aainstitutetext: London Institute for Mathematical Sciences, Royal
Institution,
21 Albemarle St, London W1S 4BS, UKbbinstitutetext: Wu Wen-Tsun Key Lab of
Mathematics, Chinese Academy of Sciencesccinstitutetext: National Research
University Higher School of Economics, Moscow, Russiaddinstitutetext: William
I. Fine Theoretical Physics Institute, University of Minnesota,
Minneapolis, MN 55455, USA

# Peculiarities of beta functions in sigma models

Oleksandr Gamayun,111Corresponding author. b,c Andrei Losev d Mikhail Shifman
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we consider perturbation theory in generic two-dimensional sigma
models in the so-called first-order formalism, using the coordinate
regularization approach. Our goal is to analyze the first-order formalism in
application to $\beta$ functions and compare its results with the standard
geometric calculations. Already in the second loop, we observe deviations from
the geometric results that cannot be explained by the
regularization/renormalization scheme choices. Moreover, in certain cases the
first-order calculations produce results that are not symmetric under the
classical diffeomorphisms of the target space. Although we could not present
the full solution to this remarkable phenomenon, we found some indirect
arguments indicating that an anomaly similar to that established in
supersymmetric Yang-Mills theory might manifest itself starting from the
second loop. We discuss why the difference between two answers might be an
infrared effect, similar to that in $\beta$ functions in supersymmetric Yang-
Mills theories.

In addition to the generic Kähler target spaces we discuss in detail the so-
called Lie-algebraic sigma models. In particular, this is the case when the
perturbed field $G^{i\bar{j}}$ is a product of the holomorphic and
antiholomorphic currents satisfying two-dimensional current algebra.

††preprint: FTPI-MINN-23-11, UMN-TH-4218/23

##  1 Introduction

Sigma models on Kählerian and super-Kählerian target spaces play an important
role in strings and various aspects of quantum field theory – from mathematics
to phenomenology. If supersymmetrized in two dimensions, they have extendended
supersymmetries, ${\mathcal{N}}=2$ or ${\mathcal{N}}=4$ 222 In addition to
${\mathcal{N}}$=(2,2), the Kähler spaces allow for chiral spersymmetrizations
${\mathcal{N}}$=(0,2) or (2,0). . In generic Kählerian models with non-
Einstein target space each order of perturbation theory brings in new
geometric structures, (see e.g. Grah ; Foakes1987 ; Foakes1988 ; Ketov2000 ;
Naka ), say, in the first- and second-loop orders we have

${\mathcal{L}}_{\rm
eff}=\left\\{G_{i\bar{j}}+\left(\alpha^{\prime}R_{i\bar{j}}+\frac{(\alpha^{\prime})^{2}}{2}R_{i\bar{k}\ell\bar{m}}\,R_{\bar{j}}^{\,\,\bar{k}\ell\bar{m}}\right)L+...\right\\}\partial_{\mu}\varphi^{i}\partial_{\mu}\bar{\varphi}^{\,\bar{j}}$
(1)

where $L$ in Eq. (1) is defined as

$L=-\log\frac{M_{\rm uv}}{\mu}\,.$ (2)

Neither $R_{i\bar{j}}$ nor
$R_{i\bar{k}\ell\bar{m}}\,R_{\bar{j}}^{\,\,\bar{k}\ell\bar{m}}$ reduce to
$G_{i\bar{j}}$ in the general case.

In this paper, we first consider renormalization aspects of the so-called
first order formalism in generic sigma models Losev2006  (for brevity referred
to as the first-order sigma models). We treat these models as deformations of
the conformal field theories of the simple bosonic “$\beta$-$\gamma$” systems
by using the so-called coordinate approach to compute the beta function. We
consider correlation functions in the real space (opposite to the traditional
momentum) and employ the point-splitting regularization. This construction is
a priori expected to produce beta functions of a certain self-repeating
pattern of the (differential) polynomial type for the inverse metric
$G^{i\bar{j}}$. Comparing with Eq. (1) we see that the expected pattern does
emerge at one loop (see below). Although generalizing our analysis to higher
loops we continue to observe the polynomial pattern, the $\beta$ function
obtained in this way does not coincide with the result of the standard
geometric calculation presented in (1) starting from the second loop Grah ;
Foakes1987 ; Foakes1988 ; Tsey . Three-loop contribution is RG-scheme
dependent – therefore one might blame the difference in the renormalization
schemes for the disagreement.333 It was calculated by Foakes and Mohammedi,
and Graham in a particular scheme Grah ; Foakes1987 ; Foakes1988 .

However, the disagreement we observe appears already in the second loop. This
leads us to the hypothesis that starting from the second loop a certain
anomaly manifests itself, perhaps, similar to that established in
supersymmetric Yang-Mills theory NSZV ; NSZV1 ; CS ; AH . Some indirect
arguments in favor of this hypothesis will be given in Sec. 4.

Besides the generic Kähler target spaces we discuss in detail the so-called
Lie-algebraic sigma models444The quantum-mechanical Hamiltonians with the Lie-
algebraic structure led to the discovery of quasi-exactly solvable systems,
see reviews tur ; shif .. In paricular, this is the case when the perturbed
field $G^{i\bar{j}}$ is a product of the holomorphic and antiholomorphic
currents satisfying two-dimensional current algebra. We obtain exact formulas
for the one and two-loop beta functions using only algebraic data and observe
the full correspondence with the known results for current-current
deformations of the conformal field theories Kutasov1989 ; PhysRevLett.86.4753
; Ludwig2003  (but not with the geometric formulas).

The paper is organized as follows. In Sec. (2) we introduce first-order sigma
model, outline the two-loop computation of the $\beta$ function and discuss
why the answer is different from the one in Eq. (1). In Sec. (3) we discuss an
important example of the current-current perturbation, which we also dub as
Lie-algebraic sigma models, and show why the answer in Sec. (2) makes sense
from the algebraic point of view. In Sec. (4) we indicate that discrepancies
between the“geometric” and “algebraic” $\beta$ functions can be an IR effect.
Illuminating instanton calculations are presented in Sec. (5).

##  2 Coordinate $\beta$ function

###  2.1 Model and derivation

The first order sigma model proposed in Losev2006  and further developed in
Zeitlin2006 ; Zeitlin2008 ; Zeitlin2009 , was initially introduced as a tool
to address singular background in string theory. More specifically, the bare
action is described by the following conformal field theory

$S_{0}=\frac{1}{\alpha^{\prime}}\int\frac{d^{2}z}{\pi}\left(p_{a}\bar{\partial}\varphi^{a}+\bar{p}_{\bar{a}}\partial\bar{\varphi}^{\bar{a}}\right).$
(3)

Here $d^{2}z=dxdy=idz\wedge d\bar{z}/2$, the index $a$ runs from $1$ to $D/2$
(here we have assumed that the target space is a complex manifold with local
coordinates $\varphi^{a}$ and $\bar{\varphi}^{\bar{a}}$). The fields $p_{a}$,
$p_{\bar{a}}$ are $(1,0)$ and $(0,1)$ forms on $\Sigma$ correspondingly, and
the fields $\varphi^{a}$, $\varphi^{\bar{a}}$ are scalars. For our purposes,
it will be enough to consider $\Sigma=\mathds{C}P^{1}$. For more precise
definitions and discussion about the choice of complex structure see
nekrasov2005lectures . To simplify notations in what follows we put
$\alpha^{\prime}=1$.

The action (3) is equivalent to the following operator product expansions,

$p_{a}(z)\varphi^{b}(w)=\frac{\delta_{a}^{b}}{z-w}+{\rm
reg},\qquad\bar{p}_{\bar{a}}(\bar{z})\bar{\varphi}^{\bar{b}}(\bar{w})=\frac{\delta_{\bar{a}}^{\bar{b}}}{\bar{z}-\bar{w}}+{\rm
reg}.$ (4)

To turn to the non-singular sigma models one considers perturbation of the
action (3) by four possible classically marginal terms

$S=S_{0}-\int\frac{d^{2}z}{\pi}(O_{G}+O_{\mu}+O_{\bar{\mu}}+O_{b})$ (5)

where

$O_{G}=G^{a\bar{a}}p_{a}\bar{p}_{\bar{a}},\qquad
O_{\mu}=p_{a}\mu^{a}_{\bar{a}}\bar{\partial}\bar{\varphi}^{\bar{a}},\qquad
O_{\bar{\mu}}=\bar{p}_{\bar{a}}\bar{\mu}_{a}^{\bar{a}}\partial\varphi^{a},\qquad
O_{b}=b_{a\bar{a}}\partial\varphi^{a}\bar{\partial}\bar{\varphi}^{\bar{a}}.$
(6)

Assuming that $G^{a\bar{a}}$ is not degenerate one can integrate over $p$ and
$\bar{p}$ and obtain the second-order sigma model with the metric and Kalb-
Ramond fields parametrized in terms of the data $G$, $b$, $\mu$ and
$\bar{\mu}$ (see Losev2006 ). Besides, an additional field of a dilaton
appears due to the anomaly in the path integral measure.

For operators (6) to be primary, in the first order of perturbation theory one
has to impose transversality condition

$\partial_{a}G^{a\bar{a}}=\partial_{\bar{a}}G^{a\bar{a}}=\partial_{a}\mu^{a}_{\bar{a}}=\partial_{\bar{a}}\bar{\mu}_{a}^{\bar{a}}=0.$
(7)

This in particular allows one to avoid possible self-contractions of $p$ and
$\varphi$ inside operators. In what follows, we will not insist on the
transversality condition and presume that self-contractions are subtracted, in
other words the operators are normal-ordered.

In this paper we focus mainly on the case when only $O_{G}$ perturbations are
present – then the metric and Kalb-Ramond fields are, respectively, symmetric
and antisymmetric forms of inverse of the bivector $G^{a\bar{a}}$, which we
denote by $G_{a\bar{a}}$. The dilaton field $\Phi$ is proportional to the
logarithm of the determinant

$\Phi\sim\log\det G\,.$

Note that for Kählerian metrics with constant determinant the effects due to
the Kalb-Ramond fields and dilatons are irrelevant.

Here again, we stress that the first order system makes sense even when the
inverse of $G^{a\bar{a}}$ does not exist. Our point of view is that the
original sigma model is represented by Lagrangian (5). In this paper we focus
on making sense of (5) as a quantum field theory.

First, let us focus on the case when
$b_{a\bar{a}}=\bar{\mu}_{a}^{\bar{a}}=\mu^{a}_{\bar{a}}=0$, i.e. on the action

$S=\int\frac{d^{2}z}{\pi}\left(p_{a}\bar{\partial}\varphi^{a}+\bar{p}_{\bar{a}}\partial\bar{\varphi}^{\bar{a}}-G^{a\bar{a}}(\varphi,\bar{\varphi})p_{a}\bar{p}_{\bar{a}}\right).$
(8)

We can regard $G$ as a perturbation for the conformal field theory given by
$S_{0}$ in Eq. (3). Let us demonstrate the necessity of renormalization using
the example of the perturbative calculation of the following two-point
function,

$C_{2}(x,y)=\Big{\langle}B(x)O_{g}(y)\Big{\rangle}_{G}\equiv\Big{\langle}B(x)O_{g}(y)\exp\left(\int_{|z|<R}\frac{d^{2}z}{\pi}O_{G}\right)\Big{\rangle}_{0}.$
(9)

Here

$B(x)\equiv
b_{c\bar{c}}\partial\varphi^{c}\bar{\partial}\bar{\varphi}^{\bar{c}}$

is a composite field which is denoted differently from $O_{b}$ to emphasize
the fact that the field $b_{a\bar{a}}$ is an observable (and not the one in
the action). By $O_{g}$ we denote $g^{a\bar{a}}p_{a}\bar{p}_{\bar{a}}$ again
to emphasize different role of the observable field $g^{a\bar{a}}$ as opposed
to to $G^{a\bar{a}}$ in the exponential. All observables are considered to be
normal ordered, so we don’t have to worry about self-contractions. The effect
of non-zero $G$ is included order by order in the perturbation theory which is
symbolized by the last equality in (9). The subscript there reflects that
computations are carried out in the free theory (3). Here we have also
introduced an IR cut-off $R$ and assume that $|x|\ll R$, $y\ll R$. In the
zeroth order using OPE (4) we conclude

$C_{2}^{(0)}(x,y)=\frac{g^{a\bar{a}}b_{a\bar{a}}}{|x-y|^{4}}.$ (10)

At this point, we have to clarify what we mean by correlation functions as
functions of the target space fields. Strictly speaking instead of
$g^{a\bar{a}}b_{a\bar{a}}$ one has to consider an integral

$\int\,g^{a\bar{a}}(\varphi^{(0)})b_{a\bar{a}}(\varphi^{(0)})\,d^{D}\varphi^{(0)}$
(11)

over the modes $\varphi^{(0)}$ of the $\varphi$-fields. We assume that the
target space is compact, or that the observables $g^{a\bar{a}}$,
$b_{a\bar{a}}$ are functions rapidly vanishing at the “space-time infinity,”
so that all integrals are convergent.

We assume this prescription for all the correlation functions in this paper.

In particular, this will allow us to integrate by parts in some of the
examples below. Note also that we use Wick’s theorem to contract derivatives
of the field, which would, otherwise, vanish after the integration.

The next order is given by an integral of the three-point function

$C_{2}^{(1)}(x,y)=\int\limits_{|z|<R}\frac{d^{2}z}{\pi}\Big{\langle}B(x)O_{g}(y)O_{G}(z)\Big{\rangle}_{0}.$
(12)

In this case the computation of the correlation function is a bit more
complicated, but still can be performed explicitly. Namely, after contracting
the $p$ field with the derivatives we arrive at

$\Big{\langle}B(x)O_{g}(y)O_{G}(z)\Big{\rangle}_{0}=\frac{\langle
b_{a\bar{a}}(x)g^{a\bar{a}}(y)G^{c\bar{c}}p_{c}\bar{p}_{\bar{c}}(z)\rangle}{|x-y|^{4}}+\frac{\langle
b_{a\bar{a}}(x)g^{a\bar{b}}\bar{p}_{\bar{b}}(y)G^{c\bar{a}}p_{c}(z)\rangle}{(x-y)^{2}(\bar{x}-\bar{z})^{2}}+\left(\begin{array}[]{c}y\leftrightarrow
z\\\ G\leftrightarrow g\end{array}\right).$ (13)

Contracting the remaining $p$ field we obtain

$\frac{\langle
b_{a\bar{a}}(x)g^{a\bar{a}}(y)G^{c\bar{c}}p_{c}\bar{p}_{\bar{c}}(z)\rangle}{|x-y|^{4}}=\frac{G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}b_{a\bar{a}}(x)g^{a\bar{a}}(y)}{|x-y|^{4}|z-x|^{2}}+\frac{G^{c\bar{c}}\partial_{c}b_{a\bar{a}}(x)\partial_{\bar{c}}g^{a\bar{a}}(y)}{|x-y|^{4}(z-x)(\bar{z}-\bar{y})}+\left(\begin{array}[]{c}y\leftrightarrow
x\\\ g\leftrightarrow b\end{array}\right),$ (14) $\langle
b_{a\bar{a}}(x)g^{a\bar{b}}\bar{p}_{\bar{b}}(y)G^{c\bar{a}}p_{c}(z)\rangle=\frac{\partial_{\bar{b}}\partial_{c}b_{a\bar{a}}g^{a\bar{b}}G^{c\bar{a}}}{(\bar{y}-\bar{x})(z-x)}+\frac{\partial_{\bar{b}}b_{a\bar{a}}\partial_{c}g^{a\bar{b}}G^{c\bar{a}}}{(\bar{y}-\bar{x})(z-y)}+\frac{b_{a\bar{a}}\partial_{c}g^{a\bar{b}}\partial_{\bar{b}}G^{c\bar{a}}}{(\bar{y}-\bar{z})(z-y)}+\frac{\partial_{c}b_{a\bar{a}}g^{a\bar{b}}\partial_{\bar{b}}G^{c\bar{a}}}{(\bar{y}-\bar{z})(z-x)}$
(15)

Assembling everything together we obtain an expression for the correlator that
does not look like a typical three point function in CFT. This happened
because we relaxed the transversality condition, which on this level is
equivalent to the statement that operators are primary. Anyhow, from the
obtained expression one might expect possible divergences when the integration
comes close to $x$ or $y$, therefore we add UV regularization prescription
that small disks of the size $\epsilon$ are excluded from the integration
domain, namely $|z-x|<\epsilon$, $|z-y|<\epsilon$. Performing integration with
such a prescription we obtain

$C_{2}^{(1)}(x,y)=\frac{\log\frac{|x-y|^{2}}{\epsilon^{2}}}{|x-y|^{4}}\Big{(}g^{a\bar{a}}\bm{[}b,G\bm{]}_{a\bar{a}}+b_{a\bar{a}}\bm{[}g,G\bm{]}^{a\bar{a}}\Big{)}+\delta
C_{2}^{(1)}(x,y)$ (16)

where $\delta C_{2}^{(1)}$ is present only due to effects of the fields being
non-primary (non-tranversal in the sense of Eq. (7)),

$\delta
C_{2}^{(1)}(x,y)=\frac{G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}(b_{a\bar{a}}g^{a\bar{a}})}{|x-y|^{4}}\log\frac{R^{2}}{|x-y|^{2}}+\frac{4}{\epsilon^{2}}\frac{g^{c\bar{c}}\partial_{c}\partial_{\bar{c}}(b_{a\bar{a}}G^{a\bar{a}})}{|x-y|^{2}}\\\
+\frac{\partial_{c}(b_{a\bar{a}}G^{a\bar{c}})\partial_{\bar{c}}g^{c\bar{a}}+\partial_{c}g^{a\bar{c}}\partial_{\bar{c}}(b_{a\bar{a}}G^{c\bar{a}})-g^{c\bar{c}}\partial_{\bar{c}}(b_{a\bar{a}}\partial_{c}G^{a\bar{a}})-g^{c\bar{c}}\partial_{c}(b_{a\bar{a}}\partial_{\bar{c}}G^{a\bar{a}})}{|x-y|^{4}}.$
(17)

Notice that if trasversality condition (7) is satisfied then $\delta
C_{2}^{(1)}$ vanishes after integration by parts in the target space (see Eq.
(11)).

The coordinate part of the first term in Eq. (16) is universal for the
correlation functions in deformed CFT Cardy:1989da ; Ginsparg:1988ui ;
Zamolodchikov:1989we , while the field dependent coefficient represents the
OPE data. The expression in the bold squared parentheses in (16) are bilinear
operations, which are explicitly given by

$\bm{[}b,G\bm{]}_{a\bar{a}}=\partial_{\bar{c}}b_{c\bar{a}}\partial_{a}G^{c\bar{c}}+b_{c\bar{c}}\partial_{a}\partial_{\bar{a}}G^{c\bar{c}}+G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}b_{a\bar{a}}+\partial_{c}b_{a\bar{c}}\partial_{\bar{a}}G^{c\bar{c}}$
(18)
$\bm{[}g,G\bm{]}^{a\bar{a}}=g^{c\bar{c}}\partial_{c}\partial_{\bar{c}}G^{a\bar{a}}-\partial_{\bar{c}}g^{c\bar{a}}\partial_{c}G^{a\bar{c}}+(G\leftrightarrow
g).$ (19)

In (18), (19) and in what follows the brackets $\bm{[\,\,]}$ typeset in bold
acting on two tensors are to be understood as a tensor product of the Lie
derivatives acting in holomorphic and antiholomorphic sectors. They also have
a close relation to the Courant/Dorfman brackets Zeitlin2008 .

Therefore, to remove UV divergences we can perform the following
renormalization,

$B\to
B^{(1)}=\left(b_{a\bar{a}}-\bm{[}b,G\bm{]}_{a\bar{a}}\log\frac{\rho^{2}}{\epsilon^{2}}\right)\partial\varphi^{a}\bar{\partial}\bar{\varphi}^{\bar{a}}-\frac{4}{\epsilon^{2}}b_{a\bar{a}}G^{a\bar{a}},$
(20) $O_{g}\to
O^{(1)}_{g}=\left(g^{a\bar{a}}-\bm{[}g,G\bm{]}^{a\bar{a}}\log\frac{\rho^{2}}{\epsilon^{2}}\right)p_{a}\bar{p}_{\bar{a}}.$
(21)

Here the logarithm plays the same role as in Eqs. (1), (2) upon the
replacement of the momentum cutoff by the coordinate regulator
$M_{uv}\to\epsilon^{-1}$ and the renormalization point $\mu\to\rho^{-1}$. In
this way, the renormalized correlation function reduces to

$\langle
B^{(1)}(x)O^{(1)}_{g}(y)\rangle=\frac{\log\frac{|x-y|^{2}}{\rho^{2}}}{|x-y|^{4}}\left(g^{a\bar{a}}\bm{[}b,G\bm{]}_{a\bar{a}}+b_{a\bar{a}}\bm{[}g,G\bm{]}^{a\bar{a}}\right)+\frac{G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}(b_{a\bar{a}}g^{a\bar{a}})}{|x-y|^{4}}\log\frac{R^{2}}{|x-y|^{2}}\\\\[5.69054pt]
+\frac{\partial_{c}(b_{a\bar{a}}G^{a\bar{c}})\partial_{\bar{c}}g^{c\bar{a}}+\partial_{c}g^{a\bar{c}}\partial_{\bar{c}}(b_{a\bar{a}}G^{c\bar{a}})-g^{c\bar{c}}\partial_{\bar{c}}(b_{a\bar{a}}\partial_{c}G^{a\bar{a}})-g^{c\bar{c}}\partial_{c}(b_{a\bar{a}}\partial_{\bar{c}}G^{a\bar{a}})}{|x-y|^{4}}+O(G^{2}).$
(22)

Proceeding in the same way step by step one can derive renormalization in the
higher orders. Unfortunately, all expressions become more and more bulky;
besides, as we see, a lot of other terms are produced that are not responsible
for the renormalization. To avoid at least part of these problems we change
our approach a bit.

Namely, since we are mostly interested in the renormalization of $O_{g}$, we
limit integration to $|z|<R$ and assume that $|y|<R$ but $|x|\gg R$. In this
sense the perturbation does not “touch” observable $B(x)$. Then in the leading
order we immediately obtain the $O_{g}$ renormalization directly

$\int\limits_{|z|<R}\frac{d^{2}z}{\pi}\Big{\langle}B(x)O_{g}(0)O_{G}(z)\Big{\rangle}_{0}=\frac{b_{a\bar{a}}\bm{[}g,G\bm{]}^{a\bar{a}}}{|x|^{4}}\int\limits_{|z|<R}\frac{d^{2}z}{\pi}\frac{1}{|z|^{2}}=\frac{b_{a\bar{a}}\bm{[}g,G\bm{]}^{a\bar{a}}}{|x|^{4}}\log\frac{R^{2}}{\epsilon^{2}}$
(23)

Here we have put coordinate of $O_{g}$ to zero $y=0$ because in the leading
order of $|x|\to\infty$ the rest of the correlation function retains
transnational invariance. In the next order we have to compute

$C_{2}^{(2)}(x)=\frac{1}{2}\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\Big{\langle}B(x)O_{g}(0)O_{G}(z_{1})O_{G}(z_{2})\Big{\rangle}_{0}$
(24)

The integration domain $\Sigma_{\epsilon}$ is defined as

$\epsilon\leq|z_{1}|\leq R,\qquad\epsilon\leq|z_{2}|\leq
R,\qquad\epsilon\leq|z_{1}-z_{2}|\leq R.$ (25)

Note extra IR regularization $|z_{1}-z_{2}|\leq R$; it does not affect the UV
divergences but significantly simplifies computations as it makes
$\Sigma_{\epsilon}$ invariant under various changes of variables , for
example, $z_{1}\leftrightarrow z_{2}$; $z_{1}\to\bar{z}_{1}$,
$z_{2}\to\bar{z_{2}}$; $z_{1}\to z_{2}-z_{1}$, $z_{2}\to z_{2}$ etc. Using
this invariance and routinely contracting $p$ and $\varphi$ using OPE (4) and
the Wick’s theorem we observe that only four possible structures appears in
$C_{2}^{(2)}(x)$

$C_{2}^{(2)}(x)=\frac{1}{|x|^{4}}\sum\limits_{i=1}^{4}\mathcal{A}_{i}I_{i}$
(26)

with

$I_{1}=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{|z_{1}|^{2}|z_{2}|^{2}},\qquad
I_{2}=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{z_{2}^{2}\bar{z}_{1}(\bar{z}_{2}-\bar{z}_{1})}$
(27)
$I_{3}=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{z_{1}^{2}\bar{z}_{2}^{2}},\qquad
I_{4}=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{|z_{1}|^{4}}.$
(28)

In the appendix (A) we demonstrate that the integrals can be expressed as

$I_{1}=L_{\epsilon}^{2},\qquad I_{2}=L_{\epsilon},\qquad I_{3}=0,\qquad
I_{4}=R^{2}/\epsilon^{2}.$ (29)

Here

$L_{\epsilon}\equiv\log R^{2}/\epsilon^{2}\,.$

The corresponding tensor structures $\mathcal{A}_{i}$ read as

$\mathcal{A}_{1}=\frac{b_{a\bar{a}}\bm{[}g,\bm{[}G,G\bm{]}\bm{]}^{a\bar{a}}}{4}+\frac{b_{a\bar{a}}\bm{[}\bm{[}g,G\bm{]},G\bm{]}^{a\bar{a}}}{2},\qquad\mathcal{A}_{2}=\frac{1}{2}b_{a\bar{a}}\bm{[}g,G,G\bm{]}^{a\bar{a}}$
(30)
$\mathcal{A}_{3}=b_{a\bar{a}}(\partial_{c}G^{k\bar{a}}\partial_{k}\partial_{\bar{k}}G^{c\bar{c}}\partial_{\bar{c}}g^{a\bar{k}}+\partial_{c}g^{k\bar{a}}\partial_{k}\partial_{\bar{k}}G^{c\bar{c}}\partial_{\bar{c}}G^{a\bar{k}}),$
(31)
$\mathcal{A}_{4}=\frac{b_{a\bar{a}}}{2}(g^{a\bar{a}}\partial_{k}\partial_{\bar{k}}G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}G^{k\bar{k}}+2G^{a\bar{a}}\partial_{k}\partial_{\bar{k}}G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}g^{k\bar{k}})$
(32)

where quadratic brackets are the operations defined as above (18), (19) that
the qubic bracket is given by

$\bm{[}g,G,G\bm{]}^{a\bar{a}}=\partial_{\bar{k}}G^{c\bar{c}}\partial_{\bar{c}}G^{k\bar{k}}\partial_{c}\partial_{k}g^{a\bar{a}}+\partial_{k}G^{c\bar{c}}\partial_{c}G^{k\bar{k}}\partial_{\bar{c}}\partial_{\bar{k}}g^{a\bar{a}}+2\partial_{k}G^{c\bar{c}}\partial_{c}g^{k\bar{k}}\partial_{\bar{k}}\partial_{\bar{c}}G^{a\bar{a}}+\\\\[5.69054pt]
2\partial_{\bar{k}}G^{c\bar{c}}\partial_{\bar{c}}g^{k\bar{k}}\partial_{k}\partial_{c}G^{a\bar{a}}+\partial_{c}\partial_{\bar{c}}G^{k\bar{k}}(\partial_{k}g^{c\bar{c}}\partial_{\bar{k}}G^{a\bar{a}}+\partial_{\bar{k}}g^{c\bar{c}}\partial_{k}G^{a\bar{a}}+\partial_{k}g^{a\bar{a}}\partial_{\bar{k}}G^{c\bar{c}}+\\\\[5.69054pt]
\partial_{\bar{k}}g^{a\bar{a}}\partial_{k}G^{c\bar{c}}+g^{a\bar{c}}\partial_{k}\partial_{\bar{k}}G^{c\bar{a}}+G^{a\bar{c}}\partial_{k}\partial_{\bar{k}}g^{c\bar{a}}+g^{c\bar{a}}\partial_{k}\partial_{\bar{k}}G^{a\bar{c}}+G^{c\bar{a}}\partial_{k}\partial_{\bar{k}}g^{a\bar{c}})\\\\[5.69054pt]
+\partial_{c}\partial_{\bar{c}}g^{k\bar{k}}(\partial_{k}G^{c\bar{c}}\partial_{\bar{k}}G^{a\bar{a}}+\partial_{\bar{k}}G^{c\bar{c}}\partial_{k}G^{a\bar{a}}+G^{c\bar{a}}\partial_{k}\partial_{\bar{k}}G^{a\bar{c}}+G^{a\bar{c}}\partial_{k}\partial_{\bar{k}}G^{c\bar{a}})\\\\[5.69054pt]
-(g^{k\bar{a}}\partial_{\bar{k}}G^{c\bar{c}}+G^{k\bar{a}}\partial_{\bar{k}}g^{c\bar{c}})\partial_{k}\partial_{c}\partial_{\bar{c}}G^{a\bar{k}}-(g^{a\bar{k}}\partial_{k}G^{c\bar{c}}+G^{a\bar{k}}\partial_{k}g^{c\bar{c}})\partial_{\bar{k}}\partial_{c}\partial_{\bar{c}}G^{k\bar{a}}\\\\[5.69054pt]
-2\partial_{k}\partial_{\bar{c}}G^{a\bar{k}}(\partial_{\bar{k}}g^{c\bar{c}}\partial_{c}G^{k\bar{a}}+\partial_{\bar{k}}G^{c\bar{c}}\partial_{c}g^{k\bar{a}})-2\partial_{c}\partial_{\bar{k}}G^{k\bar{a}}(\partial_{k}g^{c\bar{c}}\partial_{\bar{c}}G^{a\bar{k}}+\partial_{k}G^{c\bar{c}}\partial_{\bar{c}}g^{a\bar{k}})\\\\[5.69054pt]
-G^{k\bar{a}}\partial_{\bar{k}}G^{c\bar{c}}\partial_{k}\partial_{c}\partial_{\bar{c}}g^{a\bar{k}}-G^{a\bar{k}}\partial_{k}G^{c\bar{c}}\partial_{\bar{k}}\partial_{c}\partial_{\bar{c}}g^{k\bar{a}}\\\\[5.69054pt]
-2\partial_{\bar{k}}G^{c\bar{c}}\partial_{c}G^{k\bar{a}}\partial_{k}\partial_{\bar{c}}g^{a\bar{k}}-2\partial_{\bar{c}}G^{a\bar{k}}\partial_{k}G^{c\bar{c}}\partial_{c}\partial_{\bar{k}}g^{k\bar{a}}.$
(33)

Now we have to deform $O_{g}$ and $O_{G}$ by addition of some counterterms to
make the correlation function finite when $\epsilon\to 0$. In particular, to
remove contribution from $\mathcal{A}_{4}$ we have to shift $O_{g}$ by the
scalar observable

$O_{g}\to
O_{g}-\frac{\partial_{k}\partial_{\bar{k}}G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}g^{k\bar{k}}}{\epsilon^{2}}.$
(34)

Note that the first term in $\mathcal{A}_{4}$ has a disconnected character and
disappears after taking into account contributions from the partition
function. The remaining part of the correlation function reads.

$C_{2}^{(1)}(x)+C_{2}^{(2)}(x)=\frac{b_{a\bar{a}}}{|x|^{4}}\left(g^{a\bar{a}}+L_{\epsilon}\bm{[}g,G\bm{]}^{a\bar{a}}+\frac{L_{\epsilon}^{2}}{4}\bm{[}g,\bm{[}G,G\bm{]}\bm{]}^{a\bar{a}}\right.\\\
\left.+\frac{L_{\epsilon}^{2}}{2}\bm{[}\bm{[}g,G\bm{]},G\bm{]}^{a\bar{a}}+\frac{L_{\epsilon}\bm{[}g,G,G\bm{]}^{a\bar{a}}}{2}\right)$
(35)

To understand renormalization further we have to separate deformation of
$O_{g}$ and $O_{G}$. In typical Callan-Symanzik approach that would correspond
to $\beta$ and $\gamma$ functions zJJ . For this we consider the following
one-point function,

$C_{1}(x)=\Big{\langle}B(x)\exp\left(\int_{|z|<R}\frac{d^{2}z}{\pi}G^{a\bar{a}}(\varphi,\bar{\varphi})p_{a}\bar{p}_{\bar{a}}\right)\Big{\rangle}_{0}$
(36)

Again here it is assumed that $|x|\gg R$. Its expansion up to the fourth order
in $G^{a\bar{a}}$ can be deduced from (35)

$|x|^{4}C_{1}(x)=b_{a\bar{a}}\left(G+\frac{L_{\epsilon}}{2}\bm{[}G,G\bm{]}+\frac{L_{\epsilon}^{2}}{4}\bm{[}G,\bm{[}G,G\bm{]}\bm{]}+\frac{L_{\epsilon}}{3!}\bm{[}G,G,G\bm{]}\right)^{a\bar{a}}$
(37)

To make it finite as $\epsilon\to 0$ we have to deform $G$ in the following
way

$G\to\tilde{G}=G-\frac{\bm{[}G,G\bm{]}}{2}\ell+\frac{(\ell)^{2}}{4}\bm{[}G,\bm{[}G,G\bm{]}\bm{]}-\frac{\ell}{3!}\bm{[}G,G,G\bm{]}+O(G^{4})$
(38)

with

$\ell=\log\frac{\rho^{2}}{\epsilon^{2}}\,.$

Then

$\tilde{G}+\frac{L_{\epsilon}}{2}\bm{[}\tilde{G},\tilde{G}\bm{]}+\frac{L_{\epsilon}^{2}}{4}\bm{[}\tilde{G},\bm{[}\tilde{G},\tilde{G}\bm{]}\bm{]}+\frac{L_{\epsilon}}{3!}\bm{[}\tilde{G},\tilde{G},\tilde{G}\bm{]}=\\\
G-\frac{\bm{[}G,G\bm{]}}{2}L_{\rho}+\frac{(L_{\rho})^{2}}{4}\bm{[}G,\bm{[}G,G\bm{]}\bm{]}-\frac{L_{\rho}}{3!}\bm{[}G,G,G\bm{]}+O(G^{4}),$
(39)

where now

$L_{\rho}=\log R^{2}/\rho^{2}\,.$

Now we can study the flow of the deformed observable

$\frac{\partial\tilde{G}}{\partial\ell}=-\frac{\bm{[}G,G\bm{]}}{2}+\frac{\ell}{2}\bm{[}G,\bm{[}G,G\bm{]}\bm{]}-\frac{1}{3!}\bm{[}G,G,G\bm{]}+O(G^{4})=-\frac{\bm{[}\tilde{G},\tilde{G}\bm{]}}{2}-\frac{\bm{[}\tilde{G},\tilde{G},\tilde{G}\bm{]}}{3!}+O(\tilde{G}^{4})$
(40)

We see that contrary to the deformation (38) the flow (40) does not contain
logarithms but only the observables. Following the traditional notation we
call this flow a beta function

$-\frac{\partial\tilde{G}}{\partial\ell}=\beta_{2}(\tilde{G})+\beta_{3}(\tilde{G})+O(\tilde{G}^{4})$
(41)

with

$\boxed{\beta_{2}^{a\bar{a}}(G)=\frac{\bm{[}G,G\bm{]}^{a\bar{a}}}{2}=G^{c\bar{c}}\partial_{c}\partial_{\bar{c}}G^{a\bar{a}}-\partial_{c}G^{a\bar{c}}\partial_{\bar{c}}G^{c\bar{a}}}$
(42)

$\displaystyle\beta_{3}^{a\bar{a}}$
$\displaystyle=\frac{\bm{[}G,G,G\bm{]}^{a\bar{a}}}{3!}=\frac{1}{2}\left(\partial_{\bar{k}}G^{c\bar{c}}\partial_{c}(\partial_{k}G^{a\bar{a}}\partial_{\bar{c}}G^{k\bar{k}}-G^{k\bar{a}}\partial_{k}\partial_{\bar{c}}G^{a\bar{k}})\right.$
$\displaystyle+$
$\displaystyle\partial_{k}G^{c\bar{c}}\partial_{\bar{c}}(\partial_{\bar{k}}G^{a\bar{a}}\partial_{c}G^{k\bar{k}}-G^{a\bar{k}}\partial_{c}\partial_{\bar{k}}G^{k\bar{a}})+$
$\displaystyle\left.\partial_{k}\partial_{\bar{k}}G^{a\bar{c}}(G^{c\bar{a}}\partial_{c}\partial_{\bar{c}}G^{k\bar{k}}-\partial_{\bar{c}}G^{c\bar{k}}\partial_{c}G^{k\bar{a}})+\partial_{k}\partial_{\bar{k}}G^{c\bar{a}}(G^{a\bar{c}}\partial_{c}\partial_{\bar{c}}G^{k\bar{k}}-\partial_{c}G^{k\bar{c}}\partial_{\bar{c}}G^{a\bar{k}})\right)$
(43)

Once we have figured out renormalization of $G^{a\bar{a}}$ we can determine
how to renormalize $g^{a\bar{a}}$ using Eq. (35)

$g\to\tilde{g}=g-\ell\bm{[}g,G\bm{]}+\frac{\ell^{2}\bm{[}\bm{[}g,G\bm{]},G\bm{]}}{2}+\frac{\ell^{2}\bm{[}g,\bm{[}G,G\bm{]}\bm{]}}{4}-\frac{\ell}{2}\bm{[}g,G,G\bm{]}+O(G^{3}).$
(44)

The corresponding flow is

$-\frac{\partial
g}{\partial\ell}=\bm{[}g,G\bm{]}+\frac{1}{2}\bm{[}g,G,G\bm{]}+O(G^{3}).$ (45)

###  2.2 Results and discussion

Now let us discuss our results (42) and (43). It was shown already in
Losev2006  that the formula (42) leads to results identical to the one-loop
expressions obtained in the background field method Callan1985 ; Callan1986 ;
Fradkin1985 . For a Kähler metric and in the absence of a dilaton the two-loop
$\beta$ function reduces to the simple geometric flow that was already
mentioned in the introduction Friedan1980 ; Friedan1985

$\mu\frac{\partial
g_{ij}}{\partial\mu}=\beta^{(1)}_{ij}+\beta^{(2)}_{ij}+\dots$ (46)

with

$\beta^{(1)}_{ij}=R_{ij},\qquad\beta^{(2)}_{ij}=\frac{1}{2}R_{iklm}R_{j}\,^{klm}\equiv
S_{ij}.$ (47)

Here the superscripts indicate the number of loops, while the subscripts in
(42), (43) reflect the degree of $G$.

For a Kähler metric one can easily demonstrate that
$R^{a\bar{a}}=\beta_{2}^{a\bar{a}}$ (in Eq. (42)). Note that contrary to
Losev2006  we do not require neither a dilaton or the transversality condition
(7).

To compare the second loop answer (43) we need to find
$S^{ij}=G^{ik}S_{kl}G^{lj}$, which for the generic Kähler metrics is

$S^{a\bar{a}}=\frac{1}{2}G_{s\bar{s}}\left[G^{k\bar{k}}G^{c\bar{c}}\partial_{c}\partial_{\bar{k}}G^{a\bar{s}}\partial_{k}\partial_{\bar{c}}G^{s\bar{a}}-G^{c\bar{c}}\partial_{\bar{k}}G^{k\bar{s}}\partial_{c}G^{a\bar{k}}\partial_{k}\partial_{\bar{c}}G^{s\bar{a}}\right.\\\
\left.-G^{c\bar{c}}\partial_{k}G^{s\bar{k}}\partial_{\bar{c}}G^{k\bar{a}}\partial_{c}\partial_{\bar{k}}G^{a\bar{s}}+\partial_{c}G^{a\bar{k}}\partial_{\bar{c}}G^{c\bar{a}}\partial_{k}G^{s\bar{c}}\partial_{\bar{k}}G^{k\bar{s}}\right].$
(48)

This expression does not  coincide with (43). This fact is immediately obvious
for the one-dimensional target space, when $G^{1\bar{1}}\equiv G$. In this
case (42) and (43) read

$\beta_{2}=G\partial\bar{\partial}G-\partial G\bar{\partial}G$ (49)
$2\beta_{3}=(\bar{\partial}G)^{2}\partial^{2}G+(\partial
G)^{2}\bar{\partial}^{2}G-G\bar{\partial}G\partial^{2}\bar{\partial}G-G\partial
G\bar{\partial}^{2}\partial G-2\partial\bar{\partial}G\partial
G\bar{\partial}G+2G(\partial\bar{\partial}G)^{2}$ (50)

They are obviously, (differential) polynomials in $G$. The geometric answer,
however, reads as

$R^{1\bar{1}}=G\partial\bar{\partial}G-\partial G\bar{\partial}G,$ (51)
$S^{1\bar{1}}=\frac{(G\partial\bar{\partial}G-\partial
G\bar{\partial}G)^{2}}{G}=G(\partial\bar{\partial}G)^{2}-2\partial
G\bar{\partial}G\partial\bar{\partial}G+\frac{(\partial
G\bar{\partial}G)^{2}}{G}$ (52)

which is not even polynomial because of the last term in $S^{1\bar{1}}$.

It is interesting to note that for $\mathds{C}P^{1}$ model, which corresponds
to $G=\mathcal{R}(1+\varphi\bar{\varphi})^{2}/2$, both approaches give the
same answer

$\beta_{2}=\mathcal{R}^{2}(1+\varphi\bar{\varphi})^{2}/2,\qquad\beta_{3}=\mathcal{R}^{3}(1+\varphi\bar{\varphi})^{2}/2.$
(53)

Here parameter $\mathcal{R}$ is a scalar curvature, which is constant for
$\mathds{C}P^{1}$, the RG flow basically expresses how $\mathcal{R}$ depends
on $\ell$. In the general case, it is no longer true, and beta functions in
different approaches start to be different. More specifically, if we focus on
a slightly deformed case (see Sec. 3).

$G=n_{1}+n_{2}\varphi\bar{\varphi}+n_{3}(\varphi\bar{\varphi})^{2},$ (54)

Then geometric approach gives

$-\frac{\partial
G}{\partial\ell}=G\left(\frac{\mathcal{R}}{2}+\frac{\mathcal{R}^{2}}{4}+\dots\right).$
(55)

Here

$\frac{\mathcal{R}}{2}=\frac{n_{1}n_{2}+4n_{1}n_{3}\varphi\bar{\varphi}+n_{2}n_{3}(\varphi\bar{\varphi})^{2}}{n_{1}+n_{2}\varphi\bar{\varphi}+n_{3}(\varphi\bar{\varphi})^{2}}$

is a non-constant curvature for the metric with the lower indices (i.e. $1/G$,
with $G$ given in (54)). One can even conjecture that in the appropriate
scheme the flow can be resummed and in this way is defined only by a one-loop
result

$-\frac{\partial G}{\partial\ell}=\frac{G\mathcal{R}}{2-\mathcal{R}}$ (56)

In any case, we see that the structure (55) is not reproduced by Eqs. (42) and
(43) beyond one loop. On the contrary Eqs. (42), (43) show that the model is
renormalizable and the corresponding flow reads

$-\frac{\partial
n_{1}}{\partial\ell}=n_{1}n_{2}+n_{1}n_{2}^{2},\qquad-\frac{\partial
n_{2}}{\partial\ell}=4n_{1}n_{3}+4n_{1}n_{2}n_{3},\qquad-\frac{\partial
n_{3}}{\partial\ell}=n_{2}n_{3}+n_{3}(8n_{1}n_{3}-n_{2}^{2}).$ (57)

This flow is interesting by itself as the stable point there depends on
initial conditions.

For two-dimensional complex target-space we see that similar to
$\mathds{C}P^{1}$ case of Fubini-Study (Bergman) metrics Eq. (48) and (43)
gives identical results. However, already for Eguchi-Hanson metric555We use
notations of lye2023detailed ., which reads explicitly as

$G_{a\bar{c}}=\sqrt{1+\frac{\alpha}{u^{2}}}\left(\delta_{a\bar{c}}-\frac{\alpha}{\alpha+u^{2}}\frac{\bar{z}_{a}z_{c}}{u}\right),\qquad
u=|z_{1}|^{2}+|z_{2}|^{2},$ (58)

one can see the difference between (57) and (56). Notice that metric (58) is a
Kähler metric, it has unit determinant and obeys transversality condition
$\partial_{a}G^{a\bar{a}}=\partial_{\bar{a}}G^{a\bar{a}}=0$, which immediately
rules out ambiguities with a dilaton and Kalb-Ramond field.

Finally, let us discuss the transformational properties of the obtained
answers (42), (43). These expressions are manifestly not symmetric with
respect to classical diffeomorphisms of the target. For example the change
$\varphi\to 1/\varphi$ and $\bar{\varphi}\to 1/\bar{\varphi}$ would result in
the exchange $n_{1}\leftrightarrow n_{3}$ in (54) which obviously is not a
symmetry of Eq. (57) (but it is for the flow (56)). Alternatively, $n_{1}$ can
be brought to the same value as $n_{3}$ by rescaling
$\varphi\to\lambda\varphi$, which is also not reflected in (57).

To quantify general case a bit more we focus on the holomorphic
transformations

$\varphi^{a}\to\varphi^{a}-V^{a}(\varphi),\qquad\bar{\varphi}^{\bar{a}}\to\bar{\varphi}^{\bar{a}}.$
(59)

The classical diffieormorphisms

$G^{a\bar{a}}\to
G^{a\bar{a}}+V^{k}\partial_{k}G^{a\bar{a}}-\partial_{k}V^{a}G^{k\bar{a}}$ (60)

preserves the one-loop result (42). Namely, the transformation properties of
(42) are the same as for the metric

$\beta_{2}^{a\bar{a}}\to\beta_{2}^{a\bar{a}}+V^{k}\partial_{k}\beta_{2}^{a\bar{a}}-\partial_{k}V^{a}\beta_{2}^{k\bar{a}}.$
(61)

while $\beta_{3}$ in Eq. (43) transforms as

$\beta_{3}^{a\bar{a}}\to\beta_{3}^{a\bar{a}}+V^{k}\partial_{k}\beta_{3}^{a\bar{a}}-\partial_{k}V^{a}\beta_{3}^{k\bar{a}}-\frac{1}{2}\partial_{m}\partial_{c}V^{k}L_{k}^{mc;a\bar{a}}$
(62)

where

$L_{k}^{mc;a\bar{a}}=\partial_{k}\partial_{\bar{k}}G^{c\bar{a}}(G^{a\bar{c}}\partial_{\bar{c}}G^{m\bar{k}}-G^{m\bar{c}}\partial_{\bar{c}}G^{a\bar{k}})+\partial_{\bar{k}}G^{m\bar{a}}(G^{a\bar{c}}\partial_{k}\partial_{\bar{c}}G^{cd\bar{k}}-\partial_{k}G^{c\bar{c}}\partial_{\bar{c}}G^{a\bar{k}})\\\
+\partial_{k}G^{m\bar{k}}\partial_{\bar{k}}(G^{c\bar{c}}\partial_{\bar{c}}G^{a\bar{a}}-G^{a\bar{c}}\partial_{\bar{c}}G^{c\bar{a}})+G^{m\bar{k}}\partial_{\bar{k}}(\partial_{\bar{c}}G^{a\bar{a}}\partial_{k}G^{c\bar{c}}-G^{a\bar{c}}\partial_{k}\partial_{\bar{c}}G^{c\bar{a}}),$
(63)

which is another manifestation that (43) cannot be expressed via geometric
structures (metric and curvature). Nevertheless, in the next section we
discuss that for the $G^{a\bar{a}}$ that corresponds to current-current
deformations satisfying the 2D current algebras, both (42), (43), and (63) can
be neatly expressed via the algebraic data.

Finally, let us note that usually the ambiguities related to different choices
of the regularization/renormalization scheme correspond to changes Hull:1986hn

$\beta_{\mu\nu}\sim\beta_{\mu\nu}+\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}.$
(64)

In the problem under consideration, it is impossible to choose a vector
$\xi_{\mu}$ to cancel non-tensor structures, i.e. those that do not transform
as tensors under the diffeormophisms. Moreover, it is even expected that at
the quantum level, the diffeomorphisms are deformed BLN . The full
clarification of this issue requires a separate investigation.

##  3 Lie-Algebraic Sigma models

One of the important examples of the perturbed first-order sigma models comes
from the parametrization of the $G^{a\bar{a}}$ with holomorphic and
antiholomorphic vector fields, namely

$G^{a\bar{a}}=G^{A\bar{A}}V^{a}_{A}V^{\bar{a}}_{\bar{A}},$ (65)

or equivalently $O_{G}=G^{A\bar{A}}J_{A}\bar{J}_{\bar{A}}$, which is a
current-current deformation with currents given by $J_{A}=p_{a}V_{A}^{a}$ (and
similarly for antiholomorphic). For a linear realization of the vector fields
we are in a position of the so-called bosonic Gross-Neveu model Affleck2022 ;
Bykov2021 ; bykov2022betafunction . The current algebra immediately follows
from the OPE

$J_{A}(z)J_{B}(0)=-\frac{\eta_{AB}}{z^{2}}+\frac{f^{C}_{AB}J_{C}(0)-\Omega_{AB}}{z}+{\rm
reg}$ (66)

and similarly for the antiholomorphic components. Here we assume that vector
fields form an algebra with the structure constants $f_{AB}^{C}$, i.e.
$V^{c}_{[A}\partial_{c}V^{a}_{B]}\equiv f^{C}_{AB}V_{C}^{a}$, the rest of the
structures are defined as

$\eta_{AB}=\partial_{k}V^{c}_{A}\partial_{c}V_{B}^{k},\qquad\Omega_{AB}=\partial_{c}V_{B}^{k}\partial_{m}\partial_{k}V^{c}_{A}\partial\varphi^{m}.$
(67)

With such expressions the $\beta$ functions (42) and (43) reads

$\boxed{\beta_{2}=\frac{1}{2}G^{A\bar{A}}G^{B\bar{B}}f^{C}_{AB}f^{\bar{C}}_{\bar{A}\bar{B}}}$
(68)
$\boxed{\beta_{3}=\frac{1}{2}G^{A\bar{A}}G^{B\bar{B}}G^{C\bar{C}}\left(f_{AC}^{D}f_{BD}^{E}\eta_{\bar{A}\bar{B}}+{\rm
c.c.}\right)}$ (69)

More specifically, to get (42) ((43)) from (68) ((69)) one has to contract
with the corresponding vector fields
$\beta^{a\bar{a}}=\beta_{C\bar{C}}V_{C}^{a}V_{\bar{C}}^{\bar{a}}$. These
expressions correctly reproduce answers of known current-current deformations
PhysRevLett.86.4753 ; Ludwig2003  (see Appendix (B)). They were also obtained
in Sfetsos2014  using techniques developed in Sfetsos2014a  for some special
choices of current algebras (see also Sagkrioti2018 ). Moreover the expression
(33) also simplifies significantly

$\bm{[}g,G,G\bm{]}^{a\bar{a}}=\left(G^{A\bar{A}}G^{B\bar{B}}g^{C\bar{C}}+G^{A\bar{A}}g^{B\bar{B}}G^{C\bar{C}}+g^{A\bar{A}}G^{B\bar{B}}G^{C\bar{C}}\right)\eta_{\bar{A}\bar{B}}f_{AC}^{D}f^{E}_{BD}V_{E}^{a}V_{\bar{C}}^{\bar{a}}+{\rm
c.c.}$ (70)

Much more important is that formulas (68) and (69) follow immediately from the
operator algebra and are universal. To be more precise and introduce the rest
of the field in (6) we assume that fields $V_{A}$ ($V_{\bar{A}}$) form a basis
in all operator fields and introduce a conjugate basis of 1-forms normalized
such that

$\omega_{a}^{A}V^{a}_{B}=\delta^{A}_{B}.$ (71)

This is equivalent for a target space to be a complex Lie group. The basis
condition can be relaxed by considering only a certain subset of all vector
fields. In this case we limit ourselves to consideration of very specific
perturbations.

Provided with the basis of one-forms we introduce additional currents

$I^{A}=\omega^{A}_{a}\partial\varphi^{a},\qquad\bar{I}^{\bar{A}}=\omega^{\bar{A}}_{\bar{a}}\bar{\partial}\varphi^{\bar{a}}.$
(72)

This way the rest of the fields in (6) can be presented as

$O_{\mu}=m^{A}_{\bar{A}}J_{A}\bar{I}^{\bar{A}},\qquad
O_{\bar{\mu}}=\bar{m}_{A}^{\bar{A}}\bar{J}_{\bar{A}}I^{A},\qquad
O_{b}=b_{A\bar{A}}I^{A}\bar{I}^{\bar{A}}.$ (73)

Additionally to (66) the current algebra reads

$J_{A}(z)I^{B}(0)=\frac{\delta_{A}^{B}}{z^{2}}+\frac{f_{CA}^{B}I^{C}(0)}{z}+{\rm
reg},$ (74)
$I^{B}(z)J_{A}(0)=\frac{\delta_{A}^{B}}{z^{2}}-\frac{f_{CA}^{B}I^{C}(0)}{z}+{\rm
reg},$ (75) $I^{B}(z)I^{C}(0)={\rm reg},$ (76)

and similarly in the antiholomorphic sector.

To calculate the corresponding $\beta$ functions in the same coordinate
approach as in the previous section one has to consider in particular the
correlation function

$C_{1}(x)=\Big{\langle}I^{C}(x)\bar{I}^{\bar{C}}(x)\exp\left(\int_{|z|<R}\frac{d^{2}z}{\pi}G^{A\bar{A}}J_{A}J_{\bar{A}}(z)\right)\Big{\rangle}_{0}$
(77)

instead of (36). The current correlation functions readily follow from the
operator algebra. Namely, the two-point correlation function are just the
$1/z^{2}$ coefficients,

$\langle J_{A}(z)J_{B}(0)\rangle_{0}=-\frac{\eta_{AB}}{z^{2}},\qquad\langle
J_{A}(z)I^{B}(0)\rangle_{0}=\frac{\delta_{A}^{B}}{z^{2}},\qquad\langle
I^{B}(z)I^{C}(0)\rangle_{0}=0.$ (78)

Higher order correlation functions can be computed using a typical trick with
the contour integrations (see Appendix (C)). In particular, three and four
point correlation functions read

$\langle
I^{C}(x)J_{A}(z)J_{B}(w)\rangle_{0}=\frac{f_{AB}^{C}}{(x-z)(x-w)(z-w)}$ (79)
$\langle
I^{C}(x)J_{A_{1}}(z_{1})J_{A_{2}}(z_{2})J_{A_{3}}(z_{3})\rangle_{0}=-\frac{\delta^{C}_{A_{1}}\eta_{A_{2}A_{3}}}{(x-z_{1})^{2}z_{23}^{2}}+\frac{f_{DA_{1}}^{C}f_{A_{2}A_{3}}^{D}}{(x-z_{1})z_{12}z_{13}z_{23}}+{\rm
c.p.}$ (80)

where ${\rm c.p.}$ stands for cyclic permutation of $1,2,3$. Note that since

$f_{DA_{1}}^{C}f_{A_{2}A_{3}}^{D}+{\rm c.p.}=0$ (81)

this correlation function is $O(1/x^{2})$ as should be. After this, the
computation goes identically to the previous section and one can
“independently” recover the results (68) and (69).

Now let us comment on the universal nature of the obtained results. In
particular, we can immediately get beta function of the model deformed by
$O_{\bar{\mu}}+O_{\mu}$ operators. This deformation leads to the flow of
$O_{b}$ operator Gamayun2009a ; Gamayun2009b . The exact beta function
computed within the background field methods reads

$\partial_{\ell}b_{a\bar{a}}=\bar{N}_{ak}^{\bar{c}}N^{c}_{\bar{a}\bar{k}}M_{c}^{k}\bar{M}^{\bar{k}}_{\bar{c}}$
(82)

where

$M_{c}^{k}=[\delta_{c}^{k}-\bar{\mu}^{\bar{s}}_{c}\mu_{\bar{s}}^{k}]^{-1},\qquad\bar{M}_{\bar{c}}^{\bar{k}}=[\delta_{\bar{c}}^{\bar{k}}-\mu^{s}_{\bar{c}}\bar{\mu}_{s}^{\bar{k}}]^{-1},$
(83)
$N^{c}_{\bar{a}\bar{k}}=\partial_{[\bar{a}}\mu^{c}_{\bar{k}]}-\mu^{s}_{[\bar{a}}\partial_{s}\mu^{c}_{\bar{k}]},\qquad\bar{N}^{\bar{c}}_{ak}=\partial_{[a}\bar{\mu}^{\bar{c}}_{k]}-\bar{\mu}^{\bar{s}}_{[a}\partial_{\bar{s}}\bar{\mu}^{\bar{c}}_{k]}.$
(84)

Here the brackets means antisymmetrization, for instance
$\partial_{[\bar{a}}\mu^{c}_{\bar{k}]}=\partial_{\bar{a}}\mu^{c}_{\bar{k}}-\partial_{\bar{k}}\mu^{c}_{\bar{a}}$.
Notice that the beta function (82) is in fact identical to Eq. (5.8) in
Sfetsos2014  upon identification $m\to\lambda$ and $\bar{m}\to\lambda^{T}$.
Contrary to those findings, here Eq. (82) is exact since in the background
field method only one-loop diagrams are non-zero Gamayun2009a , similar to
other “all-loop” beta functions Kutasov1989 ; PhysRevLett.86.4753 ;
Itsios:2014lca ; Georgiou:2016iom . On the other hand, the computation of the
coordinate beta-function reveals a similar disagreement between the geometric
answer (48) and (43) Gamayun2009b . In Gamayun2009b  this was attributed to
the seemingly different “IR” reason for the logarithms to appear. Here we can
highlight the differences within in algebraic data. Namely, the exact beta
function expanded to the second order gives

$\beta=\beta_{a\bar{a}}\partial\varphi^{a}\bar{\partial}\bar{\varphi}^{\bar{a}}=\left(m^{A}_{\bar{A}}\bar{m}^{\bar{B}}_{B}f_{AC}^{B}f_{\bar{B}\bar{C}}^{\bar{A}}+f_{AB}^{E}f^{D}_{EC}(m\bar{m})^{B}_{D}m^{A}_{\bar{C}}+\dots\right)I^{C}\bar{I}^{\bar{C}}.$
(85)

Repeating computation in the coordinate approach for the correlation function

$\tilde{C}_{1}(x)=\Big{\langle}J_{C}(x)\bar{J}_{\bar{C}}(x)\exp\left(\int_{|z|<R}\frac{d^{2}z}{\pi}(O_{\mu}(z)+O_{\bar{\mu}}(z))\right)\Big{\rangle}_{0}$
(86)

or using universal answer (68), (69) we obtain

$\beta_{b}=\left(m^{A}_{\bar{A}}\bar{m}^{\bar{B}}_{B}f^{B}_{AC}f^{\bar{A}}_{\bar{B}\bar{C}}+\left(f_{AB}^{E}f^{D}_{EC}+\frac{1}{2}f_{EB}^{D}f_{CA}^{E}\right)(m\bar{m})^{B}_{D}m^{A}_{\bar{C}}+\dots\right)I^{C}\bar{I}^{\bar{C}}$
(87)

We see that in the cubic order two expressions differ by a term
$\frac{1}{2}f_{EB}^{D}f_{CA}^{E}(m\bar{m})^{B}_{D}m^{A}_{\bar{C}}$. In Ref.
Gamayun2009b  it was argued that the logarithms appearing in this term have a
different nature (IR) than in front of the other terms that are responsible
for the genuine UV divergences. In the next section we exhibit that similar
phenomenon occurs in the $\beta$ functions in supersymmetric theories.

Finally, let us comment on what the transformation law (62) means in the Lie-
algebraic approach. Namely, the terms with the second derivative of the vector
field come from the transformation of $\eta_{AB}$. More specifically, for the
generic transformation (59) its transformation can be found from the explicit
form (67)

$\delta_{V}\eta_{AB}=V^{k}\partial_{k}\eta_{AB}-\partial_{a}\partial_{k}V^{c}V_{B}^{k}\partial_{c}V_{A}^{a}-\partial_{c}\partial_{k}V^{a}V_{A}^{k}\partial_{a}V_{B}^{c}$
(88)

which states basically the same as transformation (62). We can rewrite (88)
equivalently as

$\delta_{V}\eta_{AB}=\partial_{c}V_{A}^{a}\partial_{a}(V^{k}\partial_{k}V_{B}^{c}-V_{B}^{k}\partial_{k}V^{c})+(B\leftrightarrow
A)$ (89)

This presentation is especially useful if $V=V_{C}$ is one of the fields
forming the algebra. In this case

$\delta_{V_{C}}\eta_{AB}=f_{CA}^{D}\eta_{DB}+f_{CB}^{D}\eta_{DA},$ (90)

which looks more natural than the transformation (62).

Concluding this section, let us return to subsection 2.2 and explain the
emergence of the Lie-algebariac metric (54) from $sl(2)\times sl(2)$ algebra.
The non-linear realization of $sl(2)$ generators has the form

$\ell_{1}=-\phi^{2}\,\frac{\partial}{\partial\phi}\,,\quad\ell_{0}=\phi\,\frac{\partial}{\partial\phi}\,,\quad\ell_{-1}=\frac{\partial}{\partial\phi}\,\\\
$

plus a similar equation for the antiholomorphic algebra.

Lie-algebraic models based on $sl(2)$ are obtained as a linear combinations of
the operators

$l_{-1}\bar{l}_{-1},\,l_{0}\bar{l}_{0},\,l_{1}\bar{l}_{1},\,(l_{0}\bar{l}_{-1}+{\rm
H.c.}),\,(l_{0}\bar{l}_{1}+{\rm H.c.}),\,l_{1}\bar{l}_{1}\ $ (91)

with arbitrary coefficients. More precisely,

$G^{1\bar{1}}\,\frac{\partial}{\partial\phi}\frac{\partial}{\partial\bar{\phi}}\leftrightarrow\sum_{a\bar{b}=1,-1,0}\left({\mathcal{P}}_{a\bar{b}}\,l_{a}\bar{l}_{\bar{b}}+{\rm
H.c}\right)$ (92)

with a set of numeric coefficients $\\{{\mathcal{P}}_{a\bar{b}}\\}$. Linear
combinations of (91) span the set of Lie-algebraic metrics $G^{1\bar{1}}$. If
we want to preserve a U(1) symmetry we choose the combination

$n_{1}l_{-1}\bar{l}_{-1}+n_{2}l_{0}\bar{l}_{0},+n_{3}l_{1}\bar{l}_{1}$ (93)

which is equivalent to the metric (54).

##  4 Comparison with Supersymmetric sigma models

One possible way out reconciling the first-order perturbation theory with the
standard geometric formalism can be presented as follows. The OPE in the
coordinate space in the Lie-algebraic (bosonic Gross-Neveu) problem which
represents the genuine Wilsonean  OPE in one loop. Higher loops are obtained
as a result of certain “anomaly” in the measure revealing itself in the
calculation of the second and higher loops, in analogy with Yang-Mills
theory.666Also, the phenomenon we observe has a direct parallel in the chiral
${\mathcal{N}}=(0,2)$ two-dimensional $\mathds{C}P^{1}$ model CS . A general
proof is presented in NSZV ; NSZV1 ; CS ; AH  while the direct perturbative
all-loop calculations are given in KS ; KS1 ; KS2 .

One can complexify the coupling constants in the Lagrangians introducing their
holomorphic counterpartners,

$\displaystyle\frac{8\pi^{2}}{g^{2}}+i\theta\qquad{\rm super\,\,YM},$
$\displaystyle\frac{4\pi}{g^{2}}+i\theta\qquad{\mathcal{N}}=(0,2)\,\,\mathds{C}P^{1}.$
(94)

In fact, this complexification is automatic in the superfield language.
Moreover, it is maintaned in the $\beta$-function calculations at one-loop,
but the same supersymmetry forces it to be broken at higher loops NSZV ; NSZV1
; CS  due to infrared singularities for massless fields.

In other words, the bona fide Wilsonean OPE in ${\mathcal{N}}=1$ super-Yang-
Mills and (0,2) $\mathds{C}P^{1}$ model ends at one loop and is holomorphic;
the holomorphic anomaly shows up at two loops in the matrix elements of the
relevant operators rather than in the OPE coefficients. In AH  this phenomenon
in supersymmetric theories was related to an anomaly in the corresponding path
integral measure.

The exact all-order NSVZ $\beta$ function in super-Yang-Mills NSZV  and its
analog in ${{\mathcal{N}}=(0,2)}$ $\mathds{C}P^{1}$ model ensued. In the both
cases the two-loop result in the $\beta$ functions fixes all higher orders
through a geometrical progression. Of course, since the third and higher
orders are scheme dependent one must use a perturbative
regularization/renormalization scheme compatible with supersymmetry. This was
directly verified for the first time in KS ; KS1 ; KS2 .

##  5 Instanton analysis

In the previous sections, we have established that the first-order formalism
for sigma models proposed in Losev2006  when applied to $\beta$ function
calculations works perfectly in the leading order, however, leads to results
incompatible with the standard geometric (background field) calculations
starting from the second order. The hypothesis we have formulated to explain
the discrepancy is as follows: the polynomiality in the second and higher
loops is due to an infrared effect which in turn reflects the loss of symmetry
in the measure not explicitly seen in the path integral. In our subsequent
work subsequent  we will demonstrate that this is indeed the case. Starting
from a supersymmetry-based regularization it will be proven that making
superpartners’ mass large we are left with a finite effect, an anomaly. In
this section, we present a parallel of exactly the same phenomenon with the
loss of holomorphy beyond one loop which occurs in four-dimensional Yang-Mills
theory and two-dimensional $\mathds{C}P^{1}$ model. The latter example is
especially close to the problem we deal with in the present paper. Below we
use the instanton-based analysis to reveal the parallel. (It is not the only
possible demonstration, but it is the simplest).

In the instanton background a part of supersymmetry is preserved. The
symmetries broken by the given instanton generate zero modes. The exact
formula for the instanton measure in pure SUSY-Yang-Mills (without matter) is

$d\mu_{{\rm
SU}(2)}=\frac{1}{256\pi^{2}}e^{-8\pi^{2}/g^{2}}(\mu)^{n_{b}-{\textstyle\frac{n_{f}}{2}}}\left(\frac{8\pi^{2}}{g^{2}}\right)^{{{\textstyle\frac{1}{2}}(n_{b}-n_{f})}}d^{4}x_{0}\,d^{2}\theta_{0}\,d\rho^{2}\,d^{2}\,\bar{\beta}$
(95)

where $n_{b}$ and $n_{f}$ are the numbers of the boson and fermion zero modes,
respectively, $x_{0},...,\bar{\beta}$ are the instanton moduli, both boson and
fermion777Integration over the instanton three angular orientations in the
SU(2) group is performed.. In what follows there is no need to show
integration over the moduli explicitly, since we will focus on the $\beta$
function. For the SU(2) gauge group

$n_{b}=8,\,\,\,n_{f}=4\,.$ (96)

For SU$(N)$ we have

$n_{b}=4N,\,\,\,n_{f}=2N\,.$ (97)

Since the expression (95) is exact we can immediately derive the $\beta$
function for $1/g^{2}$ to all orders. The $\mu$ dependence appears in (95)
explicitly in the pre-exponent and implicitly through $g^{2}$. The combination
as a whole must be $\mu$ independent, therefore,

$\beta(g^{2})=\frac{\partial}{\partial
L}\frac{1}{g^{2}}=\left(n_{b}-\frac{n_{f}}{2}\right)\frac{1}{8\pi^{2}}\,\frac{1}{1-{\textstyle\frac{1}{2}}\,(n_{b}-n_{f}){\textstyle\frac{g^{2}}{8\pi^{2}}}}.$
(98)

The second and higher order coefficients in the $\beta$ function come from the
denominator in (98) which, in turn, is entirely determined by the factor
$\left(\frac{8\pi^{2}}{g^{2}}\right)^{{{\textstyle\frac{1}{2}}(n_{b}-n_{f})}}$
in Eq. (95) representing infrared (zero modes) effect. Note that $g^{2}$ in
the denominator in fact must be understood as Re$\,g^{2}$. The reason is
obvious: the parameter $\theta={\rm Im}\,(1/g^{2})$ does not appear in
perturbation theory. However, if we ignore the denominator (i.e. limit
ourselves to one loop) the coupling $1/g^{2}$ in the left-hand side of (98)
can be viewed as holomorphic. Equation (98) (and (104) below) is to be
compared with Eq. (56).

Let us consider extended supersymmetry. For ${\mathcal{N}}=2$ we have
$n_{f}=n_{b}$, hence, the denominator disappears, we have one-loop $\beta$
function and the holomorphy is preserved. For ${\mathcal{N}}=4$ we have
$n_{f}=2n_{b}$, and the $\beta$ function vanishes.

Adding matter fields in ${\mathcal{N}}=1$ Yang-Mills theory produces two
changes. First, the overall first coefficient in (98) must now include the
matter field contributions. Generically it still remains integer and
holomorphy is preserved. Second, the anomalous dimensions of the matter fields
appear in the NSVZ $\beta$ function starting from the second loop; they defy
holomorphy since the $Z$ factors which produce the anomalous dimensions are
non-lolomorphic.

We make a pause here to rewrite (95) in the following useful general form,

$d\mu_{{\rm SU}(N)}=\propto e^{-S_{\rm
inst}}(\mu)^{n_{b}-{\textstyle\frac{n_{f}}{2}}}\left(S_{\rm
inst}\right)^{{{\textstyle\frac{1}{2}}(n_{b}-n_{f})}}$ (99)

where $S_{\rm inst}$ is the instanton classical action $S=8\pi^{2}/g^{2}$.

Now, it is instructive to briefly review the ${\mathcal{N}}=(0,2)$
$\mathds{C}P^{1}$ model since it exhibits a novel feature absent in Yang-
Mills. This model can be formulated in terms of ${\mathcal{N}}=(0,2)$
superfieds. In its minimal version it is constructed from a single superfield
with two physical components – a complex scalar field $\phi$ and a two-
dimensional Weyl fermion $\psi_{L}$. The $\mathds{C}P^{1}$ target space is, of
course, Kählerian, which allows one to introduce a complexified coupling, see
Eq. (94). The perturbative superfield calculation yields CS  in one and two
loops

$\beta(g^{-2})_{(0,2)}=\frac{\partial}{\partial
L}\frac{1}{g^{2}}=\frac{1}{2\pi}\,\left(1+{\frac{g^{2}}{4\pi}}+...\right).$
(100)

While the one loop contribution preserves holomorphy in $1/g^{2}$, the second
loop term in the parentheses is in fact ${\textstyle\frac{{\rm
Re}\,g^{2}}{4\pi}}$ which ruins holomorphy. This is due to an infrared
singularity which can be seen both perturbatively and through the instantons.
Unlike the Yang-Mills theory the fermion contribution in perturbation theory
starts from the second loop.

Let us start from the instanton in ${\mathcal{N}}=(2,2)$ $\mathds{C}P^{1}$.
The latter is well known to produce only one-loop $\beta$ function due to the
Kähler nature of the target manifold AGF ; AGF1 ; AGF2 . The instanton measure
is

$d\mu_{(2,2)\,\mathds{C}P^{1}}\propto
e^{-4\pi/g^{2}}\left(\mu\right)^{2}\,dx_{0}\frac{d\rho}{\rho}\\{d\alpha\,d\bar{\alpha}\,d\beta\,d\bar{\beta}\\}$
(101)

Fermion and boson zero modes completely cancel, both in the pre-exponent and
in the exponent. There are four real boson zero modes and four fermion zero
modes (the fermion moduli are in braces in Eq. (103)). Integration over the
U(1) angle is carried out. The zero modes give rise to

$\left(\frac{\mu}{g}\right)^{4}\quad({\rm
boson}),\qquad\left(\frac{\mu}{g^{2}}\right)^{-2}\quad({\rm fermion}).$ (102)

All two-loop and higher loops in the instanton background cancel each other,
see e.g. obz , Sect. 6.4. Therefore there is no $g$ dependence in the pre-
exponent. Imposing the condition of $\mu$ independence in the right-hand side
of (101) we obtain one-loop $\beta$ function as was expected.

Now let us pass to ${\mathcal{N}}=(0,2)$ which is exactly half-way between
(101) and non-supersymmetric $\mathds{C}P^{1}$ shifp . In the passage we must
completely discard the $\psi_{R}$ Weyl fermion present in $(2,2)$. As a
result, if in the previous $(2,2)$ case we had four fermion zero modes, now we
have two. Thus, the fermion zero mode factor

$\left(\frac{\mu}{g^{2}}\right)^{-2}_{(2,2)}\to\left(\frac{\mu}{g^{2}}\right)^{-1}_{(0,2)},$

implying $\left(\mu\right)^{3}$ in the pre-exponent. This would change the
first coefficient in the $\beta$ function which cannot happen because fermion
contribution emerges only at the second loop. What is forgotten?

Two-loop and higher loops in the instanton background still cancel each other
CS . However, the balance between the boson and fermion non-zero modes
characteristic to the $(2,2)$ model is destroyed by elimination of the
$\psi_{R}$ Weyl fermion field from the theory. This produces extra
$\exp(\log\mu)$ factor. Assembling all these factors together we arrive at the
$(0,2)$ instanton measure,

$d\mu_{(0,2)\,\mathds{C}P^{1}}\propto
e^{-4\pi/g^{2}}\left(\mu\right)^{2}\frac{1}{g^{2}}\,.$ (103)

resulting in the following exact$\beta$ function

$\beta(g^{-2})_{(0,2)}\equiv\frac{\partial}{\partial
L}\frac{1}{g^{-2}}=\frac{1}{2\pi}\,\frac{1}{\left(1-{\frac{g^{2}}{4\pi}}\right)},$
(104)

to be compared with Eq. (100).

A couple of remarks are in order here. First, again, in the denominator we
have in fact $\frac{{\rm Re}\,g^{2}}{4\pi}$ rather than $\frac{g^{2}}{4\pi}$.
Second, in the $(0,2)$ model one can introduce “matter” fields much in the
same way as in 4D super-Yang-Mills. Their impact is similar to that of matter
in super-Yang-Mills CS . Finally, a comparison of (104) to (100) gives a
strong hint in favor of conjecture Eq. (56), for the Lie-algebraic
$sl(2)\times sl(2)$ structure (54) 888We remind that the third and higher
order coefficients are scheme dependent. Although currently they are
calculated up to four loops Grah  the renormalization schemes used do not
necessarily preserve the $sl(2)\times sl(2)$ algebraic structure..

In Section 3 we introduced the simplest Lie-algebriac model. Its Lie-algebriac
structure is presented in Eq. (93). Above in this section we studied
${\mathcal{N}}=(0,2)$ supersymmetric generalization of $\mathds{C}P^{1}$
model, in combination with the instanton analysis to argue that the $\beta$
function is of the NSVZ type. Like in ${\mathcal{N}}=1$ super-Yang-Mills
theory, $\mathds{C}P^{1}_{\,\,(0,2)}$ has only one coupling constant and one
beta function. Now let us address a problem with two coupling constants and
two coupled $\beta$ functions, with the metric $G^{1\bar{1}}$ from Eq. (54)
and ${\mathcal{N}}=(2,2)$ supersymmetry. It is known that the Kählerian nature
of the target space in this case guarantees that only one-loop result survives
in the instanton background. Let us see whether we can find both $\beta$
functions.

The metric of the ${\mathcal{N}}=(2,2)$ generalized model in Eq. (93) contains
three parameters, $n_{1,2,3}$. Rescaling the fields $\phi$ one can always
arrange $n_{1}=n_{3}$ even if in the original metric $n_{1}\neq n_{3}$. The
only condition is that both $n_{1,3}\neq 0$. Then we are left with two
parameters,999Note that Eq. (57) does not allow this rescaling because it is
incompatible with diffeomorphisms. $g^{2}$ and $k$,

$n_{1}=n_{1}=n_{3}=\frac{g^{2}}{2},\quad n_{2}=g^{2}k\,.$ (105)

Besides the moduli, the instanton measure now takes the following exact form,

$d\mu_{(2,2)}\propto\mu^{2}e^{-S_{\rm inst}}\,,\quad S_{\rm
inst}={\frac{4\pi}{g^{2}}}\frac{{\rm arccosh}(k)}{\sqrt{k^{2}-1}}\,,\quad
k\geq 1\,.$ (106)

For $k<1$ the second expression in (106) must be analytically continued. From
the above equation differentiating over $L=\log\mu$ we obtain an exact
relation

$2-\frac{\partial}{\partial L}S_{\rm inst}=0\,.$ (107)

$S_{\rm inst}$ now depends on two constant. To derive two $\beta$ functions
for two couplings we need another relation. At one loop we know that

$g^{4}\left(k^{2}-1\right)={\rm RG\,\,invariant}.$ (108)

We may or may not assume that it holds to all orders. So far it is not yet
establsihed. Equation (108) in conjunction with (107) implies

$\beta_{k}=\frac{\partial}{\partial L}k=\frac{g^{2}}{2\pi}(k^{2}-1)$ (109)

Applying ${\partial}/{\partial L}$ to (108) we arrive at

$\beta_{g}=\frac{\partial}{\partial
L}g^{2}=-\frac{k}{k^{2}-1}g^{2}\beta_{k}=-\frac{kg^{4}}{2\pi}\,$ (110)

Equations (109) and (110) coincide with the standard one-loop perturbative
calculations. For $k=1$ (i.e. undeformed $\mathds{C}P^{1}$) $\beta_{k}=0$ and
$\beta_{g}$ coincides with that of $\mathds{C}P^{1}$ (here
${\alpha^{\prime}=1/(2\pi)}$).

##  6 Conclusions

We applied the first order formalism of loop calculations in generic Kähler 2D
sigma models suggested in Losev2006  (for further developments see Zeitlin2006
; Zeitlin2008 ; Zeitlin2009 ) to multiloop calculations of $\beta$ functions,
using the coordinate space graphs and operator product expansion (OPE) rules.
We also considered the same formalism in Lie-algebraic sigma models, in which
the OPE rules are especially transparent. Starting from the bare Lagrangian we
obtained a certain pattern of loop coefficients $\beta^{(i)}$. In the one-loop
coefficient we observed that the first-order formalism and the old standard
geometric method of the $\beta$-function calculations fully coincide. However,
starting from the second loop the coincidence disappears at least for some
metrics. While the standard geometric method respects all symmetries of the
target space at the classical level, this is not the case with the first-order
formalism even when one cannot expect such a paradox. In a bid to solve the
discrepancy, we made a number of encouraging observations, but the full
solution of the issue is still evasive. We conjectured that some so far
undetected infrared anomalies might be responsible for the non-completeness of
the first-order calculations, in the same vein as it happened in the anomaly
supermultiplet in super-Yang-Mills. Our hypothesis is formulated in Eq. (56).

What remains to be done? It is highly desirable to directly verify (or
falsify) the all-order formula (56) which defies polynomiality of the first-
order formalism predictions for Lie-algebraic Kähler target spaces.
Alternatively, we could undertake a step-by-step approach and further search
for loopholes in the first-order formalism focusing on the second loop. This
problem is quite general, and so must be its solution. After all, taken
separately, theoretical elements of the first-order formalism are solid. There
is nothing wrong in Feynman graphs in the coordinate space for massless
fields, or in OPE rules. Taking into account dilaton’s contribution Callan1985
; Callan1986  might also “fix” the beta function Witten1991 . However, for the
transversal Kähler metrics these contributions are absent Losev2006  yet the
coordinate and geometric approaches give different results. Besides we have
reproduced beta functions for the current-current deformations without
invoking any dilatons. Yet systematic clarification of the dilaton role is
highly desirable.

Another question for the future is if some Lie-algebraic metrics define an
integrable model as it happens with $\mathds{C}P^{1}$ then one may expect to
determine the spectrum of the asymptotic states and observe how the model at
hand acquires a mass gap generation. The number of such states is still three
at least, in a range of parameters close to $\mathds{C}P^{1}$, but, of course,
the $O(3)$ symmetry of the spectrum must be lost as it is explicitly lost on
the target space.

## Acknowledgments

We are grateful to Adam Bzowski, Alexey Litvinov and Dmitri Bykov for useful
discussions. We would like to thank Kostas Sfetsos, Kostas Siampos, and Arkady
Tseytlin for their feedback, comments and suggestions.

The work of MS is supported in part by DOE grant DE-SC0011842.

Appendices

##  Appendix A Integrals

In this appendix we study integrals (27), (28) needed for the second-order
perturbation theory. We start with $I_{1}(\epsilon)$ defined as

$I_{1}(\epsilon)=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{|z_{1}|^{2}|z_{2}|^{2}}$
(111)

where $\Sigma_{\epsilon}$ is defined in (25). Notice that $\epsilon$
dependence comes only in the form $R/\epsilon$.

To find $\epsilon$ dependence we compute the following derivative

$-\epsilon\frac{\partial}{\partial\epsilon}I_{1}(\epsilon)=\epsilon\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{\delta(|z_{1}|-\epsilon)+\delta(|z_{2}|-\epsilon)+\delta(|z_{12}|-\epsilon)}{|z_{1}|^{2}|z_{2}|^{2}}$
(112)

The delta functions fix absolute value of the corresponding variables, however
the phase can be also fixed due to the rotational symmetry $z_{i}\to
z_{i}e^{i\theta}$ inside the integral. This way one of the integrals can be
computed completely, and the result after additional rescaling $z\to\epsilon
u$ leads to

$-\epsilon\frac{\partial}{\partial\epsilon}I_{1}(\epsilon)=2\int\limits_{\hat{\Sigma}_{1}}\frac{d^{2}u}{\pi}\left(\frac{2}{|u|^{2}}+\frac{1}{|u|^{2}|u-1|^{2}}\right)$
(113)

where $\hat{\Sigma}_{1}$ is defined as

$1\leq|u|\leq R/\epsilon,\qquad 1\leq|u-1|\leq R/\epsilon.$ (114)

we can explicitly present this integral as

$-\epsilon\frac{\partial}{\partial\epsilon}I_{1}(\epsilon)=4\log\frac{R^{2}}{\epsilon^{2}}-4\int\limits_{\Sigma_{1}^{*}}\frac{d^{2}u}{\pi}\frac{1}{|u|^{2}}+2\int\limits_{\Sigma_{1}}\frac{d^{2}u}{\pi}\frac{1}{|u|^{2}|u-1|^{2}}+O(\epsilon/R)$
(115)

where we have denoted new integration domains as

$\Sigma_{1}:|u|\geq 1,\quad|u-1|\geq
1;\qquad\Sigma_{1}^{*}:|u|>1,\quad|u-1|<1.$ (116)

Notice that since integrals are IR convergent we have effectively sent
$R\to\infty$, which resulted in terms of order $O(\epsilon/R)$, which we
further disregard. Here we have also used

$\int\limits_{R>|u|,|u-1|>R}\frac{d^{2}u}{|u|^{2}}=O(1/R).$ (117)

Integral over $\Sigma_{1}$ can be computed for instance with the help of the
Stokes’s theorem

Similarly we can present

$\int\limits_{\Sigma_{1}}\frac{d^{2}u}{\pi}\frac{1}{|u|^{2}|u-1|^{2}}=\frac{i}{2\pi}\int\limits_{\Sigma_{1}}\frac{du\wedge
d\bar{u}}{|u|^{2}|u-1|^{2}}=\frac{i}{2\pi}\int\limits_{\Sigma_{1}}d\left(\log\frac{u}{u-1}\frac{d\bar{u}}{\bar{u}(\bar{u}-1)}\right)=\\\
\frac{i}{2\pi}\int_{\varphi\in[\frac{\pi}{3},\frac{5\pi}{3}]}\left[\log\frac{u}{u-1}\frac{d\bar{u}}{\bar{u}(1-\bar{u})}\Big{|}_{u=e^{i\varphi}}+\log\frac{u}{u-1}\frac{d\bar{u}}{\bar{u}(1-\bar{u})}\Big{|}_{u=1-e^{i\varphi}}\right]$
(118)

which leads to

$\int\limits_{|u|>1,|u-1|>1}\frac{d^{2}u}{\pi}\frac{1}{|u|^{2}|u-1|^{2}}=\int\limits_{\pi/3}^{5\pi/3}\frac{d\varphi}{\pi}\frac{\log(1-e^{-i\varphi})}{1-e^{-i\varphi}}=\\\
\int\limits_{\pi/3}^{5\pi/3}\frac{d\varphi}{2\pi
i}(\cot\frac{\varphi}{2}+i)\left(\log\left(2\sin\frac{\varphi}{2}\right)+i\frac{\pi-\varphi}{2}\right)=\int\limits_{\pi/3}^{5\pi/3}\frac{d\varphi}{\pi}\log\left(2\sin\frac{\varphi}{2}\right)\approx
0.64$ (119)

The integral over $\Sigma_{1}^{*}$ can be computed in a similar manner, or
directly in the polar coordinates

$\int\limits_{\Sigma^{*}_{1}}\frac{d^{2}u}{|u|^{2}}=2\int\limits_{0}^{\pi/3}d\theta\int\limits_{1}^{2\cos\theta}\frac{dr}{r}=2\int\limits_{0}^{\pi/3}d\theta\log\left(2\cos\theta\right)=\int\limits_{\pi/3}^{\pi}d\theta\log\left(2\sin\frac{\theta}{2}\right).$
(120)

Using this result and (118) we conclude that

$2\int\limits_{\Sigma_{1}}\frac{d^{2}u}{\pi}\frac{1}{|u|^{2}|u-1|^{2}}-4\int\limits_{\Sigma_{1}^{*}}\frac{d^{2}u}{\pi}\frac{1}{|u|^{2}}=0,$
(121)

which is equivalent to

$I_{1}(\epsilon)=\log^{2}\frac{R^{2}}{\epsilon^{2}}+o(\epsilon/R).$ (122)

Next we turn into the integral

$I_{2}(\epsilon)=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{z_{2}^{2}\bar{z}_{1}(\bar{z}_{2}-\bar{z}_{1})}$
(123)

Computing derivative in the same way as previously we obtain

$-\epsilon\frac{\partial
I_{2}(\epsilon)}{\partial\epsilon}=2\int\limits_{\Sigma_{1}}\frac{d^{2}u}{\pi}\left(\frac{2}{u^{2}(\bar{u}-1)}+\frac{1}{\bar{u}(1-\bar{u})}\right)+O(\epsilon/R).$
(124)

Here we have replaced $\hat{\Sigma}_{1}$ to $\Sigma_{1}$ defined in (118) (and
correction $O(\epsilon/R)$) because integrals are now IR convergent. Applying
Stokes’s theorem we get

$\int\limits_{\Sigma_{1}}\frac{d^{2}u}{\pi}\left(\frac{2}{u^{2}(\bar{u}-1)}+\frac{1}{\bar{u}(1-\bar{u})}\right)=\int\limits_{\Sigma_{1}}d\left[\left(\frac{2}{u(1-\bar{u})}+\frac{u}{\bar{u}(1-\bar{u})}\right)\frac{d\bar{u}}{2\pi
i}\right]\\\
=\int\limits_{\pi/3}^{5\pi/3}d\varphi\frac{3i-5\cot\frac{\varphi}{2}+8\sin\varphi}{4\pi
i}=1.$ (125)

This gives the following estimate

$I_{2}(\epsilon)=\log\frac{R^{2}}{\epsilon^{2}}+o(\epsilon/R).$ (126)

The next integral to compute is

$I_{3}(\epsilon)=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{z_{1}^{2}\bar{z}_{2}^{2}}.$
(127)

Computing $\epsilon$ derivative we obtain

$-\epsilon\frac{\partial
I_{3}(\epsilon)}{\partial\epsilon}=2\int\limits_{\Sigma_{1}}\frac{d^{2}u}{\pi}\left(\frac{2}{u^{2}}+\frac{1}{(u-1)^{2}\bar{u}^{2}}\right)=2\int\limits_{\partial\Sigma_{1}}\frac{d\bar{u}}{2\pi
i}\left(\frac{2}{u}+\frac{1}{(u-1)\bar{u}^{2}}\right)=0$ (128)

Here we have applied Stokes’s theorem and obtained zero after integration over
the angle. Therefore we conclude

$I_{3}(\epsilon)=o(\epsilon/R).$ (129)

Finally, we can compute the following integral

$I_{4}(\epsilon)=\int\limits_{\Sigma_{\epsilon}}\frac{d^{2}z_{1}}{\pi}\frac{d^{2}z_{2}}{\pi}\frac{1}{z_{1}^{2}\bar{z}_{1}^{2}}.$
(130)

We compute difference

$\delta
I_{4}(\epsilon)=I_{4}(\epsilon)-\int\limits_{R\geq|z_{1}|,|z_{2}|\geq\epsilon}\frac{d^{2}z_{1}}{\pi}\int\frac{d^{2}z_{2}}{\pi}\frac{1}{z_{1}^{2}\bar{z}_{1}^{2}}=-\int\limits_{|z_{1}|\geq\epsilon}\frac{d^{2}z_{1}}{\pi}\int\limits_{|z_{2}|\geq\epsilon}\frac{d^{2}z_{2}}{\pi}\frac{\theta(\epsilon-|z_{1}-z_{2}|)}{z_{1}^{2}\bar{z}_{1}^{2}}+O(\epsilon/R)$
(131)

The obtained difference is IR convergent so we have dropped the upper limit.
Now we can rescale the whole expression

$\delta I_{4}(\epsilon)=-\int\limits_{|z_{1}|\geq
1}\frac{d^{2}z_{1}}{\pi}\int\limits_{|z_{2}|\geq
1}\frac{d^{2}z_{2}}{\pi}\frac{\theta(1-|z_{1}-z_{2}|)}{z_{1}^{2}\bar{z}_{1}^{2}}+O(\epsilon/R)$
(132)

Therefore

$I_{4}(\epsilon)=\frac{R^{2}}{\epsilon^{2}}+O(1).$ (133)

##  Appendix B Current-Current deformations

In this appendix, we provide a comparison of the general formulas for
$\beta_{2}$ and $\beta_{3}$ (Eqs. (68),(69)) with the exact all-order answers
given in PhysRevLett.86.4753 ; Ludwig2003 , for the group $SU(2)$ on the level
$k$. More precisely, following PhysRevLett.86.4753  the current-current
deformation is parametrized as

$\delta S=\int
g_{\perp}\left(J^{+}\bar{J}^{-}+J^{-}\bar{J}^{+}\right)+g_{\parallel}J_{3}\bar{J}_{3}\
,$ (134)

with the current normalization as follows

$J_{3}(z)J^{\pm}(0)\sim\pm\frac{1}{z}J^{\pm}(0),\qquad
J^{+}(z)J^{-}(0)\sim\frac{k}{2}\frac{1}{z^{2}}+\frac{1}{z}J_{3}(0),\qquad
J_{3}(z)J_{3}(0)\sim\frac{k}{2}\frac{1}{z^{2}}\,.$ (135)

We require first two orders of the claimed exact beta functions there

$\displaystyle\beta_{g_{\perp}}=\frac{g_{\perp}\left(g_{\parallel}-kg^{2}_{\perp}/4\right)}{\left(1-k^{2}g^{2}_{\perp}/16\right)\left(1+kg_{\parallel}/4\right)}$
$\displaystyle\approx$ $\displaystyle
g_{\perp}g_{\parallel}-\frac{k}{4}g^{2}_{\perp}g_{\parallel}-\frac{k}{4}g_{\perp}g^{2}_{\parallel}+O(g^{4}),$
$\displaystyle\beta_{g_{\parallel}}=\frac{g^{2}_{\perp}{\left(1-kg_{\parallel}/4\right)}^{2}}{{\left(1-k^{2}g^{2}_{\perp}/16\right)}^{2}}$
$\displaystyle\approx$ $\displaystyle
g^{2}_{\perp}-\frac{k}{2}g^{2}_{\perp}g_{\parallel}+O(g^{4}).$ (136)

In our notations for $J_{A}=(J^{-},J^{+},J_{3})$, we obtain

$G^{A\bar{A}}=\left(\begin{array}[]{ccc}0&g_{\perp}&0\\\ g_{\perp}&0&0\\\
0&0&g_{\parallel}\\\
\end{array}\right),\qquad\eta_{AB}=\frac{k}{2}\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&1\\\ \end{array}\right)$ (137)
$f^{3}_{AB}=\left(\begin{array}[]{ccc}0&1&0\\\ -1&0&0\\\ 0&0&0\\\
\end{array}\right),\qquad f_{AB}^{+}=\left(\begin{array}[]{ccc}0&0&-1\\\
0&0&0\\\ 1&0&0\\\ \end{array}\right),\qquad
f^{-}_{AB}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\ 0&-1&0\\\
\end{array}\right).$ (138)

This gives

$\beta_{2}=\frac{1}{2}G^{A\bar{A}}G^{B\bar{B}}f^{C}_{AB}f^{\bar{C}}_{\bar{A}\bar{B}}=-\left(\begin{array}[]{ccc}0&g_{\perp}g_{\parallel}&0\\\
g_{\perp}g_{\parallel}&0&0\\\ 0&0&g_{\perp}^{2}\\\ \end{array}\right)$ (139)
$\beta_{3}=\frac{1}{2}G^{A\bar{A}}G^{B\bar{B}}G^{C\bar{C}}\left(f_{AC}^{D}f_{BD}^{E}\eta_{\bar{A}\bar{B}}+{\rm
c.c.}\right)=\frac{kg_{\perp}}{2}\left(\begin{array}[]{ccc}0&g_{\perp}^{2}+g_{\parallel}^{2}&0\\\
g_{\perp}^{2}+g_{\parallel}^{2}&0&0\\\ 0&0&2g_{\perp}g_{\parallel}\\\
\end{array}\right)$ (140)

which coincides with the known answer if the coefficient in front of the
action is taken to be $\alpha^{\prime}=-1/2$.

##  Appendix C Current correlation function

Here we discuss the three-point correlation functions of the currents. For
convenience, we recall two-point correlation functions

$\langle J_{A}(z)J_{B}(0)\rangle=-\frac{\eta_{AB}}{z^{2}},\qquad\langle
J_{A}(z)I^{B}(0)\rangle=\frac{\delta_{A}^{B}}{z^{2}},\qquad\langle
I^{B}(z)I^{C}(0)\rangle=0.$ (141)

Here, in principle, $\eta_{AB}$ can be a function of the target fields. Recall
that the dependence of the target fields dependence is understood as described
in (11). As for the holomorphic fields one can integrate only over the half of
the fields.

Notice that even though the one-point function of the current vanishes
$\langle J_{A}\rangle=0$, the correlator with the non-constant function
$F(z)=F(\varphi(z))$ reads as

$\langle J_{A}(z)F(w)\rangle=\frac{V_{A}^{k}\partial_{k}F}{z-w}.$ (142)

The three-point functions can be computed by the following trick. First we
identically present

$\langle
I^{C}(x)J_{A}(z)J_{B}(w)\rangle=\oint\limits_{C_{x}}\frac{dy}{y-x}\langle
I^{C}(y)J_{A}(z)J_{B}(w)\rangle$ (143)

where integration is over a small contour around $x$. Since the correlation
function is $O(1/y^{2})$ we can deform the contour such that it encircles
points $z$ and $w$, inside the contours we can use OPE, which gives us the
following

$\langle I^{C}(x)J_{A}(z)J_{B}(w)\rangle=-\oint\limits_{C_{z}\cup
C_{w}}\frac{dy}{y-x}\langle
I^{C}(y)J_{A}(z)J_{B}(w)\rangle=\frac{f_{AB}^{C}}{(x-z)(x-w)(z-w)}$ (144)

The four-point correlation function (80) can be obtained in the similar
manner.

It is interesting that the correlation function (144) is as if currents are
primary operators, even though we have not demanded it. The non-primary
effects can be seen on the following correlation function (which luckily is
not required for the beta-function computation)

$\langle
J_{A}(x)J_{B}(y)J_{C}(z)\rangle=\frac{\gamma_{ABC}}{(y-z)^{2}(y-x)}+\frac{\gamma_{ACB}}{(y-z)^{2}(z-x)}+\frac{\gamma_{CAB}+\gamma_{CBA}}{(x-y)^{2}(y-z)}+\frac{\gamma_{BAC}+\gamma_{BCA}}{(x-z)^{2}(z-y)}$
(145)

Here

$\gamma_{ABC}=f_{AB}^{D}\eta_{DC}+\partial_{m}\partial_{a}V_{A}^{k}\partial_{k}V_{B}^{a}V_{C}^{m}.$
(146)

The symmetries of the correlation function are respected due to the following
identities

$\gamma_{CAB}+\gamma_{CBA}=\gamma_{ABC}+\gamma_{BAC}=V_{C}^{k}\partial_{k}\eta_{AB}$
(147)

Notice that contrary to the other current three-point correlation functions
(144) this one does not have a form predicted for $(1,0)$ operators. We see
that the form is restores for constant $\eta_{AB}$. Indeed, in this case
$\gamma_{ABC}$ is a completely antisymmetric tensor

$\gamma_{ABC}=\frac{1}{3!}\gamma_{[ABC]}.$ (148)

and

$\langle
J_{A}(x)J_{B}(y)J_{C}(z)\rangle=\frac{\gamma_{ABC}}{(z-y)(x-z)(x-y)}.$ (149)

##  Appendix D Second loop in non-supersymmetric $\mathds{C}P^{1}$ model is an
infrared effect

In this Appendix we argue that in perturbation theory the second loop in the
$\beta$ function of the non-supersymmetric $\mathds{C}P^{1}$ model would be
absent in the Wilsonean OPE because it is due to an infrared effect. The
argument below is from shU , page 674, Ex. 6.3.2.

Let us start from supersymmetric ${\mathcal{N}}=(2,2)$ $\mathds{C}P^{1}$
model. For our purposes we will need the following:

$\displaystyle{\mathcal{L}}$ $\displaystyle=$ $\displaystyle
G\left[\partial_{\mu}\phi^{\dagger}\,\partial^{\mu}\phi+i\bar{\psi}\gamma^{\mu}\left(\partial_{\mu}+\Gamma\partial_{\mu}\phi\right)\psi+...\right]$
(150) $\displaystyle=$ $\displaystyle
G\left[\partial_{\mu}\phi^{\dagger}\,\partial^{\mu}\phi+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-i\frac{2}{1+\phi^{\dagger}\phi}\,\,\phi^{\dagger}\partial_{\mu}\phi\,\bar{\psi}\gamma^{\mu}\psi+...\right]$

where

$G=G_{1\bar{1}}=\frac{2}{g^{2}}\,\frac{1}{(1+\phi^{\dagger}\phi)^{2}}\,,$
(151)

the dots stand for a four-fermion term irrelevant for our present purposes,
and $\psi$ is a Dirac spinor.

The overall structure of the target space is completely fixed by geometry of
$\mathds{C}P^{1}$. Therefore, to determine renormalization of $1/g^{2}$ in the
metric it is sufficient to analyze the vicinity of origin in the target space,
i.e. $\phi\approx 0$ (see shU , Sec. 6.3.3, page 264). In other words, we can
focus on the renormalization of the structure
$\partial_{\mu}\phi^{\dagger}\,\partial^{\mu}\phi$ using the last term in
(150) as the interaction vertex, namely,

$-iG\,\frac{2}{1+\phi^{\dagger}\phi}\,\,\phi^{\dagger}\partial_{\mu}\phi\,\bar{\psi}\gamma^{\mu}\psi\,.$
(152)

The only two-loop graph which has exactly the needed structure
$\partial_{\mu}\phi^{\dagger}\,\partial^{\mu}\phi$ is depicted in Fig. 1. It
has three propagators: two fermion and one boson. The denominator in (152) can
be ignored while

Figure 1: Two-loop contribution in the $\beta$ function of supersymmetric
CP(1) model.

the $\phi$ fields without derivatives in the numerator in (152) and its
complex conjugated are convoluted into the boson propagator.

Next, we split the calculation in two parts. First we evaluate the fermion
loop depicted in Fig. 2. In $\mathds{C}P^{1}$ this is the same famous loop
which appears in two-dimensional Schwinger model (Fig. 2).

Figure 2: Fermion loop subdiagram.

Using

$-i\,\frac{1}{2\pi}\,\frac{x_{\mu}\gamma^{\mu}}{x^{2}}\,\,\stackrel{{\scriptstyle\rm
Fourier}}{{\longleftrightarrow}}\,\,\frac{ip_{\mu}\gamma^{\mu}}{p^{2}}$ (153)

for the fermion propagator we obtain for the graph in Fig. 2

$\frac{1}{2\pi^{2}}\,\frac{1}{x^{2}}\left(g^{\mu\nu}-\frac{2x^{\mu}x^{\nu}}{x^{2}}\right)\,\,\stackrel{{\scriptstyle\rm
Fourier}}{{\longleftrightarrow}}\,\,\frac{i}{\pi}\left(\frac{p^{\mu}p^{\nu}}{p^{2}}-g^{\mu\nu}\right).$
(154)

This is the famous Schwinger infrared effect.

Returning to the diagram in Fig. 1 we convince ourselves that the only actions
still to be performed are to input the boson propagator $i/p^{2}$ and insert
$(2/g^{2})^{2}$ for two vertices, $(g^{2}/2)^{3}$ for three propagators,
$-2^{2}$ reflecting the factor of 2 in (152) and $-2$ in its complex
conjugated, and the overall factor $-i$. The final result for the diagram in
Fig. 1 is

$\big{(}\partial_{\mu}\phi^{\dagger}\,\partial^{\mu}\phi\big{)}\,\frac{g^{2}}{2\pi^{2}}\,\log\frac{M_{0}}{\mu}$
(155)

Since there is no second loop in ${\mathcal{N}}=(2,2)$ $\mathds{C}P^{1}$
model, this contribution must be canceled by the sum of all purely bosonic
two-loop graphs. Hence, the two-loop renormalization in non-supersymmetric
(purely bosonic) $\mathds{C}P^{1}$ model is

$\delta{\mathcal{L}}=...-\big{(}\partial_{\mu}\phi^{\dagger}\,\partial^{\mu}\phi\big{)}\,\frac{g^{2}}{2\pi^{2}}\,\log\frac{M_{0}}{\mu}+...$
(156)

where the dots stand for irrelevant contributions, $M_{0}$ is the UV cutoff
and $\mu$ is the normalization point. Comparing $\delta{\mathcal{L}}$ above
with $G\big{(}\partial_{\mu}\phi^{\dagger}\,\partial^{\mu}\phi\big{)}$ in
(150) we arrive at the two-loop renormalization of the coupling constant in
$\mathds{C}P^{1}$,

$\delta\left(\frac{1}{g^{2}}\right)_{non\,\,{\rm
SUSY\,}\mathds{C}P^{1}}=-\frac{g^{2}}{4\pi^{2}}\,\log\frac{M_{0}}{\mu}\,.$
(157)

This result is certainly well-known in the literature Grah . What was not
known is the fact that only one-loop in the graph is truly UV while the other
comes from the IR and, therefore, this contribution would not be counted in
the Wilsonean OPE.

The same conclusion is achieved from the instanton analysis.

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