Scalar-fermion analytic bootstrap in 4D

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113

217

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weightshifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and "seed" operators in 4d, and MFT for currents and stress-tensors in 3d.Mean Field Theory (MFT) provides some of the simplest examples of crossing-symmetric, conformally-invariant correlation functions. Correlators in MFT are simply sums of products of two-point functions. In theories exhibiting large-N factorization, MFT is the leading contribution at large-N . For example, in AdS/CFT, MFT is the leading contribution to correlators in bulk perturbation theory [1][2][3]. In the analytic conformal bootstrap, MFT is the leading contribution to correlators at large spin [4][5][6][7][8][9][10]. MFT provides crucial example data for the numerical bootstrap [11], especially for spinning operators [12][13][14][15][16]. Furthermore, MFT OPE coefficients form the "ladder kernel" in SYK-like models [17][18][19][20][21][22][23]. Consequently, the OPE data of MFT (i.e. the scaling dimensions and OPE coefficients) is the starting point for many computations.Although correlators in MFT are simple, the OPE data can be nontrivial. OPE coefficients for a four-point function of fundamental scalars in MFT in 2-and 4-dimensions were guessed in [24]. 1 They were subsequently generalized to d-dimensions in [25] using a technique dubbed "conglomeration." In this work, we point out that conglomeration is part of a general toolkit of harmonic analysis for the Euclidean conformal group SO(d + 1, 1) [26]. Although harmonic analysis was first applied to CFTs in the 70's, it has played an especially important role in recent developments [17][18][19][20][21][22][23][27][28][29][30][31]. In section 2, we give an introduction to harmonic analysis for (Euclidean) CFTs.The calculation of [25] can be rephrased in terms of simple ingredients: the Plancherel measure, three-point pairings, and the "shadow transform" [32,33]. In particular, the computation of MFT OPE coefficients factorizes into two independent shadow transforms of threepoint functions, which are essentially generalizations of the famous "star-triangle" integral [34,35]. Using these observations, we write a general formula for OPE data of fundamental MFT fields in arbitrary Lorentz representations in section 2.8. Along the way, we derive orthogonality relations for conformal partial waves with arbitrary (internal and external) Lorentz representations.Our derivation essentially uses a "Euclidean inversion formula" -a formula that expresses OPE data as an integral of a four-point function over Euclidean space. MFT OPE data can also in principle be computed by applying the Lorentzian i...

Section: Seed Correlator In 4dsupporting

confidence: 53%

Harmonic analysis and mean field theory

Karateev

Kravchuk

Simmons-Duffin

2019

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113

217

“…In section 3 we extend the analytic bootstrap in another direction, namely to 4-point functions of fermions in 3d. A similar analysis at leading order in the large-spin expansion was carried out in 4d in [34]. In this work we carry out the large-spin expansions for the leading twist trajectories and their OPE coefficients to subleading order, in the hope that it will provide a valuable cross-check to future derivations of the Lorentzian inversion formalism for 3d fermions and can be eventually applied to interesting 3d CFTs with fermions.…”

Section: Introductionmentioning

confidence: 87%

More analytic bootstrap: nonperturbative effects and fermions

Albayrak

Meltzer

Poland

2019

104

We develop the analytic bootstrap in several directions. First, we discuss the appearance of nonperturbative effects in the Lorentzian inversion formula, which are exponentially suppressed at large spin but important at finite spin. We show that these effects are important for precision applications of the analytic bootstrap in the context of the 3d Ising and O(2) models. In the former they allow us to reproduce the spin-2 stress tensor with error at the 10 −5 level while in the latter requiring that we reproduce the stress tensor allows us to predict the coupling to the leading charge-2 operator. We also extend perturbative calculations in the lightcone bootstrap to fermion 4-point functions in 3d, predicting the leading and subleading asymptotic behavior for the double-twist operators built out of two fermions.

“…When O is the conserved current J or the stress tensor T one can additionally use the Ward identities to relate the associated OPE coefficients to the U(1) and conformal charges of ψ. For our case this was done in [54]. 19 For the conserved current they find…”

Section: Jhep06(2019)088mentioning

confidence: 99%

Fermion conformal bootstrap in 4d

Karateev

Kravchuk

Serone

et al. 2019

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We apply numerical conformal bootstrap techniques to the four-point function of a Weyl spinor in 4d non-supersymmetric CFTs. We find universal bounds on operator dimensions and OPE coefficients, including bounds on operators in mixed symmetry representations of the Lorentz group, which were inaccessible in previous bootstrap studies. We find discontinuities in some of the bounds on operator dimensions, and we show that they arise due to a generic yet previously unobserved "fake primary" effect, which is related to the existence of poles in conformal blocks. We show that this effect is also responsible for similar discontinuities found in four-fermion bootstrap in 3d, as well as in the mixed-correlator analysis of the 3d Ising CFT. As an important byproduct of our work, we develop a practical technology for numerical approximation of general 4d conformal blocks.