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Perturbative Proximity Between Supersymmetric and Nonsupersymmetric Theories PHYSICAL REVIEW D 102, 125011 (2020) Perturbative proximity between supersymmetric and nonsupersymmetric theories Mikhail Shifman William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA (Received 7 October 2020; accepted 6 November 2020; published 1 December 2020) I argue that a certain perturbative proximity exists between some supersymmetric and nonsupersym- metric theories (namely, pure Yang-Mills and adjoint QCD with two flavors, adjQCDNf ¼2). I start with N ¼ 2 super–Yang-Mills theory built of two N ¼ 1 superfields: vector and chiral. In N ¼ 1 language, the latter presents matter in the adjoint representation of SUðNÞ. Then, I convert the matter superfield into a phantom one (in analogy with ghosts), breaking N ¼ 2 down to N ¼ 1. The global SU(2) acting between two gluinos in the original theory becomes graded. Exact results in thus deformed theory allow one to obtain insights in certain aspects of nonsupersymmetric gluodynamics. In particular, it becomes 1 clear how the splitting of the β function coefficients in pure gluodynamics, β1 ¼ð4 − 3ÞN and 1 2 β2 ¼ð6 − 3ÞN , occurs. Here, the first terms in the braces (4 and 6, always integers) are geometry 1 related, while the second terms (− 3 in both cases) are bona fide quantum effects. In the same sense, N ¼ 2 adjQCDNf ¼2 is close to SYM. Thus, I establish a certain proximity between pure gluodynamics and adjQCDNf ¼2 with supersymmetric theories. (Of course, in both cases, we loose all features related to flat directions and Higgs/Coulomb branches in N ¼ 2.) As a warmup exercise, I use this idea in the two-dimensional CP(1) sigma model with N ¼ð2; 2Þ supersymmetry, through the minimal heterotic N ¼ð0; 2Þ → bosonic CP(1). DOI: 10.1103/PhysRevD.102.125011 I. INTRODUCTION renormalization [for SU(2)] in the Coulomb gauge1 in 1969. In his calculation, the distinction in the origin of 4 vs It has long been known that the behavior of Yang-Mills − 1 theories is unique in the sense that, unlike others, they 3 is transparent: the graph determining the first term in the — possess asymptotic freedom. It is also known that the first braces does not have imaginary part and hence can and in — coefficient of the β function has a peculiar form, fact does produce antiscreening; see Fig. 1. The same 4 vs − 1 split as in (1) is also seen in instanton 3 11 1 calculus and in calculation based on the background field β ¼ ≡ 4 − ð Þ 1 3 N N 3 ; 1 method [3]. In the first case, the term 4 emerges from the zero modes and has a geometrical meaning of the number of symmetries nontrivially realized on the instanton (see where the coefficients β1;2 are defined as below). Hence, it is necessarily integer. This part of β1 is in α2 α3 ∂ essence classical. Bona fide quantum corrections due to βðαÞ¼∂ α ¼ −β − β þÁÁ ∂ ≡ ð Þ − 1 L 1 2 2 L : 2 nonzero modes yield 3. 2π 4π ∂ logμ 1 In the background field method, 4 vs − 3 split in (1) emerges as an interplay between the magnetic (spin) vs The first term in the parentheses in (1) presents the electric (charge) parts of the gluon vertex. famous antiscreening, while the second is the conventional Below, we will discuss whether a similar interpretation screening, as in QED. This was first noted by Khriplovich, exists for the second coefficient in the β function in who calculated [1] the Yang-Mills coupling constant gluodynamics (nonsupersymmetric Yang-Mills). To this end, I will start from the simplest N ¼ 2 super–Yang-Mills theory, which is built of two N ¼ 1 superfields: vector and Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 1See also 1977 papers in Ref. [2] devoted to the same the author(s) and the published article’s title, journal citation, issue. Their authors apparently were unaware of Khriplovich’s and DOI. Funded by SCOAP3. publication [1]. 2470-0010=2020=102(12)=125011(8) 125011-1 Published by the American Physical Society MIKHAIL SHIFMAN PHYS. REV. D 102, 125011 (2020) Finally, for For N ¼ 4, n ¼ f ¼ 4 ¼ 4 ð Þ nb 2 TG N: 6 As it follows from (5), the N ¼ 2 β function reduces to one-loop (nb ¼ nf). The main lesson obtained in Ref. [3] was as follows. Equation (3) makes explicit that all coefficients of the β (a) (b) functions in pure super–Yang-Mills theories have a geometric origin since they are in one-to-one correspondence with the FIG. 1. Feynman graphs for the interaction of two (infinitely) number of symmetries nontrivially realized on instantons. heavy probe quarks denoted by bold strait lines were calculated I will also need the extension of (3) including N ¼ 1 by Khriplovich in the Coulomb gauge. The dotted lines stand for matter fields. We will consider one extra N ¼ 1 chiral the (instantaneous) Coulomb interaction. Thin solid lines depict ð Þ transverse gluons. In part a, a pair of transverse gluons is matter superfield in the adjoint representation of SU N ; produced. This graph has an imaginary part seen by cutting then, the loop. As in QED, this pair produces screening. In part b, a similar cut in the loop is absent since it would go through a α2 3 − ð1 − γÞ βðαÞ¼− TG TG ð Þ transverse gluon and the Coulomb dotted line. This graph is α ; 7 2π 1 − TG responsible for antiscreening. 2π where γ is the anomalous dimension of the corresponding N ¼ 1 chiral. In the language, the latter presents matter in matter field.2 This so-called NSVZ formula appeared first ð Þ the adjoint representation of SU N . The central point is in Ref. [3] and shortly after in a somewhat more general “ ” that I convert the matter superfield into a phantom one form in Ref. [5].3 (in analogy with ghosts), i.e., replace the corresponding Needless to say, in nonsupersymmetric Yang-Mills N ¼ 2 superdeterminant by 1/superdeterminant. Then, is theory (gluodynamics), exact β function determination is N ¼ 1 broken down to . The global SU(2) acting between impossible. Moreover, only the first two coefficients in the N ¼ 2 two gluinos in the original theory becomes graded. β function are scheme independent. In supersymmetric Exact results in thus deformed theory will allow me to theories, the all-order results mentioned above are valid in a obtain insights in nonsupersymmetric gluodynamics. special scheme (usually called NSVZ) recently developed also in perturbation theory in Refs. [6,7]. So far, no analog II. SETTING THE STAGE of this special scheme exists in nonsupersymmetric theo- ries. Therefore, in discussing below geometry-related terms Supersymmetric Yang-Mills (SYM) was the first four- in the β function coefficients in pure Yang-Mills theory, dimensional theory in which exact results had been I will limit myself to β1 and β2, which are scheme obtained. In what follows, I will use one of them, namely, independent. One can think of extending these results to the so-called Novikov-Shifman-Vainshtein-Zakharov higher loops in the future. (NSVZ) beta function [3] in SYM theory (reviewed in Ref. [4]). Without matter fields, we have the following result general for N ¼ 1, N ¼ 2, and N ¼ 4 theories, III. SIMPLE MODEL TO BEGIN WITH A pedagogical example of the model where the β 2 −1 nf α ðnb − nfÞα function similar to (7) appears is the N ¼ð0; 2Þ heterotic βðαÞ¼− n − 1 − ; ð3Þ b 2 2π 4π CP(1) model in two dimensions [8].4 Since Ref. [8] remains 2 Nα where nb and nf count the gluon and gluino zero modes, For the adjoint matter superfield, γ ¼ − π . 3 respectively. For N ¼ 1, these numbers are Quite recently, derivation of the NSVZ β function in perturba- tion theory was completed in Ref. [6]. This work also completes construction of the NSVZ scheme. It contains an extensive list of nb ¼ 2nf ¼ 4TG ¼ 4N ð4Þ references, including those published after Ref. [7]. 4Of course, in CP(N − 1) models, even nonsupersymmetric, all ð Þ coefficients have a geometric meaning; see e.g., Ref. [9]. This is [I limit myself to the SU N gauge group theory in this because such models themselves are defined through target space paper]. For N ¼ 2, one obtains geometry. We will use N ¼ð0; 2Þ heterotic CP(1) model as a toy model, a warmup before addressing Yang-Mills. Note also that ¼ ¼ 4 ¼ 4 ð Þ minimal heterotic models of the type (8), (11) do not exist for CP nb nf TG N: 5 (N − 1) with N>2 because of the anomaly [10]. 125011-2 PERTURBATIVE PROXIMITY BETWEEN SUPERSYMMETRIC AND … PHYS. REV. D 102, 125011 (2020) relatively unknown, I will first briefly describe it in terms is well known, has only one-loop beta function. From this on N ¼ð0; 2Þ superfields. fact, we conclude that The Lagrangian of the N ¼ð0; 2Þ model in two dimen- N ¼ 1 sions analogous to that of 4D SYM is g2 γ ¼ − : ð14Þ Z ↔ 2π 1 A†i∂ A L ¼ d2θ RR ; ð8Þ A g2 R 1 þ A†A Let us ask the following question: Is this information sufficient to find the first and second coefficients in the where A is an N ¼ð0; 2Þ bosonic chiral superfield, nonsupersymmetric (purely bosonic) CP(1) model without pffiffiffi actual calculation of relevant Feynman graphs? ð θ† θ Þ¼ϕð Þþ 2θ ψ ð Þþ θ† θ ∂ ϕ ð Þ A x; R; R x R L x i R R LL : 9 Surprising though it is, the answer is positive.
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