:88) T + λ T g ∆ ,l ( u, v ) + 2(1 − η ) (cid:88) A − λ A g ∆ ,l ( u, v ) (cid:17) , (2.32)where A stands for operators in the rank-two antisymmetric representation of O ( n ). The firstsum runs over even spins, whereas for the second one they are odd. As before, the leading term in b η ( u, v ) for x → x is given by the exchange of the identity operator in the first two correlatorsand is independent of η . For η = 1, eq. (2.32) gives rise to the same bound given in eq. (2.28),while for η = − (cid:12)(cid:12)(cid:12) (cid:88) (∆ ≥ ∆ ∗ ) ,l λ A g ∆ ,l ( z, ¯ z ) (cid:12)(cid:12)(cid:12) ≤ R ( z, ¯ z, ∆ ∗ , ∆ φ , γ ) . (2.33)It is straightforward to see that the bounds (2.28), (2.29) and (2.33) are the best that can beobtained. Indeed, in the free-theory limit one has λ S = λ /n , λ T = λ A = λ / λ being theOPE coefficients for a single free field (see e.g. eq. (5.11) in ref. [20]). The above three boundsthen reduce to eq. (2.16) which is known to give the best bound on the r.h.s. of eq. (2.12) (seehowever footnote 7) [12]. Any potentially better bound for O ( n ) theories should in particularapply to the free theory, but would then be in contradiction with the results of ref. [12].The above bounds will be used in the next section to bound the remainder of the bootstrapequations in CFTs with an O ( n ) global symmetry. The bootstrap equation for a 4-point function with identical scalars φ with scaling dimension ∆ φ in any number of dimensions is given by the sum rule (see refs. [30, 31] for pedagogical reviews) (cid:88) ∆ ,l λ O F ∆ φ , ∆ ,l ( z, ¯ z ) = u ∆ φ − v ∆ φ , F ∆ φ , ∆ ,l ( z, ¯ z ) ≡ v ∆ φ g ∆ ,l ( u, v ) − u ∆ φ g ∆ ,l ( v, u ) . (3.1)Splitting the sum into two parts, for dimensions smaller and larger than a cutoff ∆ ∗ , we canwrite (cid:88) (∆ < ∆ ∗ ) ,l λ O F ∆ φ , ∆ ,l ( z, ¯ z ) = u ∆ φ − v ∆ φ + E ( z, ¯ z, ∆ ∗ , ∆ φ ) . (3.2) In our normalization conventions for the conformal blocks, the squared OPE coefficients λ S,T,A are all positive. E of the sum rule is bounded by |E ( z, ¯ z ) | ≤ E max ( z, ¯ z ) ≡ v ∆ φ R ( z, ¯ z ) + u ∆ φ R (1 − z, − ¯ z ) , (3.3)where we have omitted the dependence on ∆ ∗ , ∆ φ and γ . The truncated sum rule (3.2) stillinvolves a generally unknown spectrum of operators up to dimension ∆ ∗ . In order to make itamenable to numerical analysis, we discretize the spectrum and make the ansatz (cid:110) (0 , d −
22 ) , (0 , d −
22 + ∆ step ) , . . . , (0 , ∆ ∗ ) , (2 , d ) , (2 , d + ∆ step ) , (2 , ∆ ∗ ) , . . . , ( l max , ∆ ∗ ) (cid:111) (3.4)for the quantum numbers (spin,dimension) of the operators that can appear in the truncatedsum rule. For each spin l , the dimension runs in steps of size ∆ step from the unitarity bound∆ d,l min ≡ l + ( d − / (1 + δ l ) to the cutoff ∆ ∗ (or a value close to that, depending on ∆ step ).Accordingly, l max is the largest spin for which the unitarity bound is still below the cutoff,∆ d,l max min < ∆ ∗ . In practice, we vary the step size ∆ step somewhat depending on the spin anddimension. This is discussed in more detail in sec. 5. We find that the bounds converge whengoing to smaller ∆ step , meaning that the discretization does not introduce any artifacts into ourcalculation.We similarly choose a finite number of points z i in the z -plane where the sum rule is evaluated.The details of our choice for this distribution of points are discussed in sec. 3.1. Together withthe discretization of operator dimensions, this turns eq. (3.2) into the matrix equation M · (cid:126)ρ = (cid:126)σ + (cid:126)(cid:15) . (3.5)The elements of the matrix M are the functions F ∆ φ , ∆ ,l ( z, ¯ z ) evaluated for the different quantumnumbers in eq. (3.4) along the rows and for the different points z i along the columns. Further-more, the vector (cid:126)ρ consists of the squared OPE coefficients λ O of the operators correspondingto the quantum numbers in eq. (3.4) and (cid:126)σ ≡ | z | φ − | − z | φ | z | φ − | − z | φ ... and (cid:126)(cid:15) ≡ E ( z , ¯ z , ∆ ∗ , ∆ φ ) E ( z , ¯ z , ∆ ∗ , ∆ φ )... . (3.6)Using the bound (3.3), we then obtain the matrix inequality (cid:32) M−M (cid:33) (cid:126)ρ ≥ (cid:32) (cid:126)σ − (cid:126)(cid:15) max − (cid:126)σ − (cid:126)(cid:15) max (cid:33) , (3.7)where (cid:126)(cid:15) max is defined as (cid:126)(cid:15) but with E replaced by E max . This is the starting point for our numericalcalculations. In order to determine bounds on OPE coefficients, we search for vectors (cid:126)ρ whichsatisfy eq. (3.7) and extremize the entry corresponding to that OPE coefficient. For bounds onthe dimension of the lowest-lying scalar operator, on the other hand, we make an assumption on Alternatively, one could adapt the approach of ref. [6] to the multipoint method. (cid:126)ρ which satisfies eq. (3.7) withthe reduced ansatz. By trying different assumptions, we can determine the maximal allowedgap. Both problems are linear programs which can be solved using fast numerical routines. Anadvantage of solving eq. (3.7) is that the vector (cid:126)ρ gives us the spectrum of operators and theirOPE coefficients of a potential CFT living at the boundary of the allowed region. This has beenused before in ref. [6]. We also consider CFTs with an O ( n ) global symmetry. For an external scalar operator inthe fundamental representation of O ( n ), the sum rule reads [19] (cid:88) S + λ S FH + (cid:88) T + λ T F (1 − n ) F− (1 + n ) H + (cid:88) A − λ A −FF−H = u ∆ φ − v ∆ φ − u ∆ φ − v ∆ φ , (3.8)where H ∆ φ , ∆ ,l ( z, ¯ z ) ≡ v ∆ φ g ∆ ,l ( u, v ) + u ∆ φ g ∆ ,l ( v, u ) and we have suppressed the arguments ofthe functions F and H . Splitting the sums in eq. (3.8) into two parts, for dimensions smallerand larger than a cutoff ∆ ∗ , we can write (cid:88) S + ∆ < ∆ ∗ λ S FH + (cid:88) T + ∆ < ∆ ∗ λ T F (1 − n ) F− (1 + n ) H + (cid:88) A − ∆ < ∆ ∗ λ A −FF−H = E u ∆ φ − v ∆ φ + E − u ∆ φ − v ∆ φ + E . (3.9)Using eqs. (2.28), (2.29) and (2.33), we obtain the bounds on the remainders |E , ( z, ¯ z ) | ≤ E max ( z, ¯ z ) , |E ( z, ¯ z ) | ≤ E max ( z, ¯ z ) , (3.10)with E max defined as in eq. (3.3). Discretizing the space of operator dimensions as in eq. (3.4)and evaluating the sum rule at a finite set of points z i , we again obtain a matrix inequality ofthe form (3.7). This is the starting point for our numerical calculations for CFTs with O ( n )global symmetry. An important choice for the multipoint method is the distribution of points in the z -plane atwhich the bootstrap equations are evaluated. Using the symmetries z ↔ ¯ z and z ↔ (1 − z ),¯ z ↔ − ¯ z of the bootstrap equations, we can restrict these points to the region Re( z ) ≥ / z ) ≥ z -plane. The remainder of the truncated sum rule is controlled by | ρ ( z ) | and | ρ (1 − z ) | (cf. eqs. (2.18) and (3.3)). Guided by this, we introduce the measure λ ( z ) ≡ | ρ ( z ) | + | ρ (1 − z ) | , (3.11) The data of CFTs at the boundary of the allowed region can also be obtained from the ‘dual’ method originallydeveloped in ref. [4] by using the extremal functional method of ref. [32]. λ ( z ) ≤ λ c for some constant λ c . It is desirable to choose λ c and thedistribution of points within that region in such a way that the obtained bounds are as stringentas possible. We have performed extensive scans over different values for λ c and distributions withdifferent density profiles and have found that a flat profile leads to as good or better bounds thanmore complicated profiles. We therefore choose the former and put points on a grid centered at z = 1 /
2. The grid spacing is chosen such that the desired number of points is within the region λ ( z ) ≤ λ c , Re( z ) ≥ / z ) ≥
0. We have then found that λ c = 0 . In fig. 2, we show the correspondingregion in the z -plane and a sample distribution of 100 points.In order to test the influence of the choice of measure on the bounds, we have performedfurther scans with λ ( z ) ≡ max( | ρ ( z ) | , | ρ (1 − z ) | ) proposed in ref. [1] and λ ( z ) ≡ | z − / | (forthe latter we have removed points at or close to the branch-cuts). We have found that, once theoptimal λ c is chosen, the bounds obtained with these measures are indistinguishable from thoseobtained with eq. (3.11). This indicates that the precise form of the region within which pointsare sampled has only a marginal effect on the quality of the bounds. We now present the results of our numerical analysis. In subsection 4.1, we study bounds onthe dimension of the lowest-dimensional scalar operator in the OPE and bounds on the centralcharge in 3D CFTs, focusing in particular on the regions where the 3D Ising and O ( n ) modelshave been identified. In subsection 4.2 we then study the same bounds for generic 4D CFTs. Weanalyze in particular how our results depend on the number N of points chosen in the z -plane,and on the cutoff ∆ ∗ . In subsection 4.3 we give a closer look at the spectrum of the 3D O ( n )models and determine the operator dimensions of the first two scalar operators in the singletand rank-two symmetric representation of O ( n ).Before presenting our results, it is important to emphasize an important difference betweenthe multipoint and the derivative bootstrap methods. As mentioned in the introduction, in thelatter we do not have a reliable way of truncating the OPE series defining the bootstrap equationsat some intermediate dimension ∆ ∗ , because we do not have a reliable estimate of the resultingerror. We are therefore forced to have ∆ ∗ as large as possible to minimize this error and canonly check a posteriori if the chosen ∆ ∗ was sufficient. More than ∆ ∗ (or its analogue), the key In more detail, we have considered bounds on the central charge and the dimension of the lowest-dimensionalscalar operator, in 3D and 4D, with O ( n ) and without symmetry, and with different choices for the number ofpoints N and the cutoff ∆ ∗ . It is remarkable that λ c = 0 . ± .
02, the resolution of our scan) comes outas the optimal choice for such a variety of cases. We are a bit sloppy here in order to keep the discussion simple and get to the point. For instance, in numericalmethods based on semi-definite programming one is able to include all operator dimensions continuously up toinfinity. The rough analogue of our ∆ ∗ in that case is the maximum spin of the primary operators entering the ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ - - - ( z ) Im ( z ) Figure 2:
The region in the z -plane with λ ( z ) ≤ . parameter that controls the accuracy of the method is given by the total number of derivatives N D that are applied to the bootstrap equations. Of course, the larger N D is, the better arethe bounds. The accuracy is then limited by the largest N D that allows the calculation to beperformed within an acceptable amount of time with the available computing resources.In the multipoint method, on the other hand, we can reliably vary ∆ ∗ due to the bound onthe remainder of the truncation discussed in sec. 2. In addition, we can also vary the number N ofpoints in the z -plane which is the analogue of N D in the derivative method. The parameter regionfor the multipoint method corresponding to the typical bootstrap analysis with the derivativemethod is then very large ∆ ∗ and N as large as possible given the available computing resources.In this paper, on the other hand, we are mostly interested in the regime where ∆ ∗ is not verylarge, with values O (10)- O (20). We find that for this range of ∆ ∗ , the results converge for N ∼ O (100) and do not improve further if N is increased. This corresponds to the fact thatthe rank of the matrix M in the discretized bootstrap equation (3.5) is then O (100). Note thatsince CPLEX is limited to double precision, we also cannot take ∆ ∗ arbitrarily large. Due to theexcellent speed of CPLEX , on the other hand, we have found that taking N large enough so thatthe bounds converge is no limiting factor. OPE which are taken into account for the numerical implementation. .50 0.51 0.52 0.53 0.54 0.551.11.21.31.41.51.6 Δ σ Δ ϵ Figure 3:
Bounds on ∆ (cid:15) as a function of ∆ σ for N = 100 points and different values of ∆ ∗ . The regionsabove the lines are excluded. The black cross marks the precise values of ∆ σ and ∆ (cid:15) for the 3D Isingmodel as determined in ref. [6]. The curves and the labels in the legend have the same order from top tobottom. Δ σ c / c free Figure 4:
Bounds on the central charge c as a function of ∆ σ for N = 100 points and different values of∆ ∗ . A gap ∆ (cid:15) > . σ and c for the 3D Ising model as determined in ref. [6]. The curves and the labelsin the legend have the same order from top to bottom. O ( n ) Models
The most remarkable numerical results from the conformal bootstrap have been obtained in3D CFTs. One interesting bound to study is on the dimension of the lowest-dimensional scalaroperator appearing in the OPE. We denote this operator by (cid:15) and the operator that is used toderive the bootstrap equations by σ . It was noted in ref. [5] that the 3D Ising model sits at aspecial point, a kink, at the boundary of the allowed region of ∆ (cid:15) as a function of ∆ σ . The Isingmodel is similarly special with respect to the bound on the central charge c as a function of ∆ σ ,15itting again at the boundary of the excluded region, at the point where c is minimized [5, 6].Note, however, that the theory minimizing c does not actually correspond to the 3D Ising model,but rather to some exotic theory with ∆ (cid:15) <
1. Most likely this theory is unphysical (though weare not aware of a solid argument to dismiss it). In practice this theory is removed by assuminga gap in the operator spectrum such that ∆ (cid:15) >
1. Independently of the nature of this theory,the condition ∆ (cid:15) > (cid:15) as a function of ∆ σ for N = 100 points and differentvalues of ∆ ∗ . Notice how the kink shows up already for ∆ ∗ = 13 and converges quite quickly as∆ ∗ increases. In fig. 4, we show the bound on the central charge c (normalized to the centralcharge c free of a free scalar theory) as a function of ∆ σ for N = 100 points and different valuesof ∆ ∗ . The gap ∆ (cid:15) > . c is obtainedeven for ∆ ∗ = 10, but the convergence when going to larger ∆ ∗ is now much slower than for thebound on ∆ (cid:15) . A minimum is visible starting from ∆ ∗ = 16 but even at ∆ ∗ = 22 it is a bit shiftedto the right with respect to its actual value. We have still not reached the asymptotic valuefor ∆ ∗ . Unfortunately, we cannot get reliable results for much higher ∆ ∗ because the numericalaccuracy of CPLEX is limited to double precision. Nevertheless, it is clear from comparing figs. 3and 4 that the lower bound on c is more “UV sensitive” than the bound on ∆ (cid:15) . In both figures,the crosses mark the location of the 3D Ising model, as determined in ref. [6].In order to quantify the dependence of our results on the number N of points, we show infigs. 5 and 6 the bounds on respectively ∆ (cid:15) and c as a function of ∆ σ for different values of N at fixed ∆ ∗ = 16. We see that in both cases the convergence in N is quite fast, with N = 40 for∆ (cid:15) and N = 60 for c being already an excellent approximation. Notice that for increasing N ,the bound on ∆ (cid:15) converges faster than the bound on c , similar to the dependence on ∆ ∗ . Wehave studied the dependence on N also for different values of ∆ ∗ and have found as expectedthat the value N ∗ beyond which no significant improvement in the bounds is observed increaseswith ∆ ∗ . The dependence is however very mild for the central charge c and barely observablefor ∆ (cid:15) . This is still a reflection of the different “UV sensitivities” of the two quantities. In allcases, N ∗ (cid:46) O (100) up to ∆ ∗ = 24.Let us now turn to 3D CFTs with O ( n ) global symmetry. We consider a primary operator φ in the fundamental representation and denote the lowest-dimensional scalar singlet operator inthe φ × φ OPE by S . It was found in refs. [14, 16] that these CFTs have kinks in the bound on∆ S as a function of ∆ φ similar to that found for the Ising model. Moreover, the kinks coincide,for all values of n that have been studied, with the values of ∆ φ and ∆ S associated with the 3D O ( n ) models. On the other hand, a minimum in c no longer occurs for generic O ( n ) models andthe lower bound on c instead monotonically decreases for n > S and c (the latter normalized to thecentral charge nc free of n free scalars) as a function of ∆ φ for different O ( n ) symmetries, at fixed N = 80 and ∆ ∗ = 16. For the central charge, gaps ∆ S > T > S and rank-two symmetric-traceless operators T are assumed as16 .50 0.51 0.52 0.53 0.54 0.551.11.21.31.41.51.6 Δ σ Δ ϵ Figure 5:
Bounds on ∆ (cid:15) as a function of ∆ σ for fixed ∆ ∗ = 16 and different values of N . The regionsabove the lines are excluded. The black cross marks the precise values of ∆ σ and ∆ (cid:15) for the 3D Isingmodel as determined in ref. [6]. The curves and the labels in the legend have the same order from top tobottom. Δ σ c / c free Figure 6:
Bounds on the central charge c as a function of ∆ σ for fixed ∆ ∗ = 16 and different values of N . The gap ∆ (cid:15) > . σ and c for the 3D Ising model as determined in ref. [6]. The curves and the labels inthe legend have the same order from top to bottom. in ref. [14]. This assumption is satisfied for the O ( n ) models and leads to more stringent bounds.The dashed line corresponds to the leading large- n prediction. All the qualitative behavioursfound in ref. [14] are reproduced, though with milder bounds, as expected. In particular, thekinks in the (∆ φ -∆ S ) plane are not well visible at ∆ ∗ = 16. In figs. 9 and 10, we show the samebounds on ∆ S and c as a function of ∆ φ at fixed N and n , for different values of ∆ ∗ . We see the Note however that no assumption on the spectrum was made for the bounds on ∆ S presented in fig. 7, incontrast to fig. 2 of ref. [14] where ∆ T > .500 0.505 0.510 0.515 0.520 0.5251.01.21.41.61.82.02.22.4 Δ ϕ Δ S Figure 7:
Bounds on ∆ S as a function of ∆ φ for 3D CFTs with different O ( n ) symmetries, with φ inthe fundamental representation of O ( n ). The regions above the lines are excluded. All the bounds havebeen determined using N = 80 points and ∆ ∗ = 16. The curves and the labels in the legend have thesame order from top to bottom. Δ ϕ cn c free no sym.234561020 Figure 8:
Bounds on the central charge c as a function of ∆ φ for 3D CFTs with different O ( n ) symmetries,with φ in the fundamental representation of O ( n ). The regions below the lines are excluded. All the boundshave been determined using N = 80 points and ∆ ∗ = 16 with gaps ∆ S > T > n prediction. The curves and the labels in the legend have the same orderfrom top to bottom. same qualitative behaviours regarding the “UV sensitivities” found for 3D CFTs with no globalsymmetry (the Ising model). In particular, in fig. 9 we see how the kink in the bound becomeswell visible at ∆ ∗ = 18 and does not significantly improve for ∆ ∗ = 20. Its location is in verygood agreement with that found in ref. [14]. On the other hand, the central-charge bound infig. 10 is still monotonically decreasing for ∆ ∗ = 18 and a minimum appears only for ∆ ∗ = 20.There are no signs of convergence comparing the bounds at ∆ ∗ = 18 and 20, indicating the needto go to larger ∆ ∗ to approach the optimal bound.18 .500 0.505 0.510 0.515 0.520 0.5251.01.52.02.53.0 Δ ϕ Δ S Figure 9:
Bounds on ∆ S as a function of ∆ φ for N = 100 points and different values of ∆ ∗ for 3DCFTs with O (20) symmetry, with φ in the fundamental representation of O (20). The regions above thelines are excluded. The black cross marks the values of ∆ φ and ∆ S for the O (20) vector model as givenin ref. [14]. The curves and the labels in the legend have the same order from top to bottom. Δ ϕ c c free Figure 10:
Bounds on the central charge c as a function of ∆ φ for N = 100 points and different valuesof ∆ ∗ for 3D CFTs with O (2) symmetry, with φ in the fundamental representation of O (2). Gaps ∆ S > T > All the above considerations can be repeated for 4D CFTs. There are no known non-super-symmetric CFTs at benchmarks points but it is still interesting to study general bounds onoperator dimensions and OPE coefficients. See e.g. refs. [4, 10, 17–22, 33], where bounds of thiskind (and others) have been determined with the derivative method using both linear and semi-definite programming.In figs. 11 and 12, we show bounds respectively on the dimension ∆ φ of the lowest-dimensional19 .0 1.1 1.2 1.3 1.4 1.5 1.6 1.7234567 Δ ϕ Δ ϕ Figure 11:
Bounds on ∆ φ as a function of ∆ φ for N = 100 points and different values of ∆ ∗ for 4DCFTs with no global symmetry. The regions above the curves are excluded. The curves and the labels inthe legend have the same order from top to bottom. Δ ϕ c / c free Figure 12:
Bounds on the central charge c as a function of ∆ φ for N = 100 points and different valuesof ∆ ∗ for 4D CFTs with no global symmetry. The regions below the curves are excluded. The curves andthe labels in the legend have the same order from top to bottom. scalar operator in the φ × φ OPE and on the central charge c as a function of ∆ φ for differentvalues of ∆ ∗ , at fixed N . The conclusions are the same as for the 3D CFTs: the bounds on theoperator dimension converge faster than those on the central charge. The point of convergenceof the bounds in N at fixed ∆ ∗ is again N ∗ ∼ O (100) and thus also very similar to that in 3DCFTs.The analysis of 4D CFTs with O ( n ) global symmetry also closely resembles its 3D counter-part. We again take the external field φ to transform in the fundamental representation of O ( n )and denote by S the lowest-dimensional singlet scalar operator that appears in the φ × φ OPE.For illustration, we report in fig. 13 the bound on ∆ S as a function of ∆ φ for CFTs with O (4)20 .0 1.1 1.2 1.3 1.4 1.5 1.6 1.7234567 Δ ϕ Δ S Figure 13:
Bounds on ∆ S as a function of ∆ φ for N = 100 points and different values of ∆ ∗ for 4DCFTs with O (4) symmetry, with φ in the fundamental representation of O (4). The regions above thecurves are excluded. The curves and the labels in the legend have the same order from top to bottom. symmetry, at fixed N and for different values of ∆ ∗ . By comparing figs.11 and 13 we notice thatthe convergence in ∆ ∗ of the operator-dimension bound in 4D CFTs with O (4) symmetry isslower than its analogue with no global symmetry. O ( n ) Models
In the last subsections, we have shown how previously determined bounds are reproduced usingthe multipoint method. Here we present some new results for the spectrum of O ( n ) models.To this end we assume, as previous analyses indicate, that the 3D O ( n ) models sit preciselyat the kink on the boundary of the excluded region in the (∆ φ -∆ S ) plane (∆ S -maximization).The vector (cid:126)ρ that we obtain from solving the linear program (3.7) then gives us the spectrumand OPE coefficients of the operators that are exchanged in the (cid:104) φφφφ (cid:105) correlator of the O ( n )models. Here we report the scaling dimensions of the first two operators in respectively thesinglet and rank-two representation of O ( n ), S , S (cid:48) and T , T (cid:48) , for n = 2 , ,
4. Scalar operatorswith larger scaling dimensions are physically uninteresting, whereas S (cid:48) and T (cid:48) are important indetermining the stability of the fixed points of the O ( n ) models (being marginal operators in theunderlying UV 4D Landau-Ginzburg theory) [24]. Actually, one additional operator should beconsidered, denoted as P , in ref. [24], but it transforms in the rank-four representation of O ( n )and hence cannot appear in the OPE of two scalar operators φ in the fundamental representation.Its dimension might be bounded (or computed) by considering a correlator involving, e.g., four T ’s. As far as we know, the scaling dimensions of S (cid:48) and T (cid:48) have not been previously determinedusing the conformal bootstrap. The best determinations of these parameters have been made See ref. [37] for a bootstrap approach to the study of the stability of fixed points in 3D O ( n ) × O ( m ) models. ∆ φ ∆ S ∆ S (cid:48) ∆ T ∆ T (cid:48) . +0 . − . [14] 3 . . +0 . − . [14] 3 . . +0 . − . [14] 3 . . +0 . − . [14] 3 . . +0 . − . [14] 3 . . +0 . − . [14] 3 . Scaling dimensions of the first two scalar operators in the singlet ( S , S (cid:48) ) and rank-two symmetric( T , T (cid:48) ) representations of O ( n ) for n = 2 , , n ∆ φ ∆ S ∆ S (cid:48) ∆ T ∆ T (cid:48) Scaling dimensions of the first two scalar operators in the singlet ( S , S (cid:48) ) and rank-two symmetric( T , T (cid:48) ) representations of O ( n ) for n = 2 , , S -maximization, the valuesof ∆ φ previously determined in the literature (first column) and the fit procedure explained in the maintext. The quoted error corresponds to 1 σ (68% confidence level). using a five-loop computation in the (cid:15) -expansion in refs. [23] and [24]. In table 1, we report the values of ∆ φ , ∆ S , ∆ S (cid:48) , ∆ T , ∆ T (cid:48) determined in the literature, for n = 2 , ,
4. They should be compared with the values in table 2 which have been determinedin this paper as follows: We take the values of ∆ φ for O ( n ) models with n = 2 , , S , ∆ S (cid:48) , ∆ T and ∆ T (cid:48) using ∆ S -maximization. We repeat this procedure for the lower, central and upper value of ∆ φ given inthese references and for different values of the cutoff ∆ ∗ ∈ [18 ,
23] and the number of points N ∈ [60 , At fixed N and ∆ ∗ , we then take the average over the scaling dimensionsobtained with the different input values of ∆ φ . Sometimes the same operator appears twice inthe spectrum, at two different but close values of the scaling dimension. In this case we takethe average of these values, weighted by the size of the corresponding OPE coefficient. Let usdenote the resulting scaling dimensions by ∆ O ( N, ∆ ∗ ) for O = S, S (cid:48) , T, T (cid:48) . Each of these valuesis associated with an error, resulting from the averaging. The stepsize ∆ step of our discretizationhas been set to 10 − in the region where the operators were expected to be found (the resultinguncertainty in the scaling dimensions is typically negligible compared to the other errors).At fixed N , the results for different values of ∆ ∗ are fitted by a function of the form a O ( N ) + b O ( N ) exp( − c O ( N )∆ ∗ ), where a O ( N ), b O ( N ) and c O ( N ) are the fit parameters. Sucha dependence is roughly expected given the exponential convergence of the OPE. Somewhat More precisely, ∆ S (cid:48) has been determined also by other means, such as fixed-dimension expansion and MonteCarlo simulations. On the other hand, since ∆ T (cid:48) has been determined only using the (cid:15) -expansion, we havedecided to omit the other results for ∆ S (cid:48) . The interested reader can find them, e.g., in table I of ref. [24], wherethe coefficients y , and y , give ∆ S (cid:48) = 3 − y , and ∆ T (cid:48) = 3 − y , . For completeness, we also report the relationsdefining ∆ S and ∆ T in the notation of ref. [24]: ∆ S = 3 − /ν , ∆ T = 3 − y , . Our numerical precision does not allow us to take higher values of ∆ ∗ and N without having issues withnumerical stability. ∆ φ ∆ S ∆ S (cid:48) ∆ T ∆ T (cid:48) ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Upper bounds on the scaling dimensions of the first two scalar operators in the singlet ( S , S (cid:48) )and rank-two symmetric ( T , T (cid:48) ) representations of O ( n ) for n = 2 , , S -maximization and the values of ∆ φ previously determined in the literature (first column). surprisingly, this simplified function fits the results extremely well, see fig. 14 for an example ofthe extrapolation fit in 1 / ∆ ∗ . Using this fit, we have extrapolated the scaling dimensions for thedifferent operators and values of N to ∆ ∗ = ∞ . We denote the resulting scaling dimensions as∆ O ( N ) ≡ ∆ O ( N, ∞ ) = a O ( N ).We have then extrapolated to N = ∞ using a linear fit in 1 /N which seems to well describethe behaviour of ∆ O ( N ) as a function of 1 /N . An example of this extrapolation fit is shownin fig.15. We denote the resulting scaling dimensions as ∆ O ≡ ∆ O ( ∞ ). We do not have ananalytic understanding of why the results should scale as 1 /N for parametrically large ∆ ∗ .We simply take it as a working hypothesis. We expect that possible deviations from the linearbehaviour should be contained within the errors of our determination (cf. fig.15). Note thathaving N as large as possible is clearly important for high precision. However, at fixed ∆ ∗ thebounds saturate for sufficiently high N and there is no gain in taking N larger.We have noticed that, at least for n = 2 , ,
4, ∆ O ( N, ∆ ∗ ) decreases as N and/or ∆ ∗ increase(this is obvious for S , but not for the other operators). If we assume that this is true for any N and ∆ ∗ , we may then set rigorous upper bounds without using any fit extrapolation. Thesebounds are reported in table 3. Comparing them with the results in table 2 gives an idea of theimpact of the fit extrapolation on the final results. As can be seen, all the scaling dimensions thatwe have determined are compatible with previous results in the literature. The only exception is∆ T (cid:48) for the O (2) model for which our result has an approximate 3 σ tension with that of ref. [24].Our accuracy in the determinations of ∆ S and ∆ T is comparable with that achieved in ref. [14],though it should be emphasized that the results there do not rely on extrapolations. Furthermore,our accuracy in the determinations of ∆ S (cid:48) and ∆ T (cid:48) is comparable with that achieved using thefive-loop (cid:15) -expansion. This is an indication that a slightly more refined bootstrap analysis willbe able to improve the determinations of these scaling dimensions.As we mentioned at the beginning of this subsection, ∆ S -maximization also allows us todetermine the OPE coefficients λ φφ O . We have not performed a detailed analysis with fit ex-trapolations as above to determine the asymptotic values of λ φφ O as ∆ ∗ , N → ∞ . Instead wejust report λ φφS as determined with the highest values ∆ ∗ = 22 ,
23 and N = 110 ,
120 used in A similar linear dependence in 1 /N has already been noticed with great accuracy in ref. [38] for the central-charge bound in 6D N = (2 ,
0) SCFTs (see their fig. 1). .01 0.02 0.03 0.04 0.05 0.06 / Δ * Δ T ' ( Δ * ) Figure 14:
Extrapolation fit to determine the scaling dimension of the operator T (cid:48) in the O (2) modelwith N = 120 points at ∆ ∗ = ∞ from the results for that scaling dimension for different values of ∆ ∗ . Thevertical error bar associated with the extrapolated point on the left corresponds to 1 σ (68% confidencelevel). / N Δ S ' ( N ) Figure 15:
Extrapolation fit to determine the scaling dimension of the operator S (cid:48) in the O (3) modelat N = ∞ from the results for that scaling dimension for different values of 1 /N . Each point correspondsto the value of ∆ S (cid:48) ( N ) extracted from a fit in 1 / ∆ ∗ . The vertical error bar associated with each pointcorresponds to 1 σ (68% confidence level). this paper: O (2) : λ φφS ≈ . ,O (3) : λ φφS ≈ . , (4.1) O (4) : λ φφS ≈ . . We have not determined the error associated with these results and have instead rounded themto the last shown digit. The results for O (2) and O (3) are in agreement with the recent deter-mination in ref. [7], whereas the result for O (4) is new as far as we know.24 Details of the Implementation
For the conformal blocks in d = 4 dimensions, we use the closed-form expression from ref. [8],normalized as in ref. [19]. For d = 3 dimensions, on the other hand, we use the recursion relationfor the conformal blocks found in ref. [14]. To this end, we iterate the recursion relation up tosome cutoff ∆ rec . We choose this cutoff large enough such that the resulting error in the conformalblocks is smaller than the error from neglecting contributions of operators with dimensions largerthan the truncation cutoff ∆ ∗ . In practice, we find that ∆ rec = ∆ ∗ + few is sufficient to ensurethis.For the ansatz (3.4) of discretized operator dimensions, we closely follow ref. [5]. We generatethe discrete spectra T1 to T4 (the latter only for sufficiently large ∆ ∗ ) in their table 2, wherewe rescale the stepsizes δ by the factor ∆ step / (2 · − ). We then remove duplicates from thecombined spectrum and restrict to operator dimensions less than or equal to ∆ ∗ . We have per-formed extensive scans using different stepsizes ∆ step and have found that the bounds convergefor sufficiently small ∆ step . This is in particular satisfied for ∆ step = 2 · − which we choosefor all the plots in this paper. For the determination of the spectra in sec. 4.3 we add additionaloperators with stepsize ∆ step = 10 − around the previously determined scaling dimensions forthe operators S , S (cid:48) , T , T (cid:48) in the O ( n ) models. Furthermore, for bounds on operator dimensionsfor which the plots extend to bounds ∆ φ > step is used upto the largest bound on ∆ φ shown in that plot. We have also performed scans using differentparametrizations for the ansatz (3.4) and have found that the bounds become indistinguishablefrom the bounds obtained with the ansatz discussed above for sufficiently small ∆ step . This givesus confidence that the discretization does not introduce any artifacts into our calculations.We use Mathematica to evaluate the conformal blocks for the different operators that appearin the ansatz (3.4) and for the set of points in the z -plane. The linear progam (3.7) is then setup by a program written in Python and is subsequently solved with the optimizer CPLEX by IBM using the primal simplex algorithm. Since this optimizer is limited to double precision, it isimportant to reduce the spread in size of the numerical values in the problem. To this end, notethat we can rescale each row of the inequality (3.7) separately by a positive number. Denotinga given row by R , we rescale its elements by R resc i = R i (cid:113) min i |R i | · max i |R i | . (5.1)Similarly, we can rescale each column of the matrix M separately by a positive number if we Alternatively, we can use the recursion relation also in d = 4 dimensions by setting d = 4 + (cid:15) (to avoiddouble poles that appear at d = 4). However, Mathematica evaluates the closed-form expression faster than (ourimplementation of) the recursion relation and we therefore choose the former. (cid:126)ρ . We again choose M resc ij = M ij (cid:113) min i |M ij | · max i |M ij | (5.2)and correspondingly for (cid:126)ρ . This procedure is iterated three times in our Python code, usingprecision arithmetric with 120 digits to ensure that no significant rounding errors are introducedin the process (the conformal blocks have been calculated with the same precision). Since weperform our own rescaling, we switch off this option in CPLEX .We find that the above rescaling typically reduces the orders of magnitude in the ratio be-tween the largest and smallest numerical value in eq. (3.7) by about half. Nevertheless, precisionis a limiting factor and does not allow us to go to cutoffs ∆ ∗ much larger than 20. The factthat double precision is sufficent for smaller cutoffs, on the other hand, makes our calculations(combined with the excellent speed of CPLEX ) very fast.
We have implemented the method proposed in ref. [1] to numerically study the bootstrap equa-tions away from the symmetric point z = ¯ z = 1 /
2. Using this method, we have qualitativelyreproduced various results that have been determined in the bootstrap literature using the morecommon method of taking derivatives at the symmetric point. The main aim of our work wasto show that bootstrapping with multipoints works and is a valid alternative to the standardderivative method. In particular, it can be useful at a preliminary stage when one wants toqualitative bound or approximately compute some quantities using the bootstrap. By choosinga sufficiently low cutoff ∆ ∗ , one can get qualitatively good results within seconds of CPU timewith a standard laptop! Since the optimizer CPLEX that we use is limited to double precision, wecan not achieve the high precision of refined bootstrap codes such as Juliboots [9] or SDPB [11].Nevertheless we have shown how, using ∆-maximization, relatively precise results can be ob-tained for the scaling dimensions of operators (though we relied on an extrapolation procedure).In particular, for O ( n ) models with n = 2 , , S (cid:48) and T (cid:48) in the singlet and symmetric-traceless represen-tation, respectively. To our knowledge, these have not been determined before using bootstraptechniques. We believe that it should not be difficult to go to arbitrary precision and get rid of thediscretization (and the extrapolation procedure) by, for instance, adapting the algorithm devel-oped in refs. [6, 9] to multipoints. We do not exclude that bootstrapping with multipoints mightthen turn out to be comparable to (or better than) the derivative method for high-precisioncomputations. From a conceptual point of view, the multipoint method is more rigorous, sincethe crossing equations are not truncated but bounded by an error. Strictly speaking, this is true only when we are guaranteed to be in the regime where the Hardy-Littlewoodtauberian theorem applies. But all the evidence so far indicates that this is always the case for ∆ ∗ (cid:38) O (10).
26e have also discussed how the multipoint method is useful in understanding to which extenta given numerical result depends sensitively on the high-dimensional operators. In particular, wehave noticed that bounds on operator dimensions are less sensitive in this respect than boundson the central charge.Ideally, one might want to push the multipoint method to the extreme “IR limit”, by choosinga cutoff ∆ ∗ so low that an analytic approach may become possible. This is certainly a veryinteresting direction that should be explored. Among other things, it requires to improve onthe estimate of the OPE convergence given in ref. [12] that applies in the opposite regime, forparametrically large ∆ ∗ . Perhaps the results of ref. [25] might be useful in this respect. An important line of development in the numerical bootstrap is the analysis of mixed cor-relators which so far are numerically accessible only using semi-definite programming [15]. Itwould be very interesting to implement mixed correlators in the multipoint bootstrap, eitherby adapting the semi-definite programming techniques or by extending the linear programmingtechniques.
Acknowledgments
M.S. thanks Pasquale Calabrese, Sheer El-Showk, Miguel F. Paulos, Alessandro Vichi and espe-cially Slava Rychkov for useful discussions. We also thank Slava Rychkov for comments on themanuscript. We warmly acknowledge Antonio Lanza, Piero Calucci and all members of SISSA- ITCS (Information Technology and Computing Services) for their continuous availability andsupport in using the Ulysses HPC facility maintained by SISSA. The work of M.S. was supportedby the ERC Advanced Grant no. 267985 DaMESyFla. A.C.E. and M.S. gratefully acknowledgesupport from the Simons Center for Geometry and Physics at Stony Brook University, wheresome of the research for this paper was performed.
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