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 þ ∂ ≡ 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 theories. 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 perturbation 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. Indeed, let us make the B superfield “phantom,” i.e., quantize ψ R as a ψ Here, ϕ is a complex scalar, and ψ L is a left-moving Weyl bosonic field. In other words, we will treat R as a fermion fermion in two dimensions. Furthermore, the matter term is ghost field. In other words, the extra minus sign is needed ψ ψ ∈ introduced through another N ¼ð0; 2Þ superfield B, in the R loop. Then, the contributions of L A and ψ ∈ β pffiffiffi R B exactly cancel each other in the two-loop † † ð θ θ Þ¼ψ ð Þþ 2θ ð Þþ θ θ ∂ ψ ð Þ function (see Fig. 2; neither ψ L nor ψ R appears at one Bi x; R; R R;i x RFi x i R R LL R;i x ; loop). At this stage, N ¼ð0; 2Þ supersymmetry is pre- ð10Þ served. The cancellation of ψ R;L contributions to β function occurs despite the fact that one of the fermions is a left ¼ 1 2 … where i is the flavor index, i ; ; ;nf, and mover and the other is a right mover. Z Then, with the phantom B superfield, we can take X 1 † L ¼ 2θ Bi Bi Eq. (13) and formally put matter d R 2 ð1 þ † Þ2 ; i A A L ¼ L þ L ð Þ n ¼ −1: A matter: 11 f

Note that the superfield B contains only one physical As a result, we end up the with the following two-loop (dynamic) field ψ R, with no bosonic counterpart. This is answer in the bosonic CP(1) model: only possible in two dimensions. In the minimal model (8) without matter, the exact beta 4 2 β ¼ − g 1 þ g ð Þ function [8] takes the form CPð1Þ bosonic 2π 2π : 15 g4 1 βðg2Þ¼− ; ð12Þ 2π 1 − g2 With satisfaction, we observe that the above formula 4π coincides with the standard answer [9]. while including matter, we arrive at IV. SUðNÞ YANG-MILLS THEORIES n γ g4 1 þ f βðg2Þ¼− 2 ; ð13Þ Now, I return to gauge theories with the goal of 2π g2 β 1 − 4π analyzing the second coefficient of the function in pure gluodynamics, to be compared with (7). Equation (12) must be compared with (7) with the second term in the parentheses omitted. The parallel is apparent. There are two minor but techni- cally important distinctions, however. First, in the two- dimensional CP(1) model, unlike four-dimensional SYM, the fermion contribution appears only in the second loop. Second, as was mentioned, the matter superfield B contains only one physical degree of freedom, namely, ψ R. In super– Yang-Mills theory, the matter superfield contains both components, bosonic and fermionic. Now, if n ¼ 1 in the model under consideration, f FIG. 2. The phantom field ψ R cancels the contribution of ψ L in supersymmetry is extended to N ¼ð2; 2Þ. In other words, the two-loop beta function. This is the only diagram to be in this case, we deal with nonchiral N ¼ 2 CP(1) which, as considered at small ϕ.

125011-3 MIKHAIL SHIFMAN PHYS. REV. D 102, 125011 (2020) 11 α2 17 α3 α2 Nα Nα β ¼ − N − N2 þ ð Þ β ¼ − 4N 1 þ 1 þ pure YM 3 2π 3 4π2 16 ph 2π 4π 2π α2 3 α We will see that the second coefficient can be represented ¼ − 4 1 þ N þ 2π N 4π as follows, α2 n α 17 1 ↔ − n 1 þ n − f β ¼ 2 ¼ 2 6 − ð Þ 2π b b 2 4π 2 N 3 N 3 ; 17 α2 α ¼ − 4N þ 6N2 : ð21Þ to be compared with the first coefficient in Eq. (1).5 The 2π 2π term 6 in the parentheses of (17) is again related to the number of instanton zero modes in N ¼ 1 SYM. Thus, it Here, β carries a subscript ph (standing for phantom)—it 1 has a geometrical meaning. The second term − 3 has a bona does not refer to any physical theory. Exactly the same fide quantum origin. formula emerges from the instanton calculation in which The line of reasoning will be the same as in Sec. III. the instanton measure is adjusted to reflect “phantomiza- Let us start from N ¼ 1 SYM without matter. tion” of the matter field. Since N ¼ 1 is unbroken, all TheP corresponding expression is given by (7) with nonzero modes still cancel. At this level, the first and the ð1 − γÞ¼0 β i TG or, which is the same, in Eqs. (3) and second coefficients in ph are related to geometry and are (4). Then, we add one chiral superfield in the adjoint integers (see the second line above). Moreover, nb and nf in representation of SUðNÞ. By the same token as in Sec. III, (21) refer to N ¼ 1 theory; see (4). supersymmetry of the model with the N ¼ 1 adjoint matter This is not the end of the story, however. Unlike the included is automatically extended from N ¼ 1 to N ¼ 2. situation in Sec. III, in the Yang-Mills case, phantomizing In the latter, the β function is given by Eqs. (3) and (5); the theory is not enough to pass to nonsupersymmetric i.e., it reduces to one loop. gluodynamics. The next step is to declare the chiral adjoint matter Let us ask ourself what diagrams present in SYM but superfield to be phantom in N ¼ 2 super–Yang-Mills absent in nonsupersymmetric gluodynamics are canceled theory. At this point, we brake the global SU(2) acting by phantom matter. The answer is obvious and is depicted on the doublet in Fig. 3. Any number of gluons (with possible λ, λ˜ λ λ˜ λ insertions) can be drawn inside the , loops in Fig. 3; ð Þ ˜ : 18 cancellation still persists. λ What does not cancel? It is obvious that all diagrams with the adjoint scalar field ϕa from the matter N ¼ 1 More exactly, we transform it into a graded suð2Þ algebra superfield still reside in Eq. (21) and must be subtracted in with the generators T becoming odd elements, while T0 λ passing to nonsupersymmetruc gluodynamics. The sim- remains even. In Eq. (18), is the first gluino, i.e., the one plest example of such a graph is presented in Fig. 4. In fact, λ˜ from the vector superfield, while is the second gluino this graph is the only one to be dealt with at one loop. One belonging to the matter superfield. should not forget that in (21) the above diagram refers to the Declaring the chiral superfield to be phantom amounts to phantom ϕa field. Hence, its subtraction is equivalent to N ¼ 1 replacing the instanton superdeterminant for matter addition of the regular (unphantomized) ϕa loop. As for by its reverse, or, diagrammatically, we must change the two-loop diagrams—for brevity, I will present them omit- sign of the matter contribution in Eq. (7), namely, ting background field legs—they are shown in Fig. 5. The situation with Fig. 5(a) is exactly the same as with that in −T ð1 − γÞ → þT ð1 − γÞð19Þ G G Fig. 4. Both Figs. 5(a) and 5(b), taken together, present the a From the one-loop condition for N ¼ 2 SYM β function phantom ϕ field contribution, which must be subtracted using Eq. (7), we derive α γ ¼ − ð Þ TG π : 20 Equation (20) in combination with (19) is to be substituted into (7). After this is done, the β function in the “phantomized” theory takes the form [after expanding FIG. 3. Two classes of graphs canceling each other. Here, λ the denominator in (7)] marks the gluino lines, while λ˜ is the adjoint matter fermion from the phantom matter field. The gray shading indicates all possible 5For the definition, see Eq. (2). gluon insertion. The wavy line is the gluon background field.

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As is well known, in N ¼ 2 SYM, the β function is exhausted by one loop; see Eq. (3) and (5). The first coefficient in this β function is 1 ðβ Þ ¼ − ¼ 2 ð Þ N ¼2 1 nb 2 nf N: 24

The geometric origin of this coefficient is obvious. FIG. 4. The one-loop ϕa contribution to be subtracted in What should be changed in N ¼ 2 SYM to convert passing from Eq. (21) to nonsupersymmetric Yang-Mills. this theory into adjoint QCD with Nf ¼ 2? The answer is clear—one should subtract the same graphs in Figs. 4 and 5, which were added in Eq. (22),

β ¼ βN ¼2 − δ β adj QCD sc 1 α2 N2 α3 ¼ − 2 þ N − : ð Þ 3 2π 3 4π2 25

Equation (25) coincides with the known β function in (a) (b) adjoint QCD [12].

FIG. 5. Two-loop graphs to be subtracted in passing from VI. RENORMALONS AND Eq. (21) to non-SUSY Yang-Mills. Here, λ marks the gluino lines, ADIABATIC CONTINUITY while λ˜ is the adjoint matter fermion from the phantom superfield. The diagram a combined with that in Fig. 4 gives the ϕa Renormalons introduced by ’t Hooft [13] emerge from a β “ ” contribution to two-loop function in QCD with scalar quark specific narrows class of multiloop diagrams,6 the so-called ϕa (i.e., is to be considered as scalar quark in the adjoint bubble chains; see Fig. 6. Formally,7 this chain produces a representation.) factorially divergent perturbative series β X β α n from ph if we want to pass to pure gluodynamics. In fact, ∼ 1 s ! ð Þ 8π n : 26 because of their phantom sign, to pass to pure gluody- n namics, we must add these graphs as a normal (non- β phantom) contribution to ph. In the Borel plane, the above factorial divergence manifests In the Appendix, I check that the impact of the quantum itself as a singularity at corrections associated with Figs. 4 and 5 on the phantom β 8π 2π 12 function in (21) is as follows, ¼ β1 N 11 β ¼ β þ δ β; ð22Þ pure YM ph sc in pure gluodynamics. At the same time, in adjoint QCD ¼2 discussed in Sec. V, the renormalon-induced where Nf singularity in the Borel plane is at 1 α2 α3 δ β ¼ N þ N2 ; ð Þ 8π 4π 6 sc 3 2π 4π2 23 ¼ : β1 N 7 which perfectly coincides with Eqs. (1), (2), (16), and (17). The both cases are depicted by crosses in Fig. 7. Needless to say, the sign of these corrections corresponds to Simultaneously, this figure shows (by closed circles) the screening. would-be positions of the renormalon singularities if small 1 quantum terms 3 in Eqs. (1) and (25) were neglected. If V. N = 2 SYM AND ADJOINT QCD measured in the units of 2π=N, the latter are integers. WITH TWO FLAVORS Why this is important? The answer to this question is associated with the program due to Ünsal launched some Recently, a renewed interest in adjoint QCD led to some unexpected results (see e.g., Ref. [11] and an old but useful 6For a recent brief review, see Ref. [14]. for my present purposes review [12]). In this section, 7“Formally” means that in order to stay within the limits of I compare N ¼ 2 SYM with adjoint QCD with two quarks applicability of perturbation theory we have to cut off the sum in (two adjoint Weyl or Majorana fields). Eq. (26) at a certain value of n ¼ n.

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graph. We see that the adiabatic matching would be perfect 1 if we could ignore the small quantum terms 3 in Eqs. (1) and (25). The adiabatic matching is perfect in the phantom theory of Sec. IV.

VII. CONCLUSION Starting from N ¼ 2 super–Yang-Mills and phantomizing the N ¼ 1 matter superfield, I arrived at a fully ’ ’ FIG. 6. T Hooft s bubble-chain diagrams representing renor- geometric β function. This proves the statement I made malons. ph in Sec. I that the integer part of the first and second β coefficients in pure YM count the number of certain symmetry generators, both bosonic and fermionic, in N ¼ 1 SYM. The latter theory becomes relevant because my phantomization procedure breaks N ¼ 2 → N ¼ 1. Relatively small noninteger additions represent bona fide quantum corrections, which do not appear in N ¼ 1 SYM because of the Bose-Fermi cancellations. At the moment, one might think that this idea could shed light on some other intricate aspects of gauge theories, for instance on nuances of renormalons. In particular, the geometric integers 4 and 6 appearing in (1) and (17) are the dimensions of the lowest-dimension gluon operators, quadratic and cubic in the gluon field FIG. 7. Leading renormalon-related singularities in the Borel strength tensor, respectively. The standard renormalon plane (marked by crosses) for pure gluodynamics and adjoint wisdom says that the renormalon singularities in the Borel plane conspire with these operators in the operator QCDNf ¼2, respectively. The closed circles mark the would-be 1 product expansion. positions of the renormalon singularities if small quantum terms 3 in Eqs. (1) and (25) were ignored. At the same time, the current understanding of renormalons in SYM theory continues to be incomplete (see Refs. [14,16]). The gluon condensate vanishes in super- time ago [15] aimed at developing a quasiclassical picture symmetric gluodynamics. The leading renormalons have on R3 × S1 at sufficiently small radius rðS1Þ. This program nothing to conspire with. Are there undiscovered cancella- later was supplemented by the idea of adiabatic continuity tions? This suggestion does not seem likely, but we cannot stating that tending rðS1Þ → ∞ (i.e., returning to R4) one avoid providing a definite answer any longer. does not encounter phase transitions on the way. Another Summarizing, I established a certain proximity between crucial observation is that the renormalon singularity must pure gluodynamics and an N ¼ 1 theory. Moreover, in the conspire with operator product expansion (OPE). In both N ¼ 2 same sense, adjoint QCDNf¼2 is close to SYM. theories outlined in Fig. 7, the leading term in OPE is due to μν Conceptually, this is similar to the proximity of pure the operator GμνG (the gluon condensate). gluodynamics to fundamental QCD in the limit N → ∞. Now, within the program [15] a large number of new In the latter case, the parameter governing the proximity is saddle-point configurations were discovered, the so-called adjustable, 1=N. In the cases considered in this paper, it is monopole instantons, also known as bions, both magnetic rather a numerical parameter whose origin is still unclear and neutral. Their action is 2π=N, i.e., N times smaller than but is quite apparent in Eqs. (1) and (17). It seems to be that of instantons. In other words, in this picture, a single related with the dominance of magnetic interactions of instanton can be viewed as a composite state of N bions. gluons over their charge interactions. If we could tend this Then, it becomes clear why a single bion saturates the parameter to zero, we would be able to say more about the gluon condensate in gluodynamics. Moreover, in adjoint adiabatic continuity.

QCDNf¼2, similar saturation is due to an bion-antibion pair I would like to add a few remarks on questions relevant which is tied up because of the existence of the fermion for the future studies. zero modes. The zero modes produced by bion and those from antibion must be contracted to give rise to the gluon Remark 1.—Since pure gluodynamics is close to N ¼ 1 condensate. In the Borel plane (Fig. 7), the single-bion phantom theory, the gluon condensate in the former must contribution is shown by a closed circle in the upper graph, be suppressed to a certain extent since it is forbidden in while the bion pair’s contribution is depicted in the lower N ¼ 1 supersymmetry.

125011-6 PERTURBATIVE PROXIMITY BETWEEN SUPERSYMMETRIC AND … PHYS. REV. D 102, 125011 (2020) Remark 2.—My present consideration entangles renor- 11 1 α2 17 1 α3 β ¼ − − − 2−2 2 − 2 YMþSQ N N N N N 2 malons, their conspiracy with OPE, the quasiclassical 3 |ffl{zffl}3 2π 3 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}3 4π treatment on R3 × S1, and adiabatic continuity all in one Fig: 4 Fig: 5a junction both in pure gluodynamics and adjoint QCD ¼2. Nf þ ð Þ Can this line of studies be continued? A1 Remark 3.—N ¼ 1 phantom SYM theory I have dis- The scalar quark contribution is underlined by underbraces. cussed in this paper calls for further investigations. For This is not the end of the story, however. In addition, instance, I believe that, despite the presence of nonunitary I have to take into account the graph depicted in contributions, if we consider amplitudes with only gluonic Fig. 5(b). extermal legs, their inclusive imaginary parts will be The fastest and most efficient method of such calcu- positive. In this narrow sense, they will preserve unitarity. lations is the background field method; see Ref. [19]. Background field emission can occur either from the λ, λ˜ ACKNOWLEDGMENTS lines or from the ϕ line. In the first case, the relevant propagator is I am grateful to K. Chetyrkin, A. Kataev, A. Losev, 1 ˆ 1 C. H. Sheu, and K. Stepanyanz for useful communications. x xα ˜ φ 5 Sðx; 0Þ¼ − Gαφð0Þγ γ þ; ðA2Þ This work is supported in part by U.S. DOE Grant No. de- 2π2 x4 8π2 x2 sc0011842. where the ellipses denote irrelevant terms in the expansion and, moreover, 1 APPENDIX: COMPLEX SCALAR ˜ βρ Gαφ ¼ εαφβρG : ADJOINT QUARKS 2 In the second case, the relevant propagator is that of ϕ (see First, let us analyze diagrams in Figs. 4 and 5(a).Aswas Ref. [20]), mentioned, they present the “scalar quark” contributions in i 1 i QCD. We can extract them from the known result for the Gðx; 0Þ¼ þ x2G2ð0Þþ ðA3Þ QCD β function, by changing the appropriate Casimir 4π2 x2 512π2 coefficients from the fundamental representation to the The result of calculation of the diagram in Fig. 5(b) is also 8 adjoint. known in the literature. We can borrow it from Sec. 5 of The QCD β function with one adjoint scalar quark Ref. [20], extracted from Refs. [17,18] in my notation reduces to α3 Δβj ¼ −2N2 : ð Þ Fig: 5b 4π2 A4 8 δ β I thank K. Chetyrkin, A. Kataev, and K. Stepanyants for help Combining (A4) and (A1), we arrive at sc given on this point. in Eq. (23).

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