aainstitutetext: HSE University, 6 Usacheva str., Moscow 119048,
Russiabbinstitutetext: P.N. Lebedev Physical Institute of the Russian Academy
of Sciences, 53 Leninskiy pr., Moscow 119991, Russiaccinstitutetext: Landau
Institute for Theoretical Physics, 1A Akademika Semenova av., Chernogolovka
142432, Russiaddinstitutetext: Center for Advanced Studies, Skolkovo Institute
of Science and Technology, 1 Nobel str., Moscow 143026, Russia

# On loop corrections to integrable $2D$
sigma model backgrounds

Mikhail Alfimov c,d and Alexey Litvinov<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study regularization scheme dependence of $\beta$-function for sigma models
with two-dimensional target space. Working within four-loop approximation, we
conjecture the scheme in which the $\beta$-function retains only two tensor
structures up to certain terms containing $\zeta_{3}$. Using this scheme, we
provide explicit solutions to RG flow equation corresponding to Yang-Baxter-
and $\lambda$-deformed $SU(2)/U(1)$ sigma models, for which these terms
disappear.

##  1 Introduction

$\beta$-functions govern scale dependence of coupling constants in Quantum
Field Theory. In the textbook example of $\varphi^{4}$ theory in $4$
dimensions the coupling constant $g$ runs at two loops as

$\dot{g}=-\beta(g)\,,\quad\beta(g)=3g^{2}-\frac{17g^{3}}{3}+\ldots$ (1)

The terms depicted by $\dots$ in (1) correspond to higher loop corrections.
They have been computed (see Kompaniets:2016hct ) up to six loops (i.e. to the
order $g^{7}$) using dimensional regularization in the minimal subtraction
scheme tHooft:1972tcz ; tHooft:1973mfk . Higher loop coefficients in (1) are
scheme dependent. Change of regularization scheme corresponds to the
“coordinate change in the space of couplings”

$g\rightarrow\tilde{g}(g)=g+\xi_{1}g^{2}+\xi_{2}g^{3}+\ldots\,,$ (2)

where the parameters $\xi_{k}$ are the parameters of the new scheme. The
$\beta$-function transforms as a vector field

$\dot{\tilde{g}}=\frac{\partial\tilde{g}}{\partial
g}\dot{g}\implies\tilde{\beta}(\tilde{g})=\left(\frac{\partial\tilde{g}(g)}{\partial
g}\right)^{-1}\beta(\tilde{g}(g))\,.$ (3)

For the $\beta$-function, which starts from $g^{2}$ (corresponding to
classically marginal coupling constants) one has

$\beta(g)=A_{1}g^{2}+A_{2}g^{3}+A_{3}g^{4}+\ldots\implies\tilde{\beta}(\tilde{g})=A_{1}\tilde{g}^{2}+A_{2}\tilde{g}^{3}+(A_{3}+A_{2}\xi_{1}+A_{1}(\xi_{1}^{2}-\xi_{2}))\tilde{g}^{4}+\ldots$
(4)

Tuning the scheme parameters $\xi_{k}$, one can transform the $\beta$-function
to some _normal form_. In particular, one can define the scheme in which the
$\beta$-function is two-loop exact.

In case one considers theories with several coupling constants the situation
becomes more involved. If $(g_{1},\dots,g_{n})$ correspond to local
coordinates in the vicinity of some fixed point the RG (Renormalization Group)
flow equations can be written in the form Wilson:1973jj

$\dot{g}_{l}=(D-\Delta_{l})g_{l}+a^{ij}_{l}g_{i}g_{j}+b^{ijk}_{l}g_{i}g_{j}g_{k}+\ldots\,,$
(5)

where $\Delta_{l}$’s are dimensions of the operators corresponding to $g_{l}$.
All the non-linear terms in (5) can be exterminated by the appropriate
transformation of the coupling constants

$g_{l}\rightarrow
g_{l}+\xi_{1}^{ij}g_{i}g_{j}+\xi_{2}^{ijk}g_{i}g_{j}g_{k}+\ldots\,,$ (6)

provided that the so called resonance terms are absent (see for example book-
Ilyashenko ). In this case the RG flow equations (5) take very simple form:
all $\beta$-functions are reduced to their linear parts (provided that all
$\Delta_{l}\neq D$). This statement is known as (formal version of) Poincaré-
Dulac theorem.

One more example of using of Poincaré-Dulac theorem besides the one considered
above, with one marginal coupling, has been considered by Al. Zamolodchikov in
Zamolodchikov:1995xk  who applied PD theorem to the sine-Gordon theory near
the BKT transition point described by two classically marginal coupling
constants $(g_{\parallel},g_{\perp})$. He showed that there is a scheme in
which RG flow equations have the following exact form

$\dot{g}_{\parallel}=\frac{g_{\perp}^{2}}{1+\frac{g_{\parallel}}{2}}\,,\quad\dot{g}_{\perp}=\frac{g_{\parallel}g_{\perp}}{1+\frac{g_{\parallel}}{2}}\,.$
(7)

This scheme later has been successfully used in Doyon:2002ui ; Lukyanov:2002fg
for calculation of correlation functions.

In these notes we study the question of scheme dependence for two-dimensional
sigma models

$S[\boldsymbol{X}]=\frac{1}{4\pi}\int
G_{ij}(\boldsymbol{X})\partial_{a}X^{i}\partial_{a}X^{j}\,d^{2}\boldsymbol{x}\,.$
(8)

Here $\boldsymbol{X}=(X^{1},\dots,X^{D})$ are treated as coordinates on some
$D$-dimensional target manifold. The RG flow equations for the metric
$G_{ij}(\boldsymbol{X})$ have the form

$\dot{G}_{ij}+\nabla_{i}V_{j}+\nabla_{j}V_{i}=-\beta_{ij}(G)\,,$ (9)

where $V_{j}$ correspond to field renormalization terms. The $\beta$-function
$\beta_{ij}(G)$ admits covariant loop expansion

$\beta_{ij}(G)=\beta_{ij}^{(1)}(G)+\beta_{ij}^{(2)}(G)+\beta_{ij}^{(3)}(G)+\ldots\,,$
(10)

where the $L$-th loop order $\beta$-function coefficient $\beta_{ij}^{(L)}$
belongs to the finite dimensional space $\mathcal{T}_{L}$ of tensors with
given scaling properties. Namely, if we assume that the metric is proportional
to the inverse of the Planck constant

$G_{ij}\sim\hbar^{-1}\implies
G^{ij}\sim\hbar,\;\Gamma^{k}_{ij}\sim\hbar^{0},\;\nabla_{i}\sim\hbar^{0},\;R_{ijk}^{\quad
l}\sim\hbar^{0},\;R_{ij}\sim\hbar^{0}\quad\text{etc}.\,,$ (11)

then the $L$-th loop order coefficient is a linear combination of tensors

$\beta_{ij}^{(L)}=\sum_{k=1}^{\textrm{dim}\,\mathcal{T}_{L}}b_{k}^{(L)}\beta_{ij}^{k,L}(G)\,,\quad\beta_{ij}^{k,L}(G)\in\mathcal{T}_{L}\,,\quad\beta_{ij}^{k,L}(G)\sim\hbar^{L-1}\,.$
(12)

The coefficients $b_{k}^{(L)}$ were computed in the MS (Minimal Subtraction)
scheme up to four loops. In particular, the one-loop $\beta$-function is
proportional to the Ricci curvature Ecker:1971xko ; Friedan:1980jm

$\beta_{ij}^{(1)}=R_{ij}\,.$ (13)

Higher loop coefficients have been calculated: two-loop in Friedan:1980jm

$\beta_{ij}^{(2)}=\frac{1}{2}R_{iklm}R_{j}^{\,\,klm}\,,$ (14)

three-loop in Graham:1987ep ; Foakes:1987ij

$\beta_{ij}^{(3)}=\frac{1}{8}\nabla_{k}R_{ilmn}\nabla^{k}R_{j}^{\,\,lmn}-\frac{1}{16}\nabla_{i}R_{klmn}\nabla_{j}R^{klmn}-\\\
-\frac{1}{2}R_{imnk}R_{jpq}^{\quad
k}R^{mqnp}-\frac{3}{8}R_{iklj}R^{kmnp}R^{l}_{\,\,mnp}\,.$ (15)

Also the four-loop result has been obtained in Jack:1989vp , but it is too
long to be presented here111Below we present it in particular case of $D=2$
(see (19))..

The higher loop coefficients $\beta_{ij}^{(L)}$ for $L>1$ are scheme
dependent. If one stays within covariant approach, then different schemes are
related to each other by covariant metric redefinitions

$G_{ij}\rightarrow G_{ij}+\sum_{k=0}^{\infty}G_{ij}^{(k)}\,,$ (16)

where $G_{ij}^{(k)}$ is of the order $\hbar^{k}$. Using this freedom and
possibility to modify the vector $V_{i}$ in (9) one can try to bring the
$\beta$-function to some normal form. This problem is highly non-trivial and
unlikely to be solved in full generality since even the dimensions of the
spaces $\mathcal{T}_{k}$ are not known in general. Moreover, the analysis
performed at lower orders Metsaev:1987zx  shows that there are tensor
structures in the $\beta$-function beyond $L=1$ which are scheme invariant and
it is highly likely that such structures persist at higher loops as well.

Some hints to the solution of this problem can be obtained by considering
integrable sigma models. In particular it has been noticed that some of the
classically integrable sigma models are one-loop renormalizable Fateev:2019xuq
and that it might be a signal for quantum integrability. This question has
been put forward in Hoare:2019ark  and later in Hoare:2019mcc ; Hoare:2020fye
; Levine:2021fof , where it has been found that classical backgrounds require
certain $\hbar$ corrections to persist renormalizability at higher loops. In
particular, in the original paper Hoare:2019ark  the authors studied two
integrable $2D$ backgrounds, namely the sausage model Fateev:1992tk  (in
modern language the Yang-Baxter-deformed $SU(2)/U(1)$ sigma model) and the
$\lambda$-deformed $SU(2)/U(1)$ sigma model Sfetsos:2013wia ;
Hollowood:2014rla . In both cases they found $\hbar$ corrections to the
classical backgrounds which provide solution to (9) within three-loop
approximation for the sausage model and within two-loop approximation for the
$\lambda$ model. In our paper we improve the results of Hoare:2019ark .
Namely, working within four-loop approximation, we provide some universal
scheme in which both backgrounds receive corrections of the order $\hbar^{0}$
only. Moreover, both $\hbar$-corrected backgrounds correctly reproduce the UV
behaviour prescribed by dual Toda field theory corresponding to integrable
perturbations of sine-Liouville CFT Fateev:2017mug .

This paper is organized as follows. In the section 2 we study in detail the
four-loop $\beta$-function in $2D$ sigma models. We show that there exists a
scheme in which the $\beta$-function contains only two tensor structures for
certain $\hbar$-corrected versions of the sausage and $\lambda$-deformed
$SU(2)/U(1)$ sigma models, which solve the four-loop RG flow equation. The
curious property of these solutions is that they are “two-loop exact”, i.e.
contain only $\hbar^{0}$ corrections to the corresponding classical
backgrounds. In the section 3 we consider the UV expansion of our solutions.
In the section 4 we provide some remarks and directions for future work.

##  2 $\beta$-function for sigma models with $2D$ target space

Present section is devoted to the study of how the $\beta$-function changes
under covariant metric redefinitions. This is done to find if there is a
particular scheme in which it has a simple form, and then to search for the
metrics, which solve the RG flow equation in this scheme.

In the case of two-dimensional target space $D=2$ all formulae are simplified
since the Riemann tensor $R_{ijkl}$ can be algebraically expressed in terms of
the scalar curvature $R$ and the metric

$R_{ijkl}=\frac{R}{2}\left(G_{ik}G_{jl}-G_{il}G_{jk}\right)\,.$ (17)

Therefore all tensors in the $\beta$-function are expressed in terms of the
metric, the scalar curvature $R$ and its covariant derivative. Substituting
(17) into first four loop coefficients in the MS scheme (see (13) and (15)),
one finds

$\beta_{ij}^{(1)}=\frac{1}{2}RG_{ij}\,,\quad\beta_{ij}^{(2)}=\frac{1}{4}R^{2}G_{ij}\,,\quad\beta_{ij}^{(3)}=\left(\frac{5}{32}R^{3}+\frac{1}{16}\big{(}\nabla
R\big{)}^{2}\right)G_{ij}-\frac{1}{16}\nabla_{i}R\nabla_{j}R$ (18)

and

$\beta_{ij}^{(4)}=\left(\frac{23}{192}R^{4}+\frac{2+\zeta(3)}{32}R^{2}\nabla^{2}R+\right.\\\
\left.+\frac{41+12\zeta(3)}{192}R\big{(}\nabla
R\big{)}^{2}+\frac{1}{192}\big{(}\nabla^{2}R\big{)}^{2}+\frac{1}{192}\big{(}\nabla_{k}\nabla_{l}R\big{)}^{2}\right)G_{ij}-\\\
-\frac{\zeta(3)}{48}R^{2}\nabla_{i}\nabla_{j}R-\frac{25+8\zeta(3)}{192}R\nabla_{i}R\nabla_{j}R-\frac{1}{96}\big{(}\nabla^{2}R\big{)}\nabla_{i}\nabla_{j}R\,.$
(19)

Then the first three orders of the most general covariant metric redefinition
(16) take the form

$\displaystyle G_{ij}^{(0)}$ $\displaystyle=c_{1}RG_{ij}\,,$ $\displaystyle
G_{ij}^{(1)}$
$\displaystyle=\left(c_{2}R^{2}+c_{3}\nabla^{2}R\right)G_{ij}+c_{4}\nabla_{i}\nabla_{j}R\,,$
$\displaystyle G_{ij}^{(2)}$
$\displaystyle=\left(c_{5}R^{3}+c_{6}\big{(}\nabla
R\big{)}^{2}+c_{7}R\nabla^{2}R+c_{8}\nabla^{2}\nabla^{2}R\right)G_{ij}+c_{9}\nabla_{i}R\nabla_{j}R+c_{10}R\nabla_{i}\nabla_{j}R+$
$\displaystyle+c_{11}\nabla_{i}\nabla_{j}\nabla^{2}R\,.$ (20)

In appendix A we show that one can tune the scheme coefficients $c_{p}$ in
such a way, that the $\beta$-function takes the form

$\beta_{ij}=\left(\frac{R}{2}+\frac{R^{2}}{4}+\frac{3R^{3}}{16}+\frac{5R^{4}}{32}+\frac{2+\zeta_{3}}{64}\nabla^{2}\left(R^{3}+2R\nabla^{2}R-\frac{1}{2}\nabla^{2}R^{2}\right)+\dots\right)G_{ij}-\\\
-\left(\frac{1}{16}+\frac{5R}{32}+\dots\right)\nabla_{i}R\nabla_{j}R+\ldots\,,$
(21)

where $\ldots$ correspond to terms coming from the $5$-th and higher loops and
we omitted the tildes which denote the quantities in the new scheme.

It is important to emphasise that the terms in the fourth-loop
$\beta$-function coefficient proportional to $(2+\zeta_{3})$ cannot be
exterminated completely by the change of scheme. This phenomenon is well known
and has been noticed first in the supersymmetric case Grisaru:1986px ;
Grisaru:1986dk . Moreover there is some evidence that such terms persist at
higher loop orders Grisaru:1986wj ; Jack:1992ce . On the other hand, one can
argue that physically acceptable backgrounds should not contain transcendental
terms like $\zeta_{n}$. This is confirmed by the exact form of the cigar
metric Dijkgraaf:1991ba  and by the all-loop metric for the sausage model
found in Hoare:2019ark . Both of them do not contain $\zeta_{n}$ terms. Thus
the presence of the $\zeta_{n}$ terms in the $\beta$-function and their
absence in explicit solutions might seem puzzling.

We have found that the resolution of this puzzle in the fourth loop order lies
in the fact that all metrics mentioned above (and also $\lambda$-deformed
$SU(2)/U(1)$ metric) satisfy the following special relation

$\nabla^{2}\left(R^{3}+2R\nabla^{2}R-\frac{1}{2}\nabla^{2}R^{2}\right)=O(\hbar^{5})\,,$
(22)

and hence the terms in (21) proportional to $(2+\zeta_{3})$ can be dropped. We
note that in this case only two tensor structures in the $\beta$-function are
present, thus suggesting the general ansatz222We can not resist to mention
that the sequence in (21) (with the terms $\sim(2+\zeta_{3})$ dropped) can be
easily extrapolated to
$\beta_{ij}=\frac{RG_{ij}}{2(1-R)^{\frac{1}{2}}}-\frac{1}{16(1-R)^{\frac{5}{2}}}\nabla_{i}R\nabla_{j}R\,.$
(*) It is possible that this “exact” expression for $\beta_{ij}$ as well as
the generic ansatz (23) could be broken by five-loop calculations (in
particular by the presence of $\zeta_{4}$ terms), but checking this is a
tedious task going beyond the scope of this paper. However, we will find that
(* ‣ 2) is confirmed indirectly by the fact that $\hbar$-corrected backgrounds
presented below (see (28) and (36)) solve not only the four-loop RG flow
equation, but also the “exact” equation as well. We will discuss the “exact”
expression (* ‣ 2) in the section 3.

$\beta_{ij}=A(R)G_{ij}+B(R)\nabla_{i}R\nabla_{j}R\,.$ (23)

We claim that the scheme in which the $\beta$-function has the form (21) (with
the terms $\sim(2+\zeta_{3})$ dropped) is natural because both classically
integrable backgrounds, $SU(2)/U(1)$ YB and $\lambda$-deformed models, admit
simple $\hbar$ extension. We study them separately in the next subsections.

###  2.1 Sausage model

It is convenient to define the sausage model as inhomogeneous YB-deformed
coset sigma model Klimcik:2008eq ; Delduc:2013fga  (also known as
$\eta$-deformed sigma model). The classical action of the YB-deformed
$SU(2)/U(1)$ sigma model is defined by Drinfel’d-Jimbo solution $\mathcal{R}$
to the modified YB equation and has the form

$\mathcal{S}=\frac{\kappa}{4\pi\hbar}\int\textrm{Tr}\left(\left(\mathbf{g}\partial_{+}\mathbf{g}^{-1}\right)^{(\textrm{c})}\,\frac{1}{1-i\kappa\mathcal{R}_{\mathbf{g}}\circ\mathrm{P}_{\textrm{c}}}\,\left(\mathbf{g}\partial_{-}\mathbf{g}^{-1}\right)^{(\textrm{c})}\right)d^{2}x\,,$
(24)

where $\mathbf{g}\in SU(2)$,
$\mathcal{R}_{\mathbf{g}}=\textrm{Ad}\,\mathbf{g}\circ\mathcal{R}\circ\textrm{Ad}\,\mathbf{g}^{-1}$
and $\mathrm{P}_{\textrm{c}}$ is the projector on the coset space. Using the
following parametrization of the coset elements

$\mathbf{g}^{-1}=e^{\frac{i\chi}{2}\sigma_{3}}e^{\frac{i\theta}{2}\sigma_{1}}\,,$
(25)

where $\sigma_{3}$ corresponds to $U(1)$, we arrive to the sausage metric

$ds^{2}=\frac{2\kappa}{\hbar}\frac{d\theta^{2}+\cos^{2}\theta
d\chi^{2}}{1-\kappa^{2}\sin^{2}\theta}\;.$ (26)

This theory is renormalizable at one loop Fateev:1992tk  provided that $\hbar$
does not run and $\kappa$ satisfies

$\dot{\kappa}=\frac{\hbar(\kappa^{2}-1)}{2}\,.$ (27)

The question of renormalizability of the sausage model at higher loops has
been first considered in Hoare:2019ark . It has been noticed in Hoare:2019ark
that renormalizability persists to higher loops subject to the addition of
$\hbar$ corrections to the original metric (26). The authors in Hoare:2019ark
proposed some corrected metric and checked within the three loop approximation
that there exists a scheme (different from ours (21)) in which their metric
solves the RG flow equation. We propose an alternative $\hbar$ completion of
the one-loop metric (26)

$ds^{2}=\frac{2\kappa}{\hbar}\frac{\left(1-\frac{\hbar\kappa\cos^{2}\theta}{1-\kappa^{2}\sin^{2}\theta}\right)d\theta^{2}+\cos^{2}\theta
d\chi^{2}}{1-\kappa^{2}\sin^{2}\theta}\,,$ (28)

which solves the four-loop RG equation (9) with the $\beta$-function (21)
provided that the coupling constant $\kappa$ satisfies the following flow
equation

$\dot{\kappa}=\frac{\hbar(\kappa^{2}-1)}{2}\left(1+\frac{\nu}{2}\hbar+\frac{3\nu^{2}-4}{8}\hbar^{2}+\frac{5\nu^{3}-12\nu}{16}\hbar^{3}+\ldots\right)\,,$
(29)

and the vector field has the form

$\boldsymbol{V}=\hbar\left\\{\frac{\kappa(\kappa^{2}-1)\sin
2\theta}{4(1-\kappa^{2}\sin^{2}\theta)^{2}},\frac{\cos^{2}\theta}{1-\kappa^{2}\sin^{2}\theta}\right\\}\left(1+\frac{\nu}{2}\hbar+\frac{3\nu^{2}-4}{8}\hbar^{2}+\frac{5\nu^{3}-12\nu}{16}\hbar^{3}+\ldots\right)\;,$
(30)

where $\nu=\kappa+\kappa^{-1}$.

We note that the metric (28) is two-loop exact and only the flow equation (29)
and the vector field (30) acquire higher loop corrections.

###  2.2 $\lambda$-deformed sigma model

Now we present $\hbar$-corrected $\lambda$-deformed $SU(2)/U(1)$ metric. The
classical ($1$-loop) metric is governed by the Lagrangian Sfetsos:2013wia ;
Hollowood:2014rla

$\mathcal{L}=\frac{1}{2\pi\hbar}\textrm{Tr}\left(-\frac{1}{2}\big{(}g^{-1}\partial
g\big{)}\big{(}g^{-1}\bar{\partial}g\big{)}+J\bar{A}-A\bar{J}+g^{-1}Ag\bar{A}-A\bar{A}+\big{(}1-\lambda^{-1}\big{)}A\mathbb{P}\bar{A}\right)+\mathcal{L}_{\text{WZ}}\,,$
(31)

where $g\in SU(2)$

$J=g^{-1}\cdot\partial g\,,\quad\bar{J}=\bar{\partial}g\cdot
g^{-1}\quad\text{and}\quad\mathbb{P}=\mathbb{P}_{\frac{SU(2)}{U(1)}}\,,$

$\mathcal{L}_{\text{WZ}}$ is the standard Wess-Zumino term, which is not
important in the present case and $\lambda$ is the deformation parameter.
Fixing the gauge symmetry as in Hoare:2019ark ; Hoare:2019mcc

$g=e^{i\alpha\sigma_{3}}e^{i\beta\sigma_{2}}\quad\text{with}\quad\cos\alpha=\sqrt{p^{2}+q^{2}}\,,\quad\tan\beta=\frac{p}{q}\,,$
(32)

and integrating over the auxiliary fields $(A,\bar{A})$, one arrives to the
metric

$ds^{2}=\frac{2}{\hbar}\frac{\kappa
dp^{2}+\kappa^{-1}dq^{2}}{1-p^{2}-q^{2}}\,,\quad\text{where}\quad\kappa=\frac{1-\lambda}{1+\lambda}\,.$
(33)

This metric is one-loop renormalizable Itsios:2014lca ; Appadu:2015nfa  with
$\kappa$ running according to (27) and the vector field given by

$V_{p}=\frac{p}{1-p^{2}-q^{2}}\,,\quad V_{q}=\frac{q}{1-p^{2}-q^{2}}\,.$ (34)

It has been demonstrated in Hoare:2019ark  that in order to ensure the two-
loop renormalizability of the model one has to add a particular quantum
correction coming from integration over the gauge field $(A,\bar{A})$

$ds^{2}\rightarrow
ds^{2}-\frac{1}{2}\left[d\log(1-p^{2}-q^{2})\right]^{2}=\frac{2}{\hbar}\left(\frac{\kappa
dp^{2}+\kappa^{-1}dq^{2}}{1-p^{2}-q^{2}}-\hbar\frac{\big{(}pdp+qdq\big{)}^{2}}{\big{(}1-p^{2}-q^{2}\big{)}^{2}}\right)$
(35)

and properly modify the vector field.

We propose another $\hbar$ completion of (33) which is also two-loop exact
similar to (28)

$ds^{2}=\frac{2}{\hbar}\left(\frac{\big{(}\kappa-\hbar\big{)}dp^{2}+\big{(}\kappa^{-1}-\hbar\big{)}dq^{2}}{1-p^{2}-q^{2}}-\hbar\frac{\big{(}pdp+qdq\big{)}^{2}}{\big{(}1-p^{2}-q^{2}\big{)}^{2}}\right)\,.$
(36)

This metric (36) solves the RG flow equation at four loops with the
$\beta$-function (21) provided that the coupling constant $\kappa$ satisfies
(29) and the vector field has the form

$V_{p}=\frac{p}{1-p^{2}-q^{2}}\left(1+V_{p}^{(1)}\hbar+V_{p}^{(2)}\hbar^{2}+V_{p}^{(3)}\hbar^{3}+\dots\right)\,,\quad
V_{q}=\\{p\rightarrow q,\kappa\rightarrow\kappa^{-1}\\}\,,$ (37)

where the coefficients $V_{p}^{(k)}$ appear in the expansion of the function

$\left(\frac{1-\hbar\kappa}{1-\hbar\kappa^{-1}}\right)^{\frac{1}{2}}\left(1-\frac{\hbar}{2\kappa}\frac{1-\left(\frac{1-\kappa^{2}}{1-\hbar\kappa}\right)q}{1-p^{2}-q^{2}}\right)=1+V_{p}^{(1)}\hbar+V_{p}^{(1)}\hbar+V_{p}^{(2)}\hbar^{2}+V_{p}^{(3)}\hbar^{3}+\ldots$
(38)

Having now two solutions (28) and (36) it becomes possible to analyze their UV
behaviour and check out how this behaviour correlates with the screening
charges which define the sausage and $\lambda$-model. To achieve this we
explain some curious observation concerning the form of the “exact”
$\beta$-function in the next section.

##  3 UV expansion

In the previous sections we managed to find two metrics (28) and (36) which
solve the RG flow equation at four loops with the $\beta$-function (21) and
vector fields (30) and (37) respectively and now would like to study the UV
(or $t\rightarrow-\infty$) behaviour of these metrics.

As we already mentioned in the section 2 the four-loop result for the
$\beta$-function (21) can be easily extrapolated to the following “exact”
expression

$\beta_{ij}=\frac{RG_{ij}}{2(1-R)^{\frac{1}{2}}}-\frac{1}{16(1-R)^{\frac{5}{2}}}\nabla_{i}R\nabla_{j}R\,.$
(39)

This form is remarkable in a sense that both solutions that we have found,
namely (28) and (36), go through it, but with modified vector field and flow
equations. More precisely, (28) and (36) solve the RG flow equation (9) with
the $\beta$-function given by (39) provided that $\kappa$ flows according to
the “exact” equation

$\dot{\kappa}=\frac{\hbar(\kappa^{2}-1)}{2\left((1-\hbar\kappa)(1-\hbar\kappa^{-1})\right)^{\frac{1}{2}}}$
(40)

and the vector field has the form

$\boldsymbol{V}=\frac{\hbar}{\sqrt{(1-\hbar\kappa)(1-\hbar\kappa^{-1})}}\left\\{\frac{\kappa(\kappa^{2}-1)\sin
2\theta}{4(1-\kappa^{2}\sin^{2}\theta)^{2}},\frac{\cos^{2}\theta}{1-\kappa^{2}\sin^{2}\theta}\right\\}\,,$
(41)

for the sausage metric (28) and

$V_{p}=\frac{p\left(\frac{1-\hbar\kappa}{1-\hbar\kappa^{-1}}\right)^{\frac{1}{2}}}{1-p^{2}-q^{2}}\left(1-\frac{\hbar}{2\kappa}\frac{1-\left(\frac{1-\kappa^{2}}{1-\hbar\kappa}\right)q}{1-p^{2}-q^{2}}\right)\,,\quad
V_{q}=\\{p\leftrightarrow q,\kappa\rightarrow\kappa^{-1}\\}$ (42)

for the $\lambda$-deformed one.

The UV stable solution of (40) admits the following UV expansion333We note
that the differential equation (40) can be explicitly integrated
$\left(\frac{1-\rho-\hbar}{1+\rho-\hbar}\right)^{1-\hbar}\left(\frac{1+\rho+\hbar}{1-\rho+\hbar}\right)^{1+\hbar}=e^{2\hbar(t-t_{0})}\,,$
(**) where
$\rho\overset{\text{def}}{=}\sqrt{(1-\hbar\kappa)(1-\hbar\kappa^{-1})}$.

$\kappa=\frac{1-\lambda}{1+\lambda}\implies\lambda=e^{\frac{\hbar(t-t_{0})}{1-\hbar}}+\frac{\hbar}{(1-\hbar)^{2}}e^{\frac{3\hbar(t-t_{0})}{1-\hbar}}+\frac{\hbar(1+4\hbar+\hbar^{2})}{(1-\hbar)^{4}}e^{\frac{5\hbar(t-t_{0})}{1-\hbar}}+\ldots\,,$
(43)

which is qualitatively the same as the one-loop UV expansion Fateev:1992tk
with the only difference that the dimension of $\lambda$ is changed:
$\hbar\rightarrow\frac{\hbar}{1-\hbar}$.

In order to study the UV behaviour more carefully, it is convenient to go to
the other coordinate frame. Namely, for both models (28) and (36) we perform
the following change of variables (note that this change is complex)

$\displaystyle\sin\theta=\kappa^{-1}\tanh\frac{x}{2}\,,$
$\displaystyle\chi=\frac{y}{2}+\frac{i}{2}\log\left(1-\frac{1-\kappa^{2}}{1+\kappa^{2}}\cosh
x\right)$ $\displaystyle\text{for sausage model}\,,$ $\displaystyle
p^{2}+q^{2}=e^{iy}\,,$ $\displaystyle\frac{p^{2}-q^{2}}{p^{2}+q^{2}}=\cosh x$
$\displaystyle\text{for $\lambda$ model}\,.$ (44)

Then one has

$ds^{2}_{\text{sausage}}=Adx^{2}+\left(A+\frac{1}{2}\right)dy^{2}+B\left(e^{x}(dx+idy)^{2}+e^{-x}(dx-
idy)^{2}\right)$ (45)

and

$ds^{2}_{\lambda}=\frac{1}{1-e^{-iy}}\times\\\
\times\left(Adx^{2}+\frac{1-\frac{A}{A+\frac{1}{2}}e^{-iy}}{1-e^{-iy}}\left(A+\frac{1}{2}\right)dy^{2}+B\left(e^{x}(dx+idy)^{2}+e^{-x}(dx-
idy)^{2}\right)\right)\,,$ (46)

where the parameters $A$ and $B$ have the form ($\lambda$ is defined by (43))

$A=\frac{1+\kappa^{2}}{4\hbar\kappa}-\frac{1}{2}=\frac{1-\hbar}{2\hbar}+O(\lambda^{2})\,,\quad
B=\frac{1-\kappa^{2}}{8\hbar\kappa}=\frac{\lambda}{2\hbar}+O(\lambda^{2})\,.$
(47)

In this form it is clear how these two models are related. Namely, one can
read off the exponent $e^{-iy}$ in the metric (46) by shifting $y$ by an
infinite imaginary number (similar observation has been done in Hoare:2015gda
).

It is known from Litvinov:2018bou  that the theory (45) in the UV limit can be
understood as the free theory perturbed by some terms determined by certain
exponential screening charges. In order to extract this form for this theory
and try to do this for the $\lambda$ model (46) it is convenient to rescale

$x\rightarrow\frac{x}{\sqrt{2A}}\,,\quad y\rightarrow\frac{y}{\sqrt{2A+1}}$
(48)

and then we obtain for the sausage sigma model

$ds^{2}_{\text{sausage}}=\frac{1}{2}\left(dx^{2}+dy^{2}\right)+\\\
+\frac{B}{2A}\left(e^{\frac{x}{\sqrt{2A}}}\left(dx+i\frac{\sqrt{2A}}{\sqrt{2A+1}}dy\right)^{2}+e^{-\frac{x}{\sqrt{2A}}}\left(dx-i\frac{\sqrt{2A}}{\sqrt{2A+1}}dy\right)^{2}\right)$
(49)

and for the $\lambda$-deformed sigma model

$ds^{2}_{\lambda}=\frac{1}{2}\left(\frac{1}{1-e^{-i\frac{y}{\sqrt{2A+1}}}}dx^{2}+\frac{1-\frac{2A}{2A+1}e^{-i\frac{y}{\sqrt{2A+1}}}}{\left(1-e^{-i\frac{y}{\sqrt{2A+1}}}\right)^{2}}dy^{2}\right)+\\\
+\frac{\frac{B}{2A}}{1-e^{-i\frac{y}{\sqrt{2A+1}}}}\left(e^{\frac{x}{\sqrt{2A}}}\left(dx+i\frac{\sqrt{2A}}{\sqrt{2A+1}}dy\right)^{2}+e^{-\frac{x}{\sqrt{2A}}}\left(dx-i\frac{\sqrt{2A}}{\sqrt{2A+1}}dy\right)^{2}\right)\,.$
(50)

Coefficients $A$ and $B$ have the following $t\rightarrow-\infty$ expansion

$A=\frac{1-\hbar}{2\hbar}+\mathcal{O}\left(e^{\frac{2\hbar(t-t_{0})}{1-\hbar}}\right)\,,\quad
B=\frac{1-\hbar}{2\hbar}\hbar^{\frac{\hbar}{1-\hbar}}e^{\frac{\hbar(t-t_{0})}{1-\hbar}}+\mathcal{O}\left(e^{\frac{3\hbar(t-t_{0})}{1-\hbar}}\right)\,.$
(51)

Substituting the expansions (51) into the metrics (49) and (50), we are able
to extract the UV expansion of these metrics. By supplementing it with a new
parameter together with an additional imaginary shift of $y$ coordinate

$b\overset{\text{def}}{=}\sqrt{\frac{1-\hbar}{\hbar}}\,,\quad y\rightarrow
y+\frac{i(t-t_{0})}{b^{2}}$ (52)

we can derive the following expressions444We did some additional
$t$-independent shifts in $x$ and $y$ coordinates to adjust the coefficients
in front of the exponential terms.

$ds^{2}_{\text{sausage}}=\frac{1}{2}\left(dx^{2}+dy^{2}\right)+\\\
+e^{\frac{t-t_{0}}{b^{2}}}\left(e^{\frac{x}{b}}\left(dx+\frac{ib}{\sqrt{1+b^{2}}}dy\right)^{2}+e^{-\frac{x}{b}}\left(dx-\frac{ib}{\sqrt{1+b^{2}}}dy\right)^{2}\right)+\mathcal{O}\left(e^{\frac{3(t-t_{0})}{b^{2}}}\right)$
(53)

and

$ds^{2}_{\lambda}=\frac{1}{2}\left(dx^{2}+dy^{2}\right)+e^{\frac{t-t_{0}}{b^{2}}}\left(e^{\frac{x}{b}}\left(dx+\frac{ib}{\sqrt{1+b^{2}}}dy\right)^{2}+e^{-\frac{x}{b}}\left(dx-\frac{ib}{\sqrt{1+b^{2}}}dy\right)^{2}\right)+\\\
+e^{\frac{t-t_{0}}{b^{2}}}e^{-\frac{iy}{\sqrt{1+b^{2}}}}\left(dx+\frac{ib}{\sqrt{1+b^{2}}}dy\right)\left(dx-\frac{ib}{\sqrt{1+b^{2}}}dy\right)+\mathcal{O}\left(e^{\frac{3(t-t_{0})}{b^{2}}}\right)\,.$
(54)

An important observation is that the obtained UV expansions (53) and (54) are
consistent with what we expect from the screening charges, which define the
sausage and $\lambda$ models respectively (see appendix B). This fact is an
additional indication that the obtained metrics (28) and (36) provide correct
completion of the corresponding one-loop theories at least up to the four loop
order.

We note also that the operator in the last line in (54) is irrelevant with the
dimension $2+\frac{1}{1+b^{2}}$. However, making a $t$-dependent shift by an
imaginary amount in $y$ direction, as we did in (52), one can make it formally
small in the UV. Such manipulations are not well justified in QFT, however
they shed some light on the meaning and role of $\lambda$-deformed sigma
models. As YB-deformed sigma models correspond to asymptotically free
theories, their $\lambda$-deformed cousins include some irrelevant operators.
Note also that without the shift (52) the UV limit of the metric (46) is the
non-trivial background

$ds^{2}=\frac{1}{2}\left(\frac{dx^{2}}{1-e^{-\frac{iy}{\sqrt{1+b^{2}}}}}+\frac{1-\frac{b^{2}}{1+b^{2}}e^{-\frac{iy}{\sqrt{1+b^{2}}}}}{\left(1-e^{-\frac{iy}{\sqrt{1+b^{2}}}}\right)^{2}}dy^{2}\right)\,,$
(55)

which solves the conformal equation

$\beta_{ij}(G)+2\nabla_{i}\nabla_{j}\Phi=0\quad\text{with}\quad\Phi=-\frac{1}{2}\log\left(1-e^{\frac{iy}{\sqrt{1+b^{2}}}}\right)-\frac{1}{4(1+b^{2})\left(1-e^{\frac{iy}{\sqrt{1+b^{2}}}}\right)}$
(56)

with the “exact” $\beta$-function (39).

The metric (55) corresponds to $SU(2)/U(1)$ gauged WZW model. It is more
convenient to consider the related $SL(2,\mathbb{R})/U(1)$ metric which is
obtained from (55) by replacing $x\leftrightarrow y$ and
$i\sqrt{1+b^{2}}\leftrightarrow b$

$ds^{2}=\frac{1}{2}\left(\frac{1-\frac{1+b^{2}}{b^{2}}e^{\frac{x}{b}}}{\left(1-e^{\frac{x}{b}}\right)^{2}}dx^{2}+\frac{dy^{2}}{1-e^{\frac{x}{b}}}\right)\,.$
(57)

Changing the variables

$x=b\log\left(1-\coth^{2}r\right),\quad y=b\varphi$ (58)

and then replacing $b^{2}=\frac{1}{\hbar}$ one finds that (57) reduces to

$ds^{2}=\frac{2}{\hbar}\left(\Big{(}1+\frac{\hbar}{\cosh^{2}r}\Big{)}dr^{2}+\tanh^{2}rd\varphi^{2}\right).$
(59)

This metric corresponds to all-loop completion of the cigar metric
Witten:1991yr  in our scheme.

In Dijkgraaf:1991ba  Dijkgraaf, Verlinde and Verlinde suggested the following
expression for the exact cigar metric (see also Bars:1992sr )

$ds^{2}_{\text{DVV}}=\frac{2}{\hbar}\left(dr^{2}+\frac{1}{\coth^{2}r-\frac{\hbar}{\hbar+1}}d\varphi^{2}\right)$
(60)

Two metrics (59) and (60) should be related by some covariant metric
redefinition (20) together with a coordinate change

$r\rightarrow r+\hbar
f_{1}(r)+\hbar^{2}f_{2}(r)+\dots,\quad\varphi\rightarrow(1+\hbar
q_{1}+\hbar^{2}q_{2}+\dots)\varphi$ (61)

One finds that this covariant metric redefinition is very simple

$\left(1-R\right)^{\frac{1}{2}}ds^{2}=ds^{2}_{\text{DVV}}\left(\varphi\rightarrow\frac{\varphi}{\sqrt{1+\hbar}}\right)=\frac{2}{\hbar}\left(dr^{2}+\frac{\frac{1}{1+\hbar}}{\coth^{2}r-\frac{\hbar}{\hbar+1}}d\varphi^{2}\right)$
(62)

i.e. it corresponds to the following choice of the scheme parameters in (20)

$c_{1}=-\frac{1}{2},\quad c_{2}=-\frac{1}{8},\quad c_{5}=-\frac{1}{16},\quad
c_{3}=c_{4}=c_{6}=c_{7}=c_{8}=c_{9}=c_{10}=c_{11}=0\,.$ (63)

We note that applying the same scheme transformation
$ds^{2}\rightarrow(1-R)^{\frac{1}{2}}ds^{2}$ to (28) gives the metric

$ds^{2}=2\kappa\left(\big{(}\kappa-\hbar^{-1}\big{)}\big{(}\kappa^{-1}-\hbar^{-1}\big{)}\right)^{\frac{1}{2}}\left(\frac{d\theta^{2}}{1-\kappa^{2}\sin^{2}\theta}+\frac{\cos^{2}\theta
d\chi^{2}}{1-\kappa\big{(}\kappa\sin^{2}\theta+\hbar\cos^{2}\theta\big{)}}\right)$
(64)

which coincides with the metric from Hoare:2019ark  (equation (2.6) in
Hoare:2019ark  with $\varkappa=i\kappa$, $r=\sin\theta$ and
$h=\frac{2\kappa}{\hbar}$) by an overall factor

$\frac{2\kappa\left(\big{(}\kappa-\hbar^{-1}\big{)}\big{(}\kappa^{-1}-\hbar^{-1}\big{)}\right)^{\frac{1}{2}}}{\frac{2\kappa}{\hbar}-1-\kappa^{2}}=1-\frac{(\kappa^{2}-1)^{2}}{8\kappa^{2}}\hbar^{2}-\frac{(\kappa^{2}-1)^{2}(\kappa^{2}+1)}{8\kappa^{3}}\hbar^{3}+O(\hbar^{4})$
(65)

We see that these two metrics coincide at two loops, but start to differ at
the third one. However, the extra factor (64) can be interpreted as a
particular type of metric redefinitions (in general non-covariant) of the form

$G_{ij}\rightarrow\left(1+\xi_{1}(t)\hbar+\xi_{2}(t)\hbar^{2}+\dots\right)G_{ij}\,,$
(66)

which implies that both metrics are directly related.

##  4 Concluding remarks

In this paper we studied the question of renormalizability of $2D$ integrable
sigma models. Namely, we considered two integrable backgrounds, both
corresponding to the deformations of the $O(3)$ model, known as YB-deformed
and $\lambda$-deformed $SU(2)/U(1)$ sigma models. We came to the conclusion
similar to already posted in Hoare:2019ark ; Hoare:2019mcc , that higher loop
corrections to the classical metrics are needed to ensure renormalizability.
We have found the scheme with the $\beta$-function (21) in which the metric
receives only $\hbar^{0}$ correction (the metric is two-loop exact). It is
important that both backgrounds, namely (28) and (36), satisfy additional
constraint (22) making $(2+\zeta_{3})$ term in the fourth loop coefficient of
the $\beta-$function negligible. Remarkably both solutions (28) and (36) also
solve RG flow equation with the “exact” $\beta$-function (39), which we
obtained by naive extrapolation of the four-loop result (21).

It is interesting to extend our study for sigma models with the dimension of
the target space $D>2$. In general, this will require the knowledge of
$\beta$-function including the one for the Kalb-Ramond field ($B$-field),
which appears in most of deformed integrable sigma models. It makes the
analysis more problematic since the $\beta$-function in the presence of the
$B$-field is known only in two-loop approximation. However, there are models,
such as $\lambda$-deformed $SO(N+1)/SO(N)$ model, where the $B$-field vanishes
Grigoriev:2007bu ; Demulder:2015lva . We plan to consider the
renormalizability of these models in a separate publication.

Another possible direction of development of our results is to study the
$\eta$-deformed $OSp(N|2m)$ sigma models, which contain anticommuting
coordinates in their target space. The RG flow of these models depends on the
difference of $N-2m$. It was shown in Alfimov:2020jpy  that in the regime of
purely imaginary $\eta=i\kappa$ they are asymptotically free if $N-2m>2$,
conformal if $N-2m=2$ and infrared free if $N-2m<2$. There was also analyzed
the UV limit of these theories for $N-2m>2$, which can be understood as a free
theory perturbed by a set of operators assembled from certain screening
charges. Interestingly, the 1-loop RG flow equation for asymptotically free
deformed $OSP(5|2)$ sigma model leads to the same differential equation for
$\kappa$ as (27) for the deformed $O(3)$ sigma-model. This similarity to the
$O(3)$ sigma model RG flow together with the fact that we know the UV limit of
the $OSp(5|2)$ action from the screenings gives some hope to find a simple
expression like (39) for the $\beta$-function, despite the target space of the
$OSp(5|2)$ has the dimension larger than $2$.

## Acknowledgments

We are grateful to Boris Feigin, Ben Hoare, Sergei Lukyanov, Konstantinos
Sfetsos, Konstantinos Siampos, Arkady Tseytlin, Mikhail Vasiliev and Alexander
Zamolodchikov for fruitful and stimulating discussions. A.L. acknowledges the
support of Basis Foundation. M.A. would like to thank Elena M. for her good
advice at the initial stage of this project.

##  Appendix A $\beta$-function in different schemes

In the present Appendix we address the question how the $\beta$-function in
the $D=2$ case changes under covariant metric redefinitions starting from the
minimal subtraction scheme. Let us describe the way to find the contributions
to the transformed $\beta$-function order by order in the covariant expansion,
using the Mathematica package xAct Martin:xAct ; Brizuela:2008ra ;
Nutma:2013zea  for the necessary calculations.

Here we denote by tilde the metric after covariant redefinition, namely

$\tilde{G}_{ij}=G_{ij}+\sum_{k=0}^{\infty}G_{ij}^{(k)}\,,$ (67)

where $G_{ij}^{(k)}$ are given by (20). Essentially we are to determine the
$\tilde{b}_{k}^{(L)}$ coefficients in front of the tensor structures from (12)
in the new scheme. For the case of two-dimensional target space the number of
independent tensor structures is much lower and we can explicitly write them
in the first four orders we are interested in

$\displaystyle\mathcal{T}_{1}$
$\displaystyle=\left\\{RG_{ij}\right\\}\;,\quad\mathcal{T}_{2}=\left\\{R^{2}G_{ij},\nabla^{2}RG_{ij},\nabla_{i}\nabla_{j}R\right\\}\,,$
$\displaystyle\mathcal{T}_{3}$
$\displaystyle=\left\\{R^{3}G_{ij},R\nabla^{2}RG_{ij},(\nabla
R)^{2}G_{ij},\nabla^{2}\nabla^{2}RG_{ij},\nabla_{i}R\nabla_{j}R\right\\}\,,$
$\displaystyle\mathcal{T}_{4}$
$\displaystyle=\left\\{R^{4}G_{ij},R^{2}\nabla^{2}RG_{ij},R(\nabla
R)^{2}G_{ij},R\nabla^{2}\nabla^{2}RG_{ij},(\nabla^{2}R)^{2}G_{ij},(\nabla_{k}\nabla_{l}R)(\nabla^{k}\nabla^{l}R)G_{ij}\,,\right.$
$\displaystyle\left.\nabla_{k}\nabla^{2}R\nabla^{k}RG_{ij},\nabla^{2}\nabla^{2}\nabla^{2}RG_{ij}\;,R\nabla_{i}R\nabla_{j}R\,,\nabla^{2}R\nabla_{i}\nabla_{j}R\right\\}\,.$
(68)

One may notice the absence of some possible structures in the space
$\mathcal{T}_{3}$, such as $R\nabla_{i}\nabla_{j}R$ and
$\nabla_{i}\nabla_{j}\nabla^{2}R$. The reason for their exclusion is that the
sum of the first of them with $\nabla_{i}R\nabla_{j}R$ is of the form
$\nabla_{i}V_{j}+\nabla_{j}V_{i}$, i.e. can be absorbed by a diffeomorphism,
and the second one has already the form of a diffeomorphism. For the same
reason we exclude from the space $\mathcal{T}_{4}$ the structures
$R^{2}\nabla_{i}\nabla_{j}R$, $\nabla_{i}R\nabla_{j}\nabla^{2}R$,
$\nabla_{j}R\nabla_{i}\nabla^{2}R$, where the latter two can be simultaneously
eliminated due to the symmetry $i\leftrightarrow j$,
$R\nabla_{i}\nabla_{j}\nabla^{2}R$ and
$\nabla_{k}R\nabla_{i}\nabla_{j}\nabla^{k}R$.

In the 4th order there is the following identity, which also diminishes the
number of possible tensor structures in $\mathcal{T}_{4}$

$\nabla_{i}\nabla_{k}R\nabla_{j}\nabla^{k}R=\nabla^{2}R\nabla_{i}\nabla_{j}R-\frac{1}{2}(\nabla^{2}R)^{2}G_{ij}+\frac{1}{2}(\nabla_{k}\nabla_{l}R)(\nabla^{k}\nabla^{l}R)G_{ij}\,.$
(69)

Let us first rewrite the terms proportional to the metric in the following
form, utilizing the expression for Ricci tensor in 2D

$\frac{1}{2}(\nabla_{k}\nabla_{l}R)(\nabla^{k}\nabla^{l}R)G_{ij}-\frac{1}{2}(\nabla^{2}R)^{2}G_{ij}=\nabla^{k}\nabla^{l}R\left(\nabla_{k}\nabla_{l}R_{ij}-\frac{1}{2}\nabla^{2}(RG_{kl}G_{ij})\right)\,.$
(70)

For the first term in the brackets on the right hand side of (70) we use the
second contracted Bianchi identity

$\nabla_{k}\nabla_{l}R_{ij}=\nabla_{k}\nabla_{i}R_{lj}-\nabla_{k}\nabla^{m}R_{lijm}\,.$
(71)

For the second term we obtain

$\frac{1}{2}\nabla^{m}\nabla_{m}(RG_{kl}G_{ij})=\nabla^{m}\nabla_{m}R_{kjli}+\frac{1}{2}\nabla^{m}\nabla_{m}(RG_{ki}G_{lj})\;.$
(72)

By substituting (71) and (72) into (70), we derive, using the fact that
covariant derivatives commute on Riemann tensor in 2D and general symmetry
properties of the Riemann tensor together with the second Bianchi identity,

$\displaystyle\frac{1}{2}\nabla_{i}\nabla_{k}R\nabla_{i}\nabla^{k}R-\frac{1}{2}\nabla_{k}\nabla^{k}R\nabla_{i}\nabla_{j}R-\nabla^{k}\nabla^{l}R\nabla^{m}(\nabla_{k}R_{lijm}+\nabla_{m}R_{kjli})=$
$\displaystyle=$
$\displaystyle\frac{1}{2}\nabla_{i}\nabla_{k}R\nabla_{j}\nabla^{l}R-\frac{1}{2}\nabla_{k}\nabla^{k}R\nabla_{i}\nabla_{j}R+\nabla^{k}\nabla^{l}R\nabla^{m}\nabla_{j}R_{limk}=$
$\displaystyle=$
$\displaystyle\nabla_{i}\nabla_{k}R\nabla_{j}\nabla^{k}R-\nabla_{k}\nabla^{k}R\nabla_{i}\nabla_{j}R\,.$
(73)

Thus, we have proven the initial identity (69). This identity allows to
eliminate the tensor structure $\nabla_{i}\nabla_{k}R\nabla_{j}\nabla^{k}R$
from $\mathcal{T}_{4}$.

In addition we do not include the structures which differ by the permutations
of the covariant derivatives acting on the same scalar. In this way we exclude
the structures such as all the permutations of the derivatives in
$\nabla^{2}\nabla^{2}RG_{ij}$ in $\mathcal{T}_{3}$ and
$\nabla^{2}\nabla^{2}\nabla^{2}RG_{ij}$ in $\mathcal{T}_{4}$ and
$\nabla^{2}\nabla_{k}R\nabla^{k}RG_{ij}$, $\nabla_{i}R\nabla^{2}\nabla_{j}R$,
$\nabla_{j}R\nabla^{2}\nabla_{i}R$,
$\nabla^{k}R\nabla_{j}\nabla_{i}\nabla_{k}R$ and
$\nabla^{k}R\nabla_{k}\nabla_{i}\nabla_{j}R$ also in $\mathcal{T}_{4}$,
because they are linearly dependent with the remaining ones. Thus, we are able
to write the expansion (12) in the first four orders explicitly and find the
coefficients $\tilde{b}_{k}^{(L)}$, where
$k=1,\ldots,\textrm{dim}\,\mathcal{T}_{L}$ in the new scheme.

Let us now describe how the $\beta$-function changes under covariant
substitution of the metric. In the new scheme we have in general

$\dot{\tilde{G}}_{ij}=-(\tilde{\beta}_{ij}+\tilde{\nabla}_{i}\tilde{V}_{j}+\tilde{\nabla}_{j}\tilde{V}_{i})(\tilde{G}_{mn})\,.$
(74)

We substitute (67) to (74) to obtain

$\dot{G}_{ij}+\sum\limits_{k=0}^{+\infty}\dot{G}_{ij}^{(k)}=-(\tilde{\beta}_{ij}+\tilde{\nabla}_{i}\tilde{V}_{j}+\tilde{\nabla}_{j}\tilde{V}_{i})\left(G_{mn}+\sum\limits_{k=0}^{+\infty}G_{mn}^{(k)}\right)\,.$
(75)

As every $G_{ij}^{(k)}$ can be expressed solely in terms of $G_{ij}$ itself,
the left hand side of (75) is transformed to

$\dot{G}_{ij}+\sum\limits_{k=0}^{+\infty}\dot{G}_{ij}^{(k)}=\left(\delta_{i}^{l}+\sum\limits_{k=1}^{+\infty}\left(K_{i}^{l}\right)^{(k)}\right)\dot{G}_{lj}\,,$
(76)

in which each operator $(K_{i}^{l})^{(k)}$ is of order $\hbar^{k}$ and is a
function of the coefficients $c_{p}$, $p=1,\ldots,+\infty$ from (20) and
$G_{ij}$ itself, possibly acting on $\dot{G}_{lj}$ with covariant derivatives.
Combining (76) with (75), we can extract the RG flow equation

$\dot{G}_{ij}=-(\tilde{\beta}_{ij}+\tilde{\nabla}_{i}\tilde{V}_{j}+\tilde{\nabla}_{j}\tilde{V}_{i})\left(G_{mn}+\sum\limits_{k=0}^{+\infty}G_{mn}^{(k)}\right)-\sum\limits_{k=1}^{+\infty}\left(K_{i}^{l}\right)^{(k)}\dot{G}_{lj}\,,$
(77)

which should be identical to (9) after substituting $\dot{G}_{ij}$ in the
minimal subtraction scheme to the right hand side. Therefore, the difference
between the right hand side of (77) and (9) has to be $0$, which leads to the
set of equations for the coefficients $\tilde{b}_{k}^{(L)}$, $b_{k}^{(L)}$ and
$c_{p}$ order by order in $\hbar$

$(\tilde{\beta}_{ij}+\tilde{\nabla}_{i}\tilde{V}_{j}+\tilde{\nabla}_{j}\tilde{V}_{i})\left(G_{mn}+\sum\limits_{k=0}^{+\infty}G_{mn}^{(k)}\right)-\beta_{ij}(G_{mn})=0\,.$
(78)

Let us track how the $\beta$-function changes in the first 3 orders and then
present the result for the 4th order. The expressions at these orders in the
minimal subtraction scheme were written in (18) and (19), while in new scheme
they are given by the ansatz (1-loop $\beta$-function is omitted as it is
trivially non-dependent on the scheme choice)

$\displaystyle\tilde{\beta}^{(2)}_{ij}$
$\displaystyle=\left(\tilde{b}_{1}^{(2)}\tilde{R}^{2}+\tilde{b}_{2}^{(2)}\tilde{\nabla}^{2}\tilde{R}\right)\tilde{G}_{ij}+\tilde{b}_{3}^{(2)}\tilde{\nabla}_{i}\tilde{\nabla}_{j}\tilde{R}\,,$
$\displaystyle\tilde{\beta}^{(3)}_{ij}$
$\displaystyle=\left(\tilde{b}_{1}^{(3)}\tilde{R}^{3}+\tilde{b}_{2}^{(3)}\tilde{R}\tilde{\nabla}^{2}\tilde{R}+\tilde{b}_{3}^{(3)}(\tilde{\nabla}\tilde{R})^{2}+\tilde{b}_{4}^{(3)}\tilde{\nabla}^{2}\tilde{\nabla}^{2}\tilde{R}\right)\tilde{G}_{ij}+\tilde{b}_{5}^{(3)}\tilde{\nabla}_{i}\tilde{R}\tilde{\nabla}_{j}\tilde{R}\,,$
$\displaystyle\tilde{\beta}^{(4)}_{ij}$
$\displaystyle=\left(\tilde{b}_{1}^{(4)}\tilde{R}^{4}+\tilde{b}_{2}^{(4)}\tilde{R}^{2}\tilde{\nabla}^{2}\tilde{R}+\tilde{b}_{3}^{(4)}\tilde{R}(\tilde{\nabla}\tilde{R})^{2}+\tilde{b}_{4}^{(4)}\tilde{R}\tilde{\nabla}^{2}\tilde{\nabla}^{2}\tilde{R}+\tilde{b}_{5}^{(4)}(\tilde{\nabla}^{2}\tilde{R})^{2}+\right.$
$\displaystyle\left.+\tilde{b}_{6}^{(4)}(\tilde{\nabla}_{k}\tilde{\nabla}_{l}\tilde{R})(\tilde{\nabla}^{k}\tilde{\nabla}^{l}R)+\tilde{b}_{7}^{(4)}(\tilde{\nabla}_{k}\tilde{\nabla}^{2}\tilde{R})(\tilde{\nabla}^{k}\tilde{R})+\tilde{b}_{8}^{(4)}\tilde{\nabla}^{2}\tilde{\nabla}^{2}\tilde{\nabla}^{2}\tilde{R}\right)\tilde{G}_{ij}+$
$\displaystyle+\tilde{b}_{9}^{(4)}\tilde{R}\tilde{\nabla}_{i}\tilde{R}\tilde{\nabla}_{j}\tilde{R}+\tilde{b}_{10}^{(4)}\tilde{\nabla}^{2}\tilde{R}\tilde{\nabla}_{i}\tilde{\nabla}_{j}\tilde{R}\,,$
(79)

where tilde over $\tilde{\nabla}$ and $\tilde{R}$ means that they are
calculated with respect to the metric $\tilde{G}_{ij}$. Putting (79), (18),
(19) and (20) into (78), we derive a set of equations for the unknown
coefficients $\tilde{b}_{k}^{(L)}$. These equations are too cumbersome to be
written here, but can be easily solved and here we write just the solution for
the two- and three-loop orders

$\displaystyle\tilde{b}_{1}^{(2)}$
$\displaystyle=\frac{1}{4}\,,\quad\tilde{b}_{2}^{(2)}=0\,,\quad\tilde{b}_{3}^{(2)}=0\,,$
$\displaystyle\tilde{b}_{1}^{(3)}$
$\displaystyle=\frac{5}{32}+\frac{c_{1}}{4}-\frac{c_{2}}{2}\,,\quad\tilde{b}_{2}^{(3)}=\frac{1}{16}-\frac{c_{1}}{2}+c_{2}-c_{3}-\frac{c_{1}^{2}}{2}\,,\quad\tilde{b}_{3}^{(3)}=-c_{3}-\frac{c_{1}^{2}}{2}\,,$
$\displaystyle\tilde{b}_{4}^{(3)}$
$\displaystyle=0\,,\quad\tilde{b}_{5}^{(3)}=-\frac{1}{16}$ (80)

and for the four-loop order

$\displaystyle\tilde{b}_{1}^{(4)}$
$\displaystyle=\frac{23}{192}+\frac{3c_{1}}{8}-c_{5}-\frac{c_{1}^{2}}{4}\,,\quad\tilde{b}_{2}^{(4)}=\frac{2+\zeta_{3}}{32}+\frac{3c_{1}}{32}-\frac{c_{3}}{2}-\frac{3c_{7}}{2}-c_{1}^{2}-\frac{3c_{1}c_{2}}{2}+\frac{c_{1}^{3}}{2}\,,$
$\displaystyle\tilde{b}_{3}^{(4)}$
$\displaystyle=\frac{41+12\zeta(3)}{192}-\frac{15c_{1}}{16}-2c_{3}+3c_{5}-\frac{3c_{6}}{2}-c_{7}-c_{8}+\frac{c_{9}}{2}-\frac{c_{10}}{2}-\frac{c_{1}^{2}}{2}-3c_{1}c_{2}+$
$\displaystyle+\frac{5c_{1}c_{4}}{4}+c_{1}^{3}\,,$
$\displaystyle\tilde{b}_{4}^{(4)}$
$\displaystyle=-\frac{3c_{8}}{2}-\frac{c_{1}c_{3}}{2}\,,$
$\displaystyle\tilde{b}_{5}^{(4)}$
$\displaystyle=\frac{1}{192}-\frac{c_{1}}{16}-\frac{c_{3}}{2}-\frac{3c_{8}}{2}-\frac{c_{9}}{2}+\frac{c_{10}}{2}-\frac{c_{1}^{2}}{4}-\frac{c_{1}c_{3}}{2}-\frac{c_{1}c_{4}}{2}\,,$
$\displaystyle\tilde{b}_{6}^{(4)}$
$\displaystyle=\frac{1}{192}-\frac{c_{1}}{16}-c_{3}+c_{6}-2c_{8}+\frac{c_{9}}{2}-\frac{c_{10}}{2}\,,$
$\displaystyle\tilde{b}_{7}^{(4)}$
$\displaystyle=-\frac{c_{1}}{8}-2c_{3}+c_{7}-5c_{8}-c_{1}c_{3}\,,\quad\tilde{b}_{8}^{(4)}=0\,,$
$\displaystyle\tilde{b}_{9}^{(4)}$
$\displaystyle=-\frac{25}{192}-\frac{5c_{1}}{16}-\frac{5c_{9}}{2}+\frac{5c_{10}}{2}-\frac{5c_{1}c_{4}}{2}\,,\quad\tilde{b}_{10}^{(4)}=-\frac{1}{96}+\frac{c_{1}}{8}+c_{9}-c_{10}+3c_{1}c_{4}\,.$
(81)

If we want to transform to the $\beta$-function (21), then the scheme
parameters $c_{k}$ have to be chosen as

$\displaystyle c_{2}$ $\displaystyle=-\frac{1}{16}+\frac{c_{1}}{2}\,,$
$\displaystyle c_{3}$ $\displaystyle=-\frac{c_{1}^{2}}{2}\,,$ $\displaystyle
c_{5}$ $\displaystyle=-\frac{7}{192}+\frac{3c_{1}}{8}-\frac{c_{1}^{2}}{4}\,,$
$\displaystyle c_{6}$
$\displaystyle=\frac{9+5\zeta_{3}}{96}+\frac{c_{1}}{8}-\frac{c_{1}^{2}}{2}+\frac{c_{1}c_{4}}{2}+\frac{c_{1}^{3}}{3}\,,$
$\displaystyle c_{7}$
$\displaystyle=\frac{5(2+\zeta_{3})}{96}+\frac{c_{1}}{8}-c_{1}^{2}+\frac{c_{1}^{3}}{3}\,,$
$\displaystyle c_{8}$
$\displaystyle=\frac{2+\zeta_{3}}{96}+\frac{c_{1}^{3}}{6}\,,$ $\displaystyle
c_{9}-c_{10}$ $\displaystyle=\frac{1}{96}-\frac{c_{1}}{8}-c_{1}c_{4}\,.$ (82)

Given the choice (82), we arrive to particularly simple expressions for the
$\beta$-function in the first four orders

$\displaystyle\tilde{\beta}^{(1)}_{ij}$
$\displaystyle=\frac{1}{2}\tilde{R}\tilde{G}_{ij}\,,\quad\tilde{\beta}^{(2)}_{ij}=\frac{1}{4}\tilde{R}^{2}\tilde{G}_{ij}\,,\quad\tilde{\beta}^{(3)}_{ij}=\frac{3}{16}\tilde{R}^{3}\tilde{G}_{ij}-\frac{1}{16}\tilde{\nabla}_{i}\tilde{R}\tilde{\nabla}_{j}\tilde{R}\,,$
$\displaystyle\tilde{\beta}^{(4)}_{ij}$
$\displaystyle=\frac{5}{32}\tilde{R}^{4}\tilde{G}_{ij}-\frac{5}{32}\tilde{R}\tilde{\nabla}_{i}\tilde{R}\tilde{\nabla}_{j}\tilde{R}+\frac{2+\zeta_{3}}{64}\tilde{\nabla}^{2}\left(\tilde{R}^{3}+2\tilde{R}\tilde{\nabla}^{2}\tilde{R}-\frac{1}{2}\tilde{\nabla}^{2}\tilde{R}^{2}\right)\,.$
(83)

##  Appendix B Deformed $O(3)$ sigma model screening charges

In the work Litvinov:2018bou  it was shown that integrable deformed $O(N)$
sigma models in the UV limit can be understood as a Gaussian theory perturbed
by certain relevant operators (for the $OSp(N|2m)$ sigma models see the
discussion in Alfimov:2020jpy ). These operators are constrained by the
integrable structure of the sigma model and are built from the screening
charges. The screening charges determine the integrals of motion of the sigma
model and vice versa (see Litvinov:2018bou ).

The deformed $O(3)$ sigma model has two bosonic degrees of freedom and
therefore can be encoded as a two component real bosonic field. Let us recall
that the theory in question may be determined by the set of screening fields
on the figure 1 (see for example Fateev:2017mug ; Litvinov:2018bou ),

$\scriptstyle{\boldsymbol{\beta}_{12}}$$\scriptstyle{1+2b^{2}}$$\scriptstyle{1+2b^{2}}$$\scriptstyle{-1-2b^{2}}$$\scriptstyle{-1-2b^{2}}$$\scriptstyle{\boldsymbol{\beta}_{34}}$$\scriptstyle{\boldsymbol{\alpha}_{1}}$$\scriptstyle{\boldsymbol{\alpha}_{2}}$$\scriptstyle{\boldsymbol{\alpha}_{3}}$$\scriptstyle{\boldsymbol{\alpha}_{4}}$$\scriptstyle{\boldsymbol{\beta}_{23}}$$\scriptstyle{\boldsymbol{\beta}_{14}}$
Figure 1: Screening charges corresponding to the deformed $O(3)$ sigma model.

where the crosses correspond to the fermionic screening fields

$\mathcal{S}_{k}=\oint e^{(\boldsymbol{\alpha}_{k},\varphi)}dz,$ (84)

which define the dual Toda-like description of the “sausage” model, while the
circles with the bar to the so called dressed or Wakimoto ones

$\mathcal{S}_{ij}=\oint
e^{(\boldsymbol{\beta}_{ij}\cdot\boldsymbol{\varphi})}\big{(}\boldsymbol{\alpha}_{i}\cdot\partial\boldsymbol{\varphi}\big{)}dz\,,$
(85)

which comprise the relevant perturbation operators in the UV limit of the
sigma model and where $\boldsymbol{\varphi}$ in our case denotes the pair of
chiral real bosonic fields. By utilizing Cartesian coordinates as in
Litvinov:2018bou  we can parametrize the fermionic screening lengths as
follows

$\displaystyle\boldsymbol{\alpha}_{1}=bE_{1}+i\sqrt{1+b^{2}}e_{1}\,,\quad\boldsymbol{\alpha}_{2}=bE_{1}-i\sqrt{1+b^{2}}e_{1}\,,$
$\displaystyle\boldsymbol{\alpha}_{3}=-bE_{1}+i\sqrt{1+b^{2}}e_{1}\,,\quad\boldsymbol{\alpha}_{4}=-bE_{1}-i\sqrt{1+b^{2}}e_{1}\,,$
(86)

where the parameter $b$ is the same as in the formula (52) and $\beta_{ij}$
lie in the same linear space spanned by $E_{1}$ and $e_{1}$. The values on the
links on the figure 1 correspond to the scalar products of the $\alpha_{i}$’s
on its ends, being $1$ for the dashed ones. In addition, there was shown in
Litvinov:2016mgi  that each pair of fermionic screenings $\mathcal{S}_{i}$ and
$\mathcal{S}_{j}$ of the type (84) can be equivalently replaced by the dressed
or Wakimoto screening $\mathcal{S}_{ij}$ of the type (85), whose length is
determined by the formula

$\boldsymbol{\beta}_{ij}=\frac{2(\boldsymbol{\alpha}_{i}+\boldsymbol{\alpha}_{j})}{(\boldsymbol{\alpha}_{i}+\boldsymbol{\alpha}_{j})^{2}}$
(87)

and in the deformed $O(3)$ case they are given by

$\boldsymbol{\beta}_{12}=\frac{E_{1}}{b}\,,\quad\boldsymbol{\beta}_{34}=-\frac{E_{1}}{b}\,,\quad\boldsymbol{\beta}_{13}=-i\frac{e_{1}}{\sqrt{1+b^{2}}}\,,\quad\boldsymbol{\beta}_{24}=i\frac{e_{1}}{\sqrt{1+b^{2}}}\,.$
(88)

An important feature of the dressed screenings is that in their prefactor we
can always shift $\alpha_{i}\rightarrow\alpha_{i}+\xi\beta_{ij}$, because this
shift is proportional to total derivative. All this allows us to say that in
the sausage metric in the UV limit (53) we observe the screenings
$\mathcal{S}_{12}$ and $\mathcal{S}_{34}$. Namely, if we take $\xi=1$, then
the screenings $\mathcal{S}_{12}$ and $\mathcal{S}_{34}$ turn to

$\mathcal{S}_{12}=\oint
dzb\left(\partial\boldsymbol{\varphi}_{1}+\frac{ib\partial\boldsymbol{\varphi}_{2}}{\sqrt{1+b^{2}}}\right)e^{\frac{\boldsymbol{\varphi}_{1}}{b}}\,,\quad\mathcal{S}_{34}=\oint
dzb\left(\partial\boldsymbol{\varphi}_{1}-\frac{ib\partial\boldsymbol{\varphi}_{2}}{\sqrt{1+b^{2}}}\right)e^{-\frac{\boldsymbol{\varphi}_{1}}{b}}\,,$
(89)

which, after being supplemented by the antichiral part and identifying
$\boldsymbol{\varphi}_{1}$ with $x$ and $\boldsymbol{\varphi}_{2}$ with $y$,
reproduce the exponential terms in (49).

The $\lambda$ model does not flow to the free theory in the UV limit, thus we
can expect that in this limit it is given by a free theory perturbed by some
irrelevant operator. And it is indeed the case: the screening
$\mathcal{S}_{13}$ taken with the different shift parameter $\xi=-1$

$\mathcal{S}_{13}=\oint
dzb\left(\partial\boldsymbol{\varphi}_{1}-\frac{ib\partial\boldsymbol{\varphi}_{2}}{\sqrt{1+b^{2}}}\right)e^{-\frac{i\boldsymbol{\varphi}_{1}}{\sqrt{1+b^{2}}}}\,,$
(90)

after multiplication by the antichiral part and the same identification of
$\boldsymbol{\varphi}$ yields the third exponential term in (50). Therefore,
the difference between (49) and (50) is that in the latter there is the
irrelevant perturbation term, which makes the corresponding theory not free in
the UV limit.

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We have the following immediate corollary, using only the fact that the
synchronous coupling is bicausal by [7, Proposition 2.2].

######  Corollary 8.2.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be Lipschitz with Lipschitz constant
$L\in(0,\infty)$. Then, for any $\mu,\nu\in\mathcal{P}_{2}(\Omega)$, we have
the bound

(8.3) $|v^{f}(\mu)-v^{f}(\nu)|\leq
L\cdot\mathbb{E}^{\pi^{\mathrm{sync}}}\\!\left[\sup_{0\leq t\leq
T}|\omega_{t}-\bar{\omega}_{t}|\right]\\!.$

###### Proof of Theorem 8.1.

Let $\pi\in\mathrm{Cpl}_{\mathrm{c}}(\mu,\nu)$ and let $(X,Y)$ be a pair of
random variables on $\Omega\times\Omega$ with joint law $\pi$. As before, let
$\mathcal{F}^{X}$ and $\mathcal{F}^{Y}$ denote the completions of the natural
filtrations of $X$ and $Y$ under the measures $\mu$ and $\nu$, respectively.
Fix $\varepsilon>0$, and take an $\varepsilon$-optimal
$\mathcal{F}^{Y}$-stopping time $\tau^{Y}$; i.e.
$\mathbb{E}[f(Y_{\tau^{Y}})]\leq v^{f}(\nu)+\varepsilon$.

For each $u\in[0,1]$, define $\theta_{u}\coloneqq\inf\\{t\geq
0\colon\;\pi(\tau^{Y}\leq t\mid\mathcal{F}^{X}_{T})\geq u\\}$. We claim that
$\theta_{u}$ is an $\mathcal{F}^{X}$-stopping time. By continuity of the paths
of processes $X$ and $Y$, we know that, for any $u\in[0,1]$ and $t\in[0,T]$,

(8.4) $\\{\theta_{u}\leq t\\}=\\{\pi(\tau^{Y}\leq
t\mid\mathcal{F}^{X}_{T})\geq u\\}.$

So it suffices to show that, for each $t\in[0,T]$, the conditional probability
$\pi(\tau^{Y}\leq t\mid\mathcal{F}^{X}_{T})$ is
$\mathcal{F}^{X}_{t}$-measurable. Since
$\pi\in\mathrm{Cpl}_{\mathrm{c}}(\mu,\nu)$, the causality condition (2.6)
states that, under $\pi$, $\mathcal{F}^{Y}_{t}$ is conditionally independent
of $\mathcal{F}^{X}_{T}$, given $\mathcal{F}^{X}_{t}$. An equivalent
formulation of this conditional independence condition (see, e.g. Kallenberg
[34, Proposition 5.6]) is that for any $F\in\mathcal{F}^{Y}_{t}$,

(8.5)
$\pi(F\mid\mathcal{F}^{X}_{T}\vee\mathcal{F}^{X}_{t})=\pi(F\mid\mathcal{F}^{X}_{t}).$

In fact, $\mathcal{F}^{X}_{t}\subset\mathcal{F}^{X}_{T}$, and so
$\mathcal{F}^{X}_{T}\vee\mathcal{F}^{X}_{t}=\mathcal{F}^{X}_{T}$. By the
definition of a stopping time, $\\{\tau^{Y}\leq t\\}\in\mathcal{F}^{Y}_{t}$.
Therefore (8.5) gives us

(8.6) $\pi(\tau^{Y}\leq t\mid\mathcal{F}^{X}_{T})=\pi(\tau^{Y}\leq
t\mid\mathcal{F}^{X}_{t}),$

which is then $\mathcal{F}^{X}_{t}$-measurable. This proves our claim that
$\theta_{u}$ is an $\mathcal{F}^{X}$-stopping time for each $u\in[0,1]$. Thus

(8.7)
$v^{f}(\mu)\leq\inf_{u\in[0,1]}\mathbb{E}^{\pi}[f(X_{\theta_{u}})]\leq\int_{0}^{1}\mathbb{E}^{\pi}[f(X_{\theta_{u}})]\,\mathrm{d}u.$

Applying the tower property of conditional expectation and Fubini’s theorem to
exchange the order of integration, we arrive at

(8.8)
$v^{f}(\mu)\leq\mathbb{E}^{\pi}\\!\left[\int_{0}^{1}\mathbb{E}^{\pi}[f(X_{\theta_{u}})\mid\mathcal{F}^{X}_{T}]\,\mathrm{d}u\right]\\!.$

Now fix a path $\omega\in\Omega$ of the process $X$. We can define a
probability measure $\mathbb{P}_{\omega}$ on $[0,T]$ by
$\mathbb{P}_{\omega}([0,t))=\pi(\tau^{Y}\leq
t\mid\mathcal{F}^{X}_{T})(\omega)$, for all $t\in[0,T]$. Then
$u\mapsto\theta_{u}(\omega)$ is the quantile function of
$\mathbb{P}_{\omega}$, and we see that

(8.9)
$\begin{split}\mathbb{E}^{\pi}[f(X_{\tau^{Y}})\mid\mathcal{F}^{X}_{T}](\omega)&=\int_{0}^{T}f(\omega_{t})\,\mathrm{d}\mathbb{P}_{\omega}(t)=\int_{0}^{1}f(\omega_{\theta_{u}(\omega)})\,\mathrm{d}u=\int_{0}^{1}\mathbb{E}^{\pi}[f(X_{\theta_{u}})\mid\mathcal{F}^{X}_{T}](\omega)\,\mathrm{d}u.\end{split}$

Taking the expectation with respect to $\pi$ and substituting into (8.8), we
find

(8.10)
$v^{f}(\mu)\leq\mathbb{E}^{\pi}\\!\left[\mathbb{E}^{\pi}[f(X_{\tau^{Y}})\mid\mathcal{F}^{X}_{T}]\right]\\!=\mathbb{E}^{\pi}\\!\left[f(X_{\tau^{Y}})\right]\\!.$

Hence
$v^{f}(\mu)-v^{f}(\nu)\leq\mathbb{E}^{\pi}\\!\left[f(X_{\tau^{Y}})-f(Y_{\tau^{Y}})\right]\\!+\varepsilon$.
Since $f$ is Lipschitz with Lipschitz constant $L$, we have

(8.11) $\begin{split}v^{f}(\mu)-v^{f}(\nu)&\leq
L\cdot\mathbb{E}^{\pi}\\!\left[|X_{\tau^{Y}}-Y_{\tau^{Y}}|\right]\\!+\varepsilon\leq
L\cdot\mathbb{E}^{\pi}\\!\left[\|X-Y\|_{\infty}\right]\\!+\varepsilon.\end{split}$

Exchanging the roles of $\mu$ and $\nu$ and noting that $\varepsilon>0$ and
$\pi\in\mathrm{Cpl}_{\mathrm{c}}(\mu,\nu)$ were chosen arbitrarily completes
the proof. ∎

######  Remark 8.3.

The quantity on the right-hand side of (8.2) is the adapted $1$-Wasserstein
distance with respect to the $L^{\infty}$ norm on $\Omega$. In general, such a
bound does not hold for $\mathcal{A}\mathcal{W}_{1}$ defined with respect to
the $L^{1}$ norm on $\Omega$, as in Definition 2.11. For example, take $f$ to
be the identity and $\nu$ concentrated on the constant zero function. Choose
$\mu$ such that, for some $\varepsilon\in(0,1)$,
$\mathbb{E}^{\mu}[\int_{0}^{T}|\omega_{t}|\,\mathrm{d}t]\leq\varepsilon$ and,
for some $\delta\in[0,T]$, $\omega_{\delta}=1$ $\mu$-almost surely. Then
$|v^{f}(\mu)-v^{f}(\nu)|=|v^{f}(\mu)|\geq 1$, while
$\mathcal{A}\mathcal{W}_{1}(\mu,\nu)\leq\varepsilon$. Therefore the bound
(8.2) does not hold when the right-hand side is replaced by
$\mathcal{A}\mathcal{W}_{1}$ defined as in Definition 2.11.

In the case of SDEs with sufficiently regular coefficients, however,
continuity of optimal stopping does hold for the adapted Wasserstein distance
$\mathcal{A}\mathcal{W}_{p}$ defined with respect to the $L^{p}$ norm on
$\Omega$ for any $p\geq 1$. For $\Lambda>0$, define
$\mathcal{P}^{\Lambda}\coloneqq\\{\mathrm{Law}(X)\colon\;X\text{ solves
}\eqref{eq:sde}\text{ with some }\Lambda\text{-Lipschitz }(b,\sigma)\\}$.

######  Corollary 8.4.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be Lipschitz and let $\Lambda>0$. Then
$v^{f}$ is continuous on $\mathcal{P}^{\Lambda}$ with respect to
$\mathcal{A}\mathcal{W}_{p}$ for any $p\geq 1$.

###### Proof.

By [7, Theorem 1.5], the adapted Wasserstein distance defined with respect to
the $L^{\infty}$ norm on $\Omega$ induces the same topology as
$\mathcal{A}\mathcal{W}_{p}$ for all $p\geq 1$. Thus the result follows from
Theorem 8.1. ∎

Establishing conditions for optimality of the synchronous coupling for the
adapted Wasserstein distance defined with respect to the $L^{\infty}$ norm on
$\Omega$ is an open problem. Thus, the right-hand side of (8.2) is not
computationally tractable. In contrast to that, the bound in Corollary 8.2
does admit an efficient numerical approximation, following the approach of
Section 7.1. Hence, our methods provide computationally feasible bounds on the
values of optimisation problems, making such problems tractable for
applications.

##  Appendix A Proofs from Section 3

###### Proof of Proposition 3.2.

For $h>0$, (i) implies that the Knothe–Rosenblatt rearrangement is optimal for
(2.9) between $\mu^{h}$ and $\nu^{h}$, by Proposition 2.9. Supposing that the
discretised processes $(X^{h},\bar{X}^{h})$ are driven by a common Brownian
motion, (ii) implies that the Knothe–Rosenblatt rearrangement is equal to
$\mathrm{Law}(X^{h},\bar{X}^{h})$, by Proposition 2.8. This together with
(iii) allows us to conclude that the limit (3.2) holds and the synchronous
coupling attains the value (BCOT) of the continuous-time bicausal transport
problem. This final step follows along the same lines as in the proof of [7,
Proposition 3.24], and we omit the details here. ∎

###### Proof of Lemma 3.3.

Let $\pi\in\mathrm{Cpl}_{\mathrm{bc}}(\mu,\nu)$. Then by [7, Proposition 2.2]
on the relation between bicausal couplings and correlated Brownian motions,
there exists a correlated Brownian motion $(W,\bar{W})$ such that
$\pi=\mathrm{Law}(X,\bar{X})$ when (2.13) is driven by $(W,\bar{W})$. Taking
the same driving Brownian motion, we write
$\pi^{N}\coloneqq\mathrm{Law}(X^{N},\bar{X}^{N})\in\mathrm{Cpl}_{\mathrm{bc}}(\mu^{N},\nu^{N})$,
for each $N\in\mathbb{N}$. Interpreting each such system of SDEs as a two-
dimensional SDE, the stability result of Stroock and Varadhan [74, Theorem
11.1.4] implies the weak convergence $\pi^{N}\to\pi$ as $N\to\infty$.
Moreover, the uniform linear growth bound implies that, for any $p\geq 1$,
there exists a constant $C>0$ such that
$\mathbb{E}^{\pi^{N}}[\sup_{t\in[0,T]}|(\omega_{t},\bar{\omega}_{t})|^{p}]\leq
C$ for all $N\in\mathbb{N}$; see, e.g. [22, Lemma 3.8]. Thus
$\mathbb{E}^{\pi^{N}}[\phi(\omega,\bar{\omega})]\to\mathbb{E}^{\pi}[\phi(\omega,\bar{\omega})]$,
for any functional $\phi\colon\Omega\times\Omega\to\mathbb{R}$ that is
continuous with order $p$ polynomial growth with respect to the uniform norm
on $\Omega\times\Omega$; see, e.g. [78, Definition 6.8]. We claim that
$(\omega,\bar{\omega})\mapsto\int_{0}^{T}c_{t}(\omega_{t},\bar{\omega}_{t})\,\mathrm{d}t$
satisfies these conditions. Indeed the polynomial growth of order $p$ follows
immediately from the bound (2.7), which holds uniformly in $t$. Continuity
follows from (2.7) and the continuity of the maps $(x,y)\mapsto c_{t}(x,y)$
for each $t\in[0,T]$ via the Lebesgue–Vitali dominated convergence theorem.
Hence

(A.1)
$\mathbb{E}^{\pi^{N}}\\!\left[\int_{0}^{T}c_{t}(\omega_{t},\bar{\omega}_{t})\,\mathrm{d}t\right]\\!\xrightarrow{N\to\infty}\mathbb{E}^{\pi}\\!\left[\int_{0}^{T}c_{t}(\omega_{t},\bar{\omega}_{t})\,\mathrm{d}t\right]\\!.$

Now take a common one-dimensional Brownian motion driving (2.13) for each set
of coefficients. Then we have

(A.2)
$\pi^{\mathrm{sync}}_{\mu^{N},\nu^{N}}=\mathrm{Law}(X^{N},\bar{X}^{N})\xrightarrow{N\to\infty}\mathrm{Law}(X,\bar{X})=\pi^{\mathrm{sync}}_{\mu,\nu}.$

Given that $\pi^{\mathrm{sync}}_{\mu^{N},\nu^{N}}$ minimises the left-hand
side of (A.1) over $\mathrm{Cpl}_{\mathrm{bc}}(\mu^{N},\nu^{N})$ for each
$N\in\mathbb{N}$, we have that $\pi^{\mathrm{sync}}_{\mu,\nu}$ minimises the
right-hand side of (A.1) over $\mathrm{Cpl}_{\mathrm{bc}}(\mu,\nu)$ and the
required convergence holds. ∎

###### Proof of Proposition 3.5.

Under Assumption 3.4, we can show that the monotone Euler–Maruyama scheme
satisfies Assumption 3.1, following the proofs of [7, Lemma 3.15, 3.16, 3.17].
The result then follows from Proposition 3.2. ∎

###### Proof of Corollary 3.7.

For each SDE, existence of a weak solution is guaranteed by Skorokhod [72,
Chapter 3, Section 3], and given that we assumed pathwise uniqueness, Yamada
and Watanabe [79, Corollary 1] implies that there exists a unique strong
solution. We can approximate the coefficients $(b,\sigma)$ and
$(\bar{b},\bar{\sigma})$ locally uniformly in space by coefficients satisfying
Assumption 3.4 and the conditions of Lemma 3.3. We conclude by combining
Proposition 3.5 and Lemma 3.3. ∎

##  Appendix B Proofs from Section 4.1

###### Proof of Lemma 4.6.

Direct computation of the first and second derivatives of $\bar{\phi}_{k}$
yield the first three points. We also observe that the first derivative is
continuous and bounded, and hence $\bar{\phi}_{k}$ is Lipschitz. The second
derivative is defined almost everywhere and is continuous and bounded on
$(-\infty,\xi_{k})\cup(\xi_{k},\infty)$. Combined with the fact that the one-
sided second derivatives at $\xi_{k}$ exist and are finite, this shows that
$\bar{\phi}_{k}^{\prime}$ is Lipschitz. Finally, differentiating
$\bar{\phi}_{k}^{\prime\prime}$ once again on
$(-\infty,\xi_{k})\cup(\xi_{k},\infty)$ gives a bounded third derivative, and
hence $\bar{\phi}_{k}^{\prime\prime}$ is piecewise Lipschitz. ∎

###### Proof of Lemma 4.7.

By Lemma 4.6, $\bar{\phi}_{k}(x)=0$ for $|x-\xi_{k}|\geq c_{0}$ for each
$k\in\\{1,\dotsc,m\\}$, and so we have $G(x)=x$ for
$x\in\mathbb{R}\setminus\bigcup_{k\in\\{1,\dotsc,m\\}}(\xi_{k}-c_{0},\xi_{k}+c_{0})$.
Moreover, for any $k\in\\{1,\dotsc,m\\}$ and
$x\in[\xi_{k}-c_{0},\xi_{k}+c_{0}]$, $G(x)=x+\alpha_{k}\bar{\phi}_{k}(x)$ and
$G^{\prime}(x)=1+\alpha_{k}\bar{\phi}_{k}^{\prime}(x)$, implying that
$G^{\prime}(x)\in[1-6c_{0}|\alpha_{k}|,1+6c_{0}|\alpha_{k}|]$ by Lemma 4.6.
Thus, as in [42, Lemma 2.2], we observe that our choice of $c_{0}$ in (4.7)
guarantees that $G$ is strictly increasing on $\mathbb{R}$ and therefore
admits a strictly increasing inverse $G^{-1}\colon\mathbb{R}\to\mathbb{R}$.
From the definition of $G$ and the properties proved in Lemma 4.6, we
immediately deduce that $G$ is bounded and Lipschitz with bounded Lipschitz
first derivative $G^{\prime}$ and bounded piecewise Lipschitz almost-
everywhere second derivative $G^{\prime\prime}$. By the inverse function
theorem, $G^{-1}$ has the same properties. ∎

###### Proof of Lemma 4.9.

As in Assumption 4.1, let $L_{f},K_{f},\gamma,\eta\in(0,\infty)$ denote the
constants that appear in the one-sided Lipschitz condition, local Lipschitz
condition and exponential growth condition for $f$. Write $L_{g}\in(0,\infty)$
for a Lipschitz constant of $g$, and
$M_{g}\coloneqq\sup_{x\in\mathbb{R}}|g(x)|<\infty$. Let
$k\in\\{0,\dotsc,m\\}$. By (A1)(i), $f$ is absolutely continuous on
$(\xi_{k},\xi_{k+1})$ and, as noted in Remark 4.5, the one-sided limits
$f(\xi_{k}-),f(\xi_{k}+)$ exist and are finite, and so
$M_{f}\coloneqq\sup_{x\in(\xi_{k},\xi_{k+1})}|f(x)|<\infty$. Therefore,
$(f\cdot g)\colon(\xi_{k},\xi_{k+1})\to\mathbb{R}$ is absolutely continuous,
as the product of bounded absolutely continuous functions. Now, for any
$x,y\in(\xi_{k},\xi_{k+1})$,

(B.1)
$\begin{split}(x-y)(f(x)g(x)-f(y)g(y))&=(x-y)(f(x)-f(y))g(x)+(x-y)(g(x)-g(y))f(y)\\\
&\leq(L_{f}M_{g}+L_{g}M_{f})|x-y|^{2},\end{split}$

and also

(B.2)
$\begin{split}|f(x)g(x)-f(y)g(y)|&\leq|f(x)-f(y)||g(x)|+|f(y)||g(x)-g(y)|\\\
&\leq
K_{f}M_{g}(\exp\\{\gamma|x|^{\eta}\\}+\exp\\{\gamma|y|^{\eta}\\})|x-y|+L_{g}M_{f}|x-y|\\\
&\leq(K_{f}M_{g}+L_{g}M_{f})(\exp\\{\gamma|x|^{\eta}\\}+\exp\\{\gamma|y|^{\eta}\\})|x-y|.\end{split}$

Thus, $(f\cdot g)\colon\mathbb{R}\to\mathbb{R}$ satisfies (A1). ∎

###### Proof of Lemma 4.10.

By (A1) and the continuity of $f$, (A1)(i), (A1)(ii), and (A1)(iii) also hold
on the closed intervals $[\xi_{k},\xi_{k+1}]$, for all
$k\in\\{1,\dotsc,m-1\\}$. By absolute continuity on the closed intervals, we
can define $f^{\prime}\in L^{1}(\mathbb{R})$ that agrees with the almost-
everywhere derivative of $f$ on each open interval. Linearity of the integral
then implies that $f^{\prime}$ is the almost-everywhere derivative of $f$ on
$\mathbb{R}$. Therefore $f$ is absolutely continuous on $\mathbb{R}$. In this
spirit, it is straightforward to show that (A1)(ii) and (A1)(iii) also extend
to the whole of $\mathbb{R}$. ∎

###### Proof of Lemma 4.11.

First, as in [42], we note that the constants $\alpha_{k}$,
$k\in\\{1,\dotsc,m\\}$ are chosen in (4.6) such that
$bG^{\prime}+1/2\sigma^{2}G^{\prime\prime}$ is continuous on $\mathbb{R}$.
Indeed by Lemma 4.6, for each $k\in\\{1,\dotsc,m\\}$,
$G^{\prime}(\xi_{k})=1+\alpha_{k}\bar{\phi}_{k}^{\prime}(\xi_{k})=1$,
$G^{\prime\prime}(\xi_{k}+)=\alpha_{k}\bar{\phi}_{k}^{\prime\prime}(\xi_{k}+)=2\alpha_{k}$,
$G^{\prime\prime}(\xi_{k}-)=-2\alpha_{k}$, and so

(B.3)
$\begin{split}(bG^{\prime}+\frac{1}{2}\sigma^{2}G^{\prime\prime})(\xi_{k}+)-(bG^{\prime}+\frac{1}{2}\sigma^{2}G^{\prime\prime})(\xi_{k}-)&=b(\xi_{k}+)-b(\xi_{k}-)+2\alpha_{k}\sigma^{2}(\xi_{k})=0.\end{split}$

Continuity on $\mathbb{R}$ then follows from Lemma 4.7. Also by Lemma 4.7,
$G^{\prime}$ is bounded and Lipschitz. Therefore, by Lemma 4.9, $bG^{\prime}$
satisfies (A1). The product $\sigma^{2}G^{\prime\prime}$ is zero on
$\mathbb{R}\setminus\bigcup_{k\in\\{1,\dotsc,m\\}}(\xi_{k}-c_{0},\xi_{k}+c_{0})$,
and $\sigma^{2}$ and $G^{\prime\prime}$, and hence their product, are bounded
and Lipschitz on each interval $[\xi_{k}-c_{0},\xi_{k}+c_{0}]$,
$k\in\\{1,\dotsc,m\\}$. Thus $\sigma^{2}G^{\prime\prime}$ satisfies (A1).
Taking the sum, $bG^{\prime}+1/2\sigma^{2}G^{\prime\prime}$ satisfies (A1), is
continuous on $\mathbb{R}$, and hence satisfies (A1)(i), (A1)(ii), and
(A1)(iii) globally on $\mathbb{R}$, by Lemma 4.10. Taking the composition with
$G^{-1}$, which is also Lipschitz by Lemma 4.7, preserves these properties,
and we conclude that $\tilde{b}$ satisfies statements (i)–(iii) of the lemma.

Now, $\sigma G^{\prime}$ is globally Lipschitz, as the product of the
Lipschitz function $\sigma$ and the bounded Lipschitz function $G^{\prime}$
that is zero on
$\mathbb{R}\setminus\bigcup_{k\in\\{1,\dotsc,m\\}}(\xi_{k}-c_{0},\xi_{k}+c_{0})$.
Taking the composition with the Lipschitz function $G^{-1}$ preserves the
Lipschitz property, and this concludes the proof of statement (iv).

Finally, statement (v) of the lemma follows from statements (ii) and (iv). ∎

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