Convergent momentum-space OPE and bootstrap equations in conformal field theory

Gillioz, Marc; Lu, Xiaochuan; Luty, Markus A.; Mikaberidze, Guram

doi:10.1007/jhep03(2020)102

Cited by 35 publications

(38 citation statements)

References 43 publications

(69 reference statements)

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“…At tree level, we confirm that our cutting rules agree with the discontinuity of the full Witten diagram. By using the momentum-space OPE [94,95], we will relate the bulk cut structure to the spectrum of the dual CFT and find agreement with previous work on the OPE limit of Witten diagrams. Finally, by studying tree and loop examples, we show that cut AdS diagrams become the corresponding cut flat space diagrams in the flat space limit.…”

Section: Applications To Witten Diagramssupporting

confidence: 71%

CFT unitarity and the AdS Cutkosky rules

Meltzer

Sivaramakrishnan

2020

J. High Energ. Phys.

141

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We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. These rules are naturally formulated and derived in Lorentzian momentum space, where the double-commutator is manifestly related to the CFT optical theorem. Finally, we study the AdS cutting rules in explicit examples at tree level and one loop. In these examples, we confirm that the rules are consistent with the OPE limit and that we recover the S-matrix optical theorem in the flat space limit. The AdS cutting rules and the CFT dispersion formula together form a holographic unitarity method to reconstruct Witten diagrams from their cuts.

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Section: Applications To Witten Diagramssupporting

confidence: 71%

CFT unitarity and the AdS Cutkosky rules

Meltzer

Sivaramakrishnan

2020

J. High Energ. Phys.

141

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“…It is known that the OPE for Wightman functions converges in the sense of distributions [65], but the series might not converge at every given configuration of momenta (see Ref. [66] for a discussion and examples in d = 2 dimensions). In any case, this problem is absent when the OPE applies to a 3-point function since the sum is given by a single term.…”

Section: Ope Limits In Momentum Spacementioning

confidence: 99%

“…Many questions about the convergence of this OPE in a 4-point function remain to be answered: it is guaranteed to converge in a distributional sense only[65], but not necessarily at every single point in momentum space. The problem has been addressed in d = 2[66], but it remains open in higher dimensions.…”

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confidence: 99%

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

Gillioz

2020

Commun. Math. Phys.

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In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell $$F_4$$ F 4 type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.

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“…With the development of the conformal bootstrap program, a lot of insight has been gained into the structure of CFT (see [1][2][3][4] and references therein). Most of the development in the conformal bootstrap program is achieved in position space and its momentum space analogue has not gained much attention [5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Momentum space spinning correlators and higher spin equations in three dimensions

Jain

John

Malvimat

2020

J. High Energ. Phys.

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In this article, we explicitly compute in momentum space the three and four-point correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that the scalar operator $$ \overline{\psi}\psi $$ ψ ¯ ψ is odd under parity. To account for this, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of $$ \overline{\psi}\psi $$ ψ ¯ ψ operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation.

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Convergent momentum-space OPE and bootstrap equations in conformal field theory

Cited by 35 publications

References 43 publications

CFT unitarity and the AdS Cutkosky rules

CFT unitarity and the AdS Cutkosky rules

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

Momentum space spinning correlators and higher spin equations in three dimensions

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