REVIEW
Anomalous dimensions in N=4 SYM theory at order g^4
High Energy Physics - Theory · Submitted 22 Mar 2000 · 43 references
Pith's verdict: A solid N=1 superfield computation whose four-point function is independently confirmed, but whose headline number, the two-loop Konishi anomalous dimension, leans on a one-loop OPE coefficient imported from the authors' own earlier paper.
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Referee report
The paper
The paper claims
By computing a four-point correlation function of scalar composite operators to order $g^4$ in $N=1$ superfield perturbation theory, the authors extract the two-loop contribution to the anomalous dimension of the Konishi supermultiplet in $
The reading
What carries the argument
The load-bearing device is a change of limit: send one external point $x_1$ to infinity after multiplying by $(x_1^2)^2$, which is legitimate because the external operator has protected dimension two. This collapses the diagrammatic sum to a handful of double integrals and lets the whole answer be written through just two functions, the box integral $g(i,j,k) = int dx_5 / (x_{i5}^2 x_{j5}^2 x_{k5}^2)$ and the ladder integral $f(i;j,k) = int dx_5 dx_6 / (x_{i5}^2 x_{j5}^2 x_{i6}^2 x_{k6}^2 x_{56}^2)$. The identities $Box_j f(i;j,k) = -4pi^2 g(i,j,k)/x_{ij}^2$ and its counterpart then convert the final expression into the box function alone, which is what makes the short-distance logarithm readable and lets the OPE comparison be made at the level of $Box^2 G$ rather than $G$ itself.
Extended reading notes
Core claim
The central claim is that the order $g^4$ four-point function $G^{(H)} = <C^{11}(x_1)C^{†11}(x_2)C^{22}(x_3)C^{†22}(x_4)>$ can be computed exactly in $N=1$ superspace and that its short-distance logarithmic behaviour is fully accounted for by the exchange of unprotected operators, with the Konishi scalar $K_1 = (1/3) tr(phi^i phi^i)$ dominating the $s$-channel. Acting with $Box^2$ on the result converts the awkward four-point integral into the simpler box function $g(2,3,4)$, and taking $x_4 -> x_3$ isolates a linear logarithm whose coefficient reproduces the known one-loop value $gamma_1 = 3N/(16 pi^2)$ squared, plus a constant term. Matching that constant against the OPE expansion, which uses the one-loop OPE coefficients $a_0$ and $a_1$ fixed in the authors' earlier work, yields the two-loop Konishi anomalous dimension $gamma_2 = -3g^4N^2/[16(4pi^2)^2]$, Eq. (50). The authors also argue that operators in the 105 of the R-symmetry stay protected at this order and that the 84 combination orthogonal to the Konishi descendant is protected too.
Load-bearing premise
Extracting the two-loop number assumes that the $s$-channel logarithmic behaviour of the four-point function is saturated by exchange of the Konishi scalar and reuses the one-loop OPE coefficients $a_0$, $a_1$ computed in the authors' own earlier paper; the authors state in Section 5 that the result 'certainly requires further investigation and some cross-checks', so the number rests on that prior input rather than on an independent determination.
Original abstract
We compute four-point correlation functions of scalar composite operators in the N=4 supercurrent multiplet at order g^4 using the N=1 superfield formalism. We confirm the interpretation of short-distance logarithmic behaviours in terms of anomalous dimensions of unprotected operators exchanged in the intermediate channels and we determine the two-loop contribution to the anomalous dimension of the N=4 Konishi supermultiplet.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We compute four-point correlation functions of scalar composite operators in the N=4 supercurrent multiplet at order g^4 using the N=1 superfield formalism. We confirm the interpretation of short-distance logarithmic behaviours in terms of anomalous dimensions of unprotected operators exchanged in the intermediate channels and we determine the two-loop contribution to the anomalous dimension of the N=4 Konishi supermultiplet.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compute four-point correlation functions of scalar composite operators in the N=4 supercurrent multiplet at order g^4 using the N=1 superfield formalism. | contribution 0.90 | 0.96 | 0.09 | asserted 1.00 |
| s1 | We confirm the interpretation of short-distance logarithmic behaviours in terms of anomalous dimensions of unprotected operators exchanged in the intermediate channels and we determine the two-loop contribution to the anomalous dimension of the N=4 Konishi supermultiplet. | result 0.92 | 0.93 | 0.75 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-th/0003203 by the authors.
References
How the literature uses it
32 arXiv papers cite this in 41 sentences
Papers citing it · 28 sentences without a confident label
Compares with it 4 papers
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The two-loop correlation function we have constructed in this way agrees with the results of the explicit perturbative calculations [ref] .
Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM · Fixing the constant from the logarithmic singularities · 0.98
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Thus, the renormalized three-point function reads ⟨O^ren_K O_J O_J ⟩/⟨O_J O_J ⟩= Jx_34^2/3x_13^2x_14^2 (1+ 24g^2/Jlogx_34^2/x_13^2x_24^2+6(c̃_K-4g^2)/J ) From this, we can read off the one-loop anomalous dimension $Δ_K=2+g_YM^2Δ_1+⋯$ and the large charge expansion of the structure constant $C_K =JC_K^(0)+C_K^(1)+C_K^(2)/J+⋯$, Δ_1=48g^2/g_YM^2 J=3/2π^2 C_K^(0)=1/3 C_K^(1)=-8g^2+4g∫_0^∞dw 4gw-J_1(8g w)/sinh^2(w) The anomalous dimension agrees with the result in the literature [ref] while the structure constant $C_K$ is a genuinely new prediction, which we will study in more detail below.
Large Charge 't Hooft Limit of $\mathcal{N}=4$ Super-Yang-Mills · Three-point function of Konishi and two large charge operators · 0.92
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They are in agreement with the known results [ref] .
A note on three-point functions of unprotected operators · Two-point functions · 0.99
Cites it as background 6 papers
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The determination of the four-point correlator in planar $𝒩=4$ SYM has spanned over two decades (and counting), with one and two loops initially obtained in the following series of papers [ref] .
Perturbative Correlation Functions and Scattering Amplitudes in Planar $\mathcal{N}=4$ Supersymmetric Yang-Mills · Introduction · 0.95
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Already in the early days of AdS/CFT, one- and two-loop correlators (both integrands and integrated results) have been determined [ref] , but it has to wait until 2011 before the planar integrand of the correlator was determined for three loops [ref] , which was followed quickly by results for four, five, six [ref] and even seven loops [ref] (the integrated results are known up to three loops [ref] ).
The Cusp Limit of Correlators and A New Graphical Bootstrap for Correlators/Amplitudes to Eleven Loops · Introduction · 0.96
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[ref] , and is contained in the $O_2^2 O_2^2$ OPE.
Four-point correlators with BPS bound states in AdS$_3$ and AdS$_5$ · $⟨O_2^2 O_2^2 O_2 O_2 ⟩$ and the twist two sector · 0.97
Discussion
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