March 2021

HU-EP-21/07-RTG

Beta functions for the duality-invariant sigma model

Roberto Bonezzi, Tomas Codina and Olaf Hohm

Institute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> The $O(d,d)$ invariant worldsheet theory for bosonic string theory with $d$
> abelian isometries is employed to compute the beta functions and Weyl
> anomaly at one-loop. We show that vanishing of the Weyl anomaly coefficients
> implies the equations of motion of the Maharana-Schwarz action. We give a
> self-contained introduction into the required techniques, including beta
> functions, the Weyl anomaly for two-dimensional sigma models and the
> background field method. This sets the stage for a sequel to this paper on
> generalizations to higher loops and $\alpha^{\prime}$ corrections.

###### Contents

  1. 1 Introduction
  2. 2 The Trace Anomaly
    1. 2.1 Generalities
    2. 2.2 String Sigma Model
  3. 3  Background Field Method
    1. 3.1 Effective Action
    2. 3.2 Nonlinear Sigma Model
    3. 3.3 One-loop Effective Action
    4. 3.4 The Dilaton
  4. 4 Duality-invariant Sigma Model
    1. 4.1 $O(d,d)$-invariant Worldsheet
    2. 4.2 Weyl Anomaly Equations
    3. 4.3 Beta Functions of Duality Invariant Sigma Model
    4. 4.4 Field Equations
  5. 5 Conclusions and Outlook
  6. A Technical details
    1. A.1 One-loop Feynman integrals
    2. A.2 Infrared regularization in the internal sector

##  1 Introduction

String theory is a theory of quantum gravity that at low energies reduces to
Einstein gravity coupled to matter fields. The physical spacetime in which
this theory is defined is usually referred to as the target space, to
distinguish it from the two-dimensional worldsheet of the fundamental string.
Physical observables such as scattering amplitudes may be computed using data
of this worldsheet theory, and the interplay of worldsheet and target space
techniques is one of the intriguing features of string theory. For instance,
demanding that the classical scale invariance of the worldsheet theory is
preserved quantum mechanically, i.e, that the Weyl anomaly vanishes, yields as
a consistency condition target space equations that, to lowest order in the
(inverse) string tension $\alpha^{\prime}$, are equivalent to the Einstein
equations. Higher loop corrections then imply that Einstein’s theory receives
(infinitely many) higher-derivative $\alpha^{\prime}$ corrections. Notably,
$\alpha^{\prime}$ corrections in the target space are a feature already of
classical string theory that can be derived from quantum considerations of the
worldsheet theory.

In this paper we compute the trace anomaly for a duality invariant worldsheet
theory at one-loop level. By duality we refer to the phenomenon that on the
space of string backgrounds (solutions of the stringy Einstein equations) with
$d$ abelian isometries there is a global $O(d,d,\mathbb{R})$ ‘T-duality’
invariance [1]. Specifically, the effective action obtained by dimensionally
reducing along these $d$ dimensions (i.e. taking the fields to be independent
of $d$ coordinates) is $O(d,d,\mathbb{R})$ invariant to all orders in
$\alpha^{\prime}$ [2]. This in turn poses strong constraints on the possible
higher-derivative corrections, which are only known to a few orders in
$\alpha^{\prime}$. For instance, the most general $O(d,d,\mathbb{R})$
invariant equations for purely time-dependent backgrounds (as relevant for
cosmological FLRW backgrounds with flat spatial metric) were recently
determined to all orders in $\alpha^{\prime}$ [3], but there is a finite
number of free parameters at each order in $\alpha^{\prime}$ that are not
constrained by duality and hence must be determined by other methods [4].

Traditionally, one would first determine the higher-derivative corrections for
the full target space string theory (say by requiring vanishing of the Weyl
anomaly) and, second, dimensionally reduce along $d$ directions. Both steps
are technically challenging (to put it mildly). Here we aim to circumvent the
need for a two-step procedure by employing a duality invariant worldsheet
theory that is directly adapted to the dimensional reduction and is manifestly
$O(d,d,\mathbb{R})$ invariant. This worldsheet action, which generalizes a
formulation due to Tseytlin [5, 6], was originally given by Schwarz and Sen
[7] and more recently re-derived in [8, 9]. In particular in [9] it was shown
that this model is a consistent truncation of the standard worldsheet theory
in which internal momentum and winding modes are set to zero, being the
worldsheet counterpart of taking the target space fields to be independent of
$d$ coordinates. Moreover, it was shown that the duality symmetries become
anomalous quantum mechanically due to the presence of chiral bosons. The
anomaly can then be cancelled by assigning a non-trivial duality
transformation to the 2-form B-field according to a Green-Schwarz mechanism.
This result gives a worldsheet interpretation for the observation that at
order $\alpha^{\prime}$ the target space equations are only
$O(d,d,\mathbb{R})$ invariant provided the singlet B-field transforms non-
trivially [10, 11], in analogy to the heterotic string [12, 13]. While this is
a one-loop effect, the anomaly itself is finite and hence does not enter the
one-loop beta function computation performed here, in agreement with the fact
that the Green-Schwarz deformation is invisible to lowest order in
$\alpha^{\prime}$. The anomaly will contribute, however, to any two-loop beta
function computation.

In the remainder of this paper we show at one loop that the duality invariant
worldsheet theory can be used to derive directly the dimensionally reduced
$O(d,d,\mathbb{R})$ invariant target space equations by demanding vanishing of
the Weyl anomaly. Specifically, we derive the $O(d,d,\mathbb{R})$ invariant
theory to zeroth order in $\alpha^{\prime}$ (second order in derivatives) and
show that it coincides with the theory derived by Maharana and Schwarz through
dimensional reduction. In this we generalize previous work in [14, 15] by
including the (doubled) Kaluza-Klein vector fields and the external B-field,
which leads to significant technical complications that we deal with in due
course. We also use the opportunity to give a self-contained introduction into
the general methods needed for this computation. While to some extent this is
textbook material, we found that at the level of detail needed for our
applications the required technology is somewhat scattered through the
literature of the 1980s, and we hope that our exposition will be helpful to
the community. This sets the stage for a sequel to this paper, in which we
extend the analysis to two loops and hence to first order in
$\alpha^{\prime}$. This will be an important additional test for the quantum
consistency of this non-standard duality invariant sigma model (whose
worldsheet diffeomorphism invariance, in particular, is realized in a non-
manifest fashion). Indeed, unexpected features may emerge given that the
target space equations cannot be written entirely in terms of the generalized
metric that enters the duality invariant sigma model [10, 11].

The rest of this paper is organized as follows. In sec. 2 we introduce the
core notions needed to compute the Weyl anomaly, in particular the beta
functions whose vanishing is related to (but not equivalent to) the vanishing
of the Weyl anomaly. In sec. 3 we explain the background field method and
clarify various issues that arise when computing the beta functions for the
conventional string worldsheet theory. These two sections are essentially
review but hopefully condense the needed techniques in a useful manner. Our
main new results are presented in sec. 4 where we display the one-loop
computation for the duality invariant worldsheet that yields the Maharana-
Schwarz theory. Some technical details are collected in an appendix.

##  2 The Trace Anomaly

Our goal is to extract target space field equations from a string sigma model
by requiring quantum consistency [16, 17, 18, 19, 20]. One ensures that the
non-linear sigma model is conformally invariant quantum mechanically by
imposing the vanishing of the Weyl anomaly. This condition is often stated to
be equivalent to the vanishing of the renormalization-group (RG) beta
functions $\beta^{i}$, but this is not quite correct for the string sigma
model: ${\beta^{i}=0}$ ensures only _rigid_ scale invariance or, equivalently,
the vanishing of the integrated trace $\int
d^{2}\sigma\sqrt{\gamma}\,T^{\alpha}{}_{\alpha}$ of the energy-momentum
tensor. The quantum string is consistent provided the stronger condition
$T^{\alpha}{}_{\alpha}=0$ holds locally. This leads to target space equations
for the true “Weyl anomaly coefficients” $\bar{\beta}^{i}$ of the schematic
form

$\bar{\beta}^{i}=\beta^{i}+\Delta_{{\cal W}}\varphi^{i}=0\;,$ (2.1)

where $\varphi^{i}$ are the target space fields for which $\beta^{i}$ are the
beta functions, and $\Delta$ is the operator implementing the gauge
transformations of $\varphi^{i}$, with field-dependent effective parameter
${\cal W}$. In this section we will review the derivation of an operator
expression for the Weyl anomaly [21, 22, 23], which will elucidate the origin
of the extra terms in (2.1). We shall follow closely the general discussion
given by Tseytlin in [22].

###  2.1 Generalities

We consider renormalizable field theories in two dimensions. The main example
relevant for our subsequent applications is the string sigma model with target
space metric $g_{\mu\nu}(X)$

$S=\frac{1}{2\lambda}\int
d^{2}\sigma\sqrt{\gamma}\,\gamma^{\alpha\beta}g_{\mu\nu}(X)\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\;,$
(2.2)

where $\gamma_{\alpha\beta}$ is the Euclidean worldsheet metric and
$\lambda=2\pi\alpha^{\prime}$. Recall that the $g_{\mu\nu}(X)$, which are
functions on the $D-$dimensional target space, are viewed as a collection of
infinitely many coupling constants. Upon quantization we have to distinguish
between bare couplings and renormalized couplings. Using the dot product

$f\cdot g=\int d^{D}x\,f(x)\,g(x)\;,$ (2.3)

we write the bare action as

$S_{0}=\int d^{n}\sigma\,A_{i0}\cdot\varphi^{i}_{0}\;,$ (2.4)

where $\varphi^{i}_{0}$ and $A_{i0}$ denote bare couplings and composite
operators, respectively, and $n=2+\epsilon$, where $\epsilon$ is the
dimensional regularization parameter. For instance, for the string sigma model
(2.2) one has

$\varphi^{i}_{0}=g_{0\mu\nu}(x)\;,\quad
A_{i0}=\frac{1}{2\lambda}\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,\delta^{D}(x-X(\sigma))\;.$
(2.5)

We denote the renormalized couplings by $\varphi^{i}$ and choose them to be
dimensionless. The bare couplings $\varphi^{i}_{0}$ are taken to have mass
dimension $\epsilon=n-2$, implying that the bare operators $A_{i0}$ have
dimension 2.

In the following we perform renormalization via minimal subtraction, in which
the counterterms are purely divergent. Expressing the counterterms as a
Laurent series in the dimensional regularization parameter $\epsilon$ one
writes the bare action (2.4) as a sum of the renormalized action and the
counterterms,

$\begin{split}S_{0}&=S_{\rm ren}+S_{\rm c.t.}\;,\\\ S_{\rm ren}&=\int
d^{n}\sigma\,\mu^{\epsilon}A_{i0}\cdot\varphi^{i}\;,\qquad S_{\rm c.t.}=\int
d^{n}\sigma\,\mu^{\epsilon}A_{i0}\cdot\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,T^{i}_{n}(\varphi)\;,\end{split}$
(2.6)

where $\mu$ is the renormalization scale that is introduced in order to keep
the renormalized couplings dimensionless. The action (2.6) is used to derive
the Feynman rules, and the counterterms $T^{i}_{n}$ are fixed by demanding
that they cancel the divergent contributions to the quantum effective action.
Comparison of (2.6) with the canonical form (2.4) of the bare action gives the
bare couplings in terms of the renormalized ones:

$\varphi^{i}_{0}=\mu^{\epsilon}\Big{[}\varphi^{i}+\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,T^{i}_{n}(\varphi)\Big{]}\;.$
(2.7)

With this relation we may compute the beta functions associated to the
couplings $\varphi^{i}$ in $n=2+\epsilon$ dimensions, which are defined by

$\beta^{i}:=\frac{d\varphi^{i}}{dt}+\epsilon\,\varphi^{i}\,,\qquad
t:=\log\mu\;,$ (2.8)

where the derivative of $\varphi$ is obtained by requiring that the bare
couplings $\varphi^{i}_{0}$ do not depend on this renormalization scale:

$\frac{d\varphi^{i}_{0}}{dt}=0\,.$ (2.9)

Differentiating (2.7) gives

$0=\epsilon\varphi^{i}+T^{i}_{1}+{\cal
O}(\epsilon^{-1})+\frac{d\varphi^{i}}{dt}+\frac{1}{\epsilon}\,\frac{d\varphi^{j}}{dt}\frac{\partial}{\partial\varphi^{j}}T_{1}^{i}+{\cal
O}(\epsilon^{-2})\;.$ (2.10)

Matching the ${\cal O}(\epsilon)$ and ${\cal O}(\epsilon^{0})$ terms one
obtains

$\hat{\beta}^{i}:=\frac{d\varphi^{i}}{dt}=-\epsilon\,\varphi^{i}+\beta^{i}\;,\quad\beta^{i}=-T^{i}_{1}+\varphi\cdot\frac{\partial}{\partial\varphi}T^{i}_{1}\;,$
(2.11)

where $\hat{\beta}^{i}$ is a short-hand notation that shall be useful in what
follows. Note that the expression for $\beta^{i}$ is exact in terms of the
counterterm $T^{i}_{1}$ and the usual perturbative evaluation arises from the
loop expansion of the latter. The higher order terms in $\epsilon^{-n}$
provide the so-called pole relations between higher order counterterms [24,
25]. Note that the above operator
$\varphi\cdot\frac{\partial}{\partial\varphi}$ has to be understood in terms
of the functional derivative as

$f\cdot\frac{\partial F}{\partial\varphi}=\int d^{D}x\,f^{i}(x)\,\frac{\delta
F}{\delta\varphi^{i}(x)}\;.$ (2.12)

The partition function of the theory is given by the path integral

$Z(\varphi,\mu)=e^{-W(\varphi,\mu)}=\int\mathcal{D}X\,e^{-S_{0}}\;,$ (2.13)

and quantum expectation values are denoted by
$\langle\cdots\rangle=\frac{1}{Z}\int\mathcal{D}X\cdots\,e^{-S_{0}}$. Given
that the theory is renormalizable, the bare action (2.6) contains all the
required counterterms to render the above path integral finite. Since the bare
action (2.4) written in terms of the bare couplings does not depend on $\mu$
we can now derive the renormalization group equation

$\frac{dW}{dt}=\frac{\partial W}{\partial t}+\hat{\beta}^{i}\,\frac{\partial
W}{\partial\varphi^{i}}=0\;,$ (2.14)

where we used (2.11). The next ingredient we need is a prescription to define
renormalized composite operators $[A_{i}]\,$ for the bare couplings in (2.4),
where by $[\cdots]$ we mean some normal-ordering prescription ensuring
finiteness. Recalling (2.4), the integral of the bare operator $A_{i0}$ is
given by111 Notice that for the worldsheet action (2.2) we have  $\frac{\delta
S_{0}}{\delta g_{0\mu\nu}(x)}=\frac{1}{2\lambda}\int
d^{n}\sigma\,\delta^{D}(x-X(\sigma))\,\partial_{\alpha}X^{\mu}\partial^{\alpha}X^{\nu}\;,$
i.e., the derivative is still integrated over the worldsheet.

$\int d^{n}\sigma\,A_{i0}=\frac{\partial S_{0}}{\partial\varphi^{i}_{0}}\;.$
(2.15)

Accordingly, we now define the renormalized composite operators $[A_{i}]$ by
demanding

$\int d^{n}\sigma[A_{i}]=\frac{\partial S_{0}}{\partial\varphi^{i}}\;.$ (2.16)

The quantum expectation value is then given by

$\Big{\langle}\int d^{n}\sigma[A_{i}]\Big{\rangle}=\frac{\partial
W}{\partial\varphi^{i}}\,,$ (2.17)

that is guaranteed to be finite, because it is the derivative of a finite
quantity by a finite parameter. Given (2.16) and (2.15) the relation between
bare and renormalized operators is known up to possible total derivative
terms:

$[A_{i}]=A_{j0}\cdot\frac{\partial\varphi^{j}_{0}}{\partial\varphi^{i}}+\partial_{\alpha}\Omega^{\alpha}_{i}\;.$
(2.18)

Assuming that the set $\\{A_{i0}\\}$ is a complete basis of dimension 2
operators (modulo the bare equations of motions $\frac{\delta S_{0}}{\delta
X^{\mu}}$ that we shall always discard, having zero expectation value), the
total derivative part can also be expanded in terms of $A_{i0}$, namely

$\partial_{\alpha}\Omega^{\alpha}_{i}=A_{j0}\cdot\Lambda^{j}_{i}\;,\quad\Lambda_{i}^{j}=\mu^{\epsilon}\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,Q^{j}_{ni}(\varphi)\;.$
(2.19)

This allows us to define the renormalization matrix as

$[A_{i}]=A_{j0}\cdot Z^{j}_{i}\;,\quad
Z^{j}_{i}=\frac{\partial\varphi^{j}_{0}}{\partial\varphi^{i}}+\Lambda^{j}_{i}\,.$
(2.20)

Given the relation (2.7) between bare and renormalized couplings, we infer

$Z^{i}_{j}=\mu^{\epsilon}\Big{[}\delta^{i}_{j}+\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,X^{i}_{nj}(\varphi)\Big{]}\;,\quad
X^{i}_{nj}=\frac{\partial T^{i}_{n}}{\partial\varphi^{j}}+Q^{i}_{nj}\;.$
(2.21)

It should be emphasized that while for ordinary field theories the $Z^{i}_{j}$
are finite-dimensional matrices, for the string sigma model they are actually
differential operators. Specifically, one has
$T^{i}_{1}=T^{g}_{1\mu\nu}=-\alpha^{\prime}R_{\mu\nu}$ at one-loop order, with
$R_{\mu\nu}$ the target space Ricci tensor, giving

$\frac{\partial T^{i}_{1}}{\partial\varphi^{j}}\;\rightarrow\frac{\delta
T_{1\mu\nu}(x)}{\delta
g_{\lambda\rho}(y)}=\frac{\alpha^{\prime}}{2}\,\Big{[}\delta^{(\lambda}_{\mu}\delta^{\rho)}_{\nu}\nabla^{2}-2\,\delta^{(\lambda}_{(\mu}\nabla^{\rho)}\nabla_{\nu)}+g^{\lambda\rho}\nabla_{(\mu}\nabla_{\nu)}\Big{]}_{x}\delta^{D}(x-y)\;,$
(2.22)

which back in (2.21) implies that $Z^{i}_{j}$ is a differential operator. It
is often more convenient to contract such operators with arbitrary functions,

$\frac{\partial T^{i}_{1}}{\partial\varphi^{j}}\cdot
F_{j}\;\rightarrow\;\frac{\alpha^{\prime}}{2}\,\Big{[}\nabla^{2}F_{\mu\nu}-2\nabla^{\lambda}\nabla_{(\mu}F_{\nu)\lambda}+\nabla_{\mu}\nabla_{\nu}F\Big{]}\;.$
(2.23)

At this point, the main goal is to find an operator expression for the trace
of the energy-momentum tensor. At the quantum level one can choose

$T^{\alpha}{}_{\alpha}=\frac{2}{\sqrt{\gamma}}\,\gamma^{\alpha\beta}\frac{\delta
S_{0}}{\delta\gamma^{\alpha\beta}}\;,$ (2.24)

which implies that the expectation value

$\big{\langle}T^{\alpha}{}_{\alpha}\big{\rangle}=\frac{2}{\sqrt{\gamma}}\,\gamma^{\alpha\beta}\frac{\delta
W}{\delta\gamma^{\alpha\beta}}$ (2.25)

is finite. The Weyl variation of the bare operators is given by

$2\,\gamma^{\alpha\beta}\frac{\partial}{\partial\gamma^{\alpha\beta}}A_{i0}=-\epsilon\,A_{i0}+\partial_{\alpha}\omega^{\alpha}_{i}\,,$
(2.26)

where, again, we took into account possible total derivative terms. For the
$A_{i0}$ defined in (2.5) this may be verified by a direct computation (in
which case there are no total derivative terms). It should be noted that since
the bare operators $A_{i0}$ enter (2.26) this total derivative is given
directly by the Weyl variation of the Lagrangian, implying that it requires no
quantum computation.

In the remainder of this subsection we derive a general expression for
$T^{\alpha}{}_{\alpha}$ and the relation between the RG beta functions
$\beta^{i}$ and the Weyl anomaly coefficient $\bar{\beta}^{i}$. We can, as
before, use the completeness of the basis $\\{A_{i0}\\}$ modulo equations of
motion to expand

$\partial_{\alpha}\omega^{\alpha}_{i}=A_{j0}\cdot\lambda^{j}_{i}(\varphi_{0})\;,\quad\lambda^{j}_{i}(\varphi_{0})=\lambda^{j}_{i}(\varphi)+{\cal
O}(\epsilon^{-1})\;.$ (2.27)

The trace operator is then computed as

$\begin{split}\sqrt{\gamma}\,T^{\alpha}{}_{\alpha}&=2\,\gamma^{\alpha\beta}\frac{\delta}{\delta\gamma^{\alpha\beta}}S_{0}=2\,\gamma^{\alpha\beta}\frac{\delta}{\delta\gamma^{\alpha\beta}}\int
d^{n}\sigma\,A_{i0}\cdot\varphi^{i}_{0}\\\
&=(-\epsilon\,A_{i0}+\partial_{\alpha}\omega^{\alpha}_{i})\cdot\varphi^{i}_{0}=A_{i0}\cdot\psi^{i}\;,\end{split}$
(2.28)

where

$\begin{split}\psi^{i}&=-\epsilon\,\varphi^{i}_{0}+\lambda^{i}_{j}(\varphi_{0})\cdot\varphi^{j}_{0}\\\
&=\mu^{\epsilon}\Big{[}-\epsilon\,\varphi^{i}-T^{i}_{1}(\varphi)+\lambda^{i}_{j}(\varphi)\cdot\varphi^{j}+{\cal
O}(\epsilon^{-1})\Big{]}\;,\end{split}$ (2.29)

thanks to (2.7). The last step consists in rewriting the bare operators in
terms of the renormalized ones, _i.e._ $A_{i0}=[A_{j}]\cdot(Z^{-1})^{j}_{i}$
where, by using (2.21),

$(Z^{-1})^{j}_{i}=\mu^{-\epsilon}\,\Big{[}\delta^{j}_{i}-\frac{1}{\epsilon}\,X^{j}_{1i}+{\cal
O}(\epsilon^{-2})\Big{]}\;.$ (2.30)

Combining the last two expressions we find

$(Z^{-1})^{i}_{j}\cdot\psi^{j}=-\epsilon\,\varphi^{i}-T^{i}_{1}+\varphi\cdot\frac{\partial}{\partial\varphi}T^{i}_{1}+\lambda^{i}_{j}\cdot\varphi^{j}+Q^{i}_{1j}\cdot\varphi^{j}\;.$
(2.31)

In the above formula all the divergent terms must cancel out, since by
definition both $T^{\alpha}{}_{\alpha}$ and $[A_{i}]$ are finite. Upon letting
$\epsilon=0$ the Weyl anomaly operator can thus be written as

$\sqrt{\gamma}\,T^{\alpha}{}_{\alpha}=[A_{i}]\cdot{\bar{\beta}}^{i}\;,$ (2.32)

where the Weyl anomaly coefficients are given by

$\bar{\beta}^{i}=\beta^{i}+\big{(}\lambda^{i}_{j}+Q^{i}_{1j}\big{)}\cdot\varphi^{j}\;.$
(2.33)

This formula is the precise form of (2.1), for which we will see in the next
subsection that the extra term encodes a target space gauge variation. One can
see that the Weyl anomaly coefficients $\bar{\beta}^{i}$ differ from the RG
$\beta-$functions by two total derivative terms: the $\lambda-$contribution
can be found simply by varying the classical action, while the
$Q-$contribution, which is much harder to compute, can be found by direct
renormalization of the dimension two operators $A_{i}\,$.

###  2.2 String Sigma Model

We can now specialize the general procedure outlined above to the case of the
string sigma model coupled to all massless background fields [22], and
evaluate the right-hand side of (2.33). In particular, we analyze the dilaton
contributions. We start from the Euclidean bare action

$\begin{split}S_{0}&=\frac{1}{2\lambda}\int
d^{n}\sigma\,\Big{\\{}\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{0\mu\nu}(X)+i\,\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}B_{0\mu\nu}(X)+\alpha^{\prime}\,\sqrt{\gamma}\bar{R}\,\Phi_{0}(X)\Big{\\}}\;,\end{split}$
(2.34)

with the loop-counting parameter ${\lambda=2\pi\alpha^{\prime}}\,$. The
dimensionally extended worldsheet curvature $\bar{R}$ is defined with the
unusual normalization $\bar{R}=\frac{1}{n-1}\,R^{(n)}$ [22, 23], which is
chosen purely for future convenience. The antisymmetric density
$\varepsilon^{\alpha\beta}$ is the most problematic object to define in
$n=2+\epsilon$ dimensions. Following [22], we choose

$\varepsilon^{\alpha\beta}=\sqrt{\gamma}\,e^{\alpha}{}_{a}\,e^{\beta}{}_{b}\,\epsilon^{ab}\;,$
(2.35)

where $e_{\alpha}{}^{a}(\sigma)$ are worldsheet (zwei$\,+\,\epsilon$)-beins,
and $\epsilon^{ab}$ is a _constant_ antisymmetric Lorentz tensor obeying
$\epsilon^{ac}\epsilon_{bc}=\delta^{a}_{b}$ in $n$ dimensions. Under a Weyl
variation one has

$2\,\gamma^{\alpha\beta}\frac{\delta}{\delta\gamma^{\alpha\beta}}\,\int
d^{n}\sigma\,\varepsilon^{\gamma\delta}B_{\gamma\delta}=e^{\alpha}{}_{a}\,\frac{\delta}{\delta
e^{\alpha}{}_{a}}\,\int
d^{n}\sigma\,\varepsilon^{\gamma\delta}B_{\gamma\delta}=-\epsilon\,\varepsilon^{\alpha\beta}B_{\alpha\beta}\;,$
(2.36)

where $B_{\alpha\beta}$ is the pullback of the target space $B-$field. The
above choice for the continuation to $n$ dimensions has an obvious problem:
the action (2.34) is not gauge-invariant under transformations of the
$B-$field $\delta B_{\mu\nu}=\partial_{[\mu}\lambda_{\nu]}$, since the
conformal factor of the metric does not drop out for $n=2+\epsilon$. For
instance, in conformal gauge $e_{\alpha}{}^{a}=e^{\rho}\delta_{\alpha}^{a}$
one has $\varepsilon^{\alpha\beta}=e^{\epsilon\rho}\epsilon^{\alpha\beta}$ and
$\partial_{\alpha}(\varepsilon^{\alpha\beta})=\epsilon\,\partial_{\alpha}\rho\,\varepsilon^{\alpha\beta}$.
This problem ultimately leads to a trace operator
$\sqrt{\gamma}T^{\alpha}{}_{\alpha}$ that violates the $B-$field gauge
invariance [22]. The proposal of [22] was to modify the definition of the
trace anomaly operator by further total derivative counterterms, whose role is
precisely to cancel non gauge-invariant contributions. In the following we
will ignore terms containing $\partial_{\alpha}\varepsilon^{\alpha\beta}$
(they do not alter the general form of $T^{\alpha}{}_{\alpha}$ when
$\epsilon\to 0$) and comment on this issue at the end of this section.

Given the sigma model action (2.34), our goal is to determine the general form
of the total derivative terms $\lambda^{i}_{j}\cdot\varphi^{j}$ and
$Q^{i}_{1j}\cdot\varphi^{j}$ appearing in the Weyl anomaly coefficients
(2.33). In order to do so, it is useful to classify the possible total
derivative terms. There are two total derivative operators of second order in
worldsheet derivatives that are worldsheet diffeomorphism invariant. We denote
these operators by $d_{\gamma}$ and $d_{\varepsilon}$, which act on target
space vector fields $N_{\mu}$ and $K_{\mu}$ as

$\begin{split}d_{\gamma}^{\mu}\cdot
N_{\mu}&:=\frac{1}{2\lambda}\,\partial_{\alpha}\,\big{[}\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\beta}X^{\mu}\,N_{\mu}(X)\big{]}\;,\\\
d^{\mu}_{\varepsilon}\cdot
K_{\mu}&:=\frac{i}{2\lambda}\,\partial_{\alpha}\,\big{[}\varepsilon^{\alpha\beta}\partial_{\beta}X^{\mu}\,K_{\mu}(X)\big{]}\;.\end{split}$
(2.37)

According to our previous discussion, it should be possible to expand these
total derivatives (modulo the bare equations of motion) in terms of a basis of
dimension 2 bare operators $\\{A_{i0}\\}$, which in the present case are given
by

$\begin{split}A_{g0}\cdot
F^{g}&=\frac{1}{2\lambda}\,\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,F^{g}_{\mu\nu}(X)\;,\quad
A_{B0}\cdot
F^{B}=\frac{i}{2\lambda}\,\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,F_{\mu\nu}^{B}(X)\;,\\\
A_{\Phi 0}\cdot
F^{\Phi}&=\frac{\alpha^{\prime}}{2\lambda}\,\sqrt{\gamma}\bar{R}\,F^{\Phi}(X)\;,\end{split}$
(2.38)

upon contraction with arbitrary functions. In order to show that any total
derivative (2.37) can be expanded in terms of (2.38), we shall need the bare
equations of motion computed from (2.34):

$\frac{\lambda}{\sqrt{\gamma}}\,g^{\mu\nu}_{0}\frac{\delta S_{0}}{\delta
X^{\nu}}=-\mathcal{D}^{\alpha}_{0}\partial_{\alpha}X^{\mu}+\frac{i}{2\sqrt{\gamma}}\,\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\nu}\partial_{\beta}X^{\lambda}\,H^{\mu}_{0}{}_{\nu\lambda}+\frac{\alpha^{\prime}}{2}\,\bar{R}\,\nabla^{\mu}_{0}\Phi_{0}=0\;,$
(2.39)

where $H_{\mu\nu\lambda}=3\,\partial_{[\mu}B_{\nu\lambda]}$. All quantities
with a subscript $0$ are constructed from the bare target space fields, and
$\mathcal{D}_{\alpha}$ is the covariant derivative w.r.t the worldsheet and
target space Levi-Civita connections, i.e.,

$\mathcal{D}_{\alpha}v_{\beta}^{\mu}=\partial_{\alpha}v_{\beta}^{\mu}-\Gamma_{\alpha\beta}^{\gamma}(\gamma)\,v_{\gamma}^{\mu}+\partial_{\alpha}X^{\nu}\,\Gamma^{\mu}_{\nu\lambda}(g)\,v_{\beta}^{\lambda}\;.$
(2.40)

Expanding the derivatives in (2.37) we obtain, for instance,

$\begin{split}2\lambda\,d_{\gamma}^{\mu}\cdot
N_{\mu}&=\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,\partial_{\mu}N_{\nu}(X)+N_{\mu}(X)\partial_{\alpha}\big{(}\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\beta}X^{\mu}\big{)}\\\
&=\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,\nabla_{0\mu}N_{\nu}(X)+\sqrt{\gamma}\,N_{\mu}(X)\,\mathcal{D}_{0}^{\alpha}\partial_{\alpha}X^{\mu}\;.\end{split}$
(2.41)

Using the bare equations of motion (2.39) one can see that any total
derivative term can be expanded in the basis (2.38) as:

$\begin{split}d_{\gamma}^{\mu}\cdot
N_{\mu}&=A^{\mu\nu}_{g0}\cdot\Big{(}\nabla_{0(\mu}N_{\nu)}\Big{)}+A^{\mu\nu}_{B0}\cdot\Big{(}\tfrac{1}{2}\,H^{\lambda}_{0\mu\nu}\,N_{\lambda}\Big{)}+A_{\Phi
0}\cdot\Big{(}\tfrac{1}{2}\,\nabla^{\mu}_{0}\Phi_{0}\,N_{\mu}\Big{)}\;,\\\
d^{\mu}_{\varepsilon}\cdot
K_{\mu}&=A^{\mu\nu}_{B0}\cdot\Big{(}\partial_{[\mu}K_{\nu]}\Big{)}\;.\end{split}$
(2.42)

Let us emphasize that the above decomposition takes the form of operators
$A_{i0}$ contracted with the target space gauge (or diffeomorphism) variation
of the corresponding field, i.e.

$2\,d^{\mu}_{\gamma}\cdot N_{\mu}=A_{i0}\cdot{\cal
L}_{N}\varphi^{i}_{0}\;,\qquad 2\,d^{\mu}_{\varepsilon}\cdot
K_{\mu}=A_{B0}\cdot\delta_{K}B_{0}\;,$ (2.43)

where the Lie derivative on the $B-$field is meant to act in covariant form,
meaning ${\cal L}_{N}B_{\mu\nu}=N^{\lambda}H_{\lambda\mu\nu}$, modulo a gauge
transformation. The general form of (2.43) is to be expected on general
grounds: focusing on target space diffeomorphisms, for instance, we recall
that the action (2.34) is invariant if one transforms _both_ the target space
couplings $\varphi^{i}=(g_{\mu\nu},B_{\mu\nu},\Phi)$ and the worldsheet fields
$X^{\mu}$. This yields the identity

$\delta_{\xi}S=\int d^{2}\sigma\,\Big{[}-\xi^{\mu}(X)\,\frac{\delta S}{\delta
X^{\mu}}+{\cal L}_{\xi}\varphi^{i}\cdot\frac{\delta
S}{\delta\varphi^{i}}\Big{]}=\int
d^{2}\sigma\,\Big{[}-\xi^{\mu}(X)\,\frac{\delta S}{\delta X^{\mu}}+{\cal
L}_{\xi}\varphi^{i}\cdot A_{i}\Big{]}=0\;,$ (2.44)

where we used that the integrated operators $A_{i}$ are obtained by
differentiating the action w.r.t. the couplings, c.f. (2.15). Since we are
discarding the equations of motion operator, i.e. $\frac{\delta S}{\delta
X^{\mu}}=0$, the integral of the second term must vanish, implying that
$A_{i}\cdot{\cal L}_{\xi}\varphi^{i}$ is a total worldsheet derivative, hence
proportional to $d^{\mu}_{\gamma}\cdot\xi_{\mu}$. The argument for the
$B-$field gauge transformation is completely analogous.

With the expansion (2.42) we are ready to determine the general form of the
$\lambda^{i}_{j}$ and $Q^{i}_{1j}$ terms in the Weyl anomaly. We start from
the $\lambda^{i}_{j}$ contribution that, recalling (2.26) and (2.27), can be
found by the Weyl variation of the Lagrangian. Upon using the general
variation of the Ricci scalar

$\delta
R=\gamma_{\alpha\beta}\nabla^{2}\delta\gamma^{\alpha\beta}-\nabla_{\alpha}\nabla_{\beta}\,\delta\gamma^{\alpha\beta}+R_{\alpha\beta}\,\delta\gamma^{\alpha\beta}\;,$
(2.45)

the Weyl variation of (2.34) yields

$\begin{split}\sqrt{\gamma}\,T^{\alpha}{}_{\alpha}&=2\,\gamma^{\alpha\beta}\frac{\delta
S_{0}}{\delta\gamma^{\alpha\beta}}=\frac{1}{2\lambda}\,\Big{[}-\epsilon\,\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{0\mu\nu}(X)-\epsilon\,i\,\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}B_{0\mu\nu}(X)\\\
&\hskip
102.42992pt-\epsilon\,\alpha^{\prime}\sqrt{\gamma}\bar{R}\,\Phi_{0}(X)\Big{]}+\frac{1}{2\pi}\,\sqrt{\gamma}\,\nabla^{\alpha}\partial_{\alpha}\Phi_{0}(X)\\\\[5.69054pt]
&=A_{g0}\cdot(-\epsilon\,g_{0})+A_{B0}\cdot(-\epsilon\,B_{0})+A_{\Phi
0}\cdot(-\epsilon\,\Phi_{0})+\frac{1}{2\pi}\,\partial_{\alpha}\big{[}\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\beta}\Phi_{0}\big{]}\;.\end{split}$
(2.46)

The total derivative term in here can be expressed in terms of (2.37):

$\frac{1}{2\pi}\partial_{\alpha}\big{[}\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\beta}\Phi_{0}\big{]}=d_{\gamma}^{\mu}\cdot
N_{\mu}\;,\qquad\text{where}\qquad
N_{\mu}=2\alpha^{\prime}\partial_{\mu}\Phi_{0}\;.$ (2.47)

By using the decomposition (2.42), the trace operator can thus be put in the
form $\sqrt{\gamma}\,T^{\alpha}{}_{\alpha}=A_{i0}\cdot\psi^{i}$, with

$\begin{split}\psi^{g}_{\mu\nu}&=-\epsilon\,g_{0\mu\nu}+2\alpha^{\prime}\,\nabla_{0\mu}\partial_{\nu}\Phi_{0}\;,\quad\psi^{B}_{\mu\nu}=-\epsilon\,B_{0\mu\nu}+\alpha^{\prime}\,H^{\lambda}_{0}{}_{\mu\nu}\,\partial_{\lambda}\Phi_{0}\;,\\\
\psi^{\Phi}&=-\epsilon\,\Phi_{0}+\alpha^{\prime}\,\nabla_{0}^{\mu}\Phi_{0}\,\partial_{\mu}\Phi_{0}\;.\end{split}$
(2.48)

Comparing the above result with the general formulas

$\psi^{i}=-\epsilon\,\varphi^{i}_{0}+\lambda^{i}_{j}(\varphi_{0})\cdot\varphi^{j}_{0}\;,\quad\bar{\beta}^{i}=\beta^{i}+\lambda^{i}_{j}(\varphi)\cdot\varphi^{j}+Q^{i}_{1j}(\varphi)\cdot\varphi^{j}\;,$
(2.49)

we can read off the $\lambda^{i}_{j}$ from the first equation, and using this
in the second equation one obtains the $\lambda-$contribution to the Weyl
anomaly coefficients in terms of the familiar dilaton terms:

$\begin{split}\bar{\beta}^{g}_{\mu\nu}&=\beta^{g}_{\mu\nu}+2\alpha^{\prime}\,\nabla_{\mu}\nabla_{\nu}\Phi+\cdots\;,\quad\bar{\beta}^{B}_{\mu\nu}=\beta^{B}_{\mu\nu}+\alpha^{\prime}\,H^{\lambda}{}_{\mu\nu}\nabla_{\lambda}\Phi+\cdots\;,\\\\[5.69054pt]
\bar{\beta}^{\Phi}&=\beta^{\Phi}+\alpha^{\prime}\,(\nabla\Phi)^{2}+\cdots\;,\end{split}$
(2.50)

where the dots stand for the contributions $Q^{i}_{1j}\cdot\varphi^{j}\,$.

Instead of trying to determine the $Q-$contributions to $\bar{\beta}^{i}$, we
shall only fix their general form. Let us recall from (2.18) and (2.19) that
objects involving $Q^{i}_{1j}$ arise from the decomposition along the basis
$\\{A_{i0}\\}$ of the total derivatives
$\partial_{\alpha}\Omega^{\alpha}_{j}$. Since any total worldsheet derivative
can only depend on two distinct target space vectors (see (2.37)), the
$Q-$contributions can be parametrized in terms of two undetermined field-
dependent vectors: $W_{\mu}$, related to the $d_{\gamma}$ structure, and
$L_{\mu}$, related to the $d_{\varepsilon}$ structure. Since the decomposition
(2.42) is valid for any pair of vector fields, the $W-$contribution to the
Weyl anomaly must be of the form of a target space diffeomorphism, c.f.
(2.43), while the $L-$contribution can only take the form of a $B-$field gauge
transformation. Including these terms in the expression for the Weyl anomaly
coefficients, we obtain the final form

$\begin{split}\bar{\beta}^{g}_{\mu\nu}&=\beta^{g}_{\mu\nu}+2\alpha^{\prime}\,\nabla_{\mu}\nabla_{\nu}\Phi+\nabla_{(\mu}W_{\nu)}\;,\quad\\\\[5.69054pt]
\bar{\beta}^{B}_{\mu\nu}&=\beta^{B}_{\mu\nu}+\alpha^{\prime}\,H^{\lambda}{}_{\mu\nu}\nabla_{\lambda}\Phi+\tfrac{1}{2}\,H^{\lambda}{}_{\mu\nu}W_{\lambda}+\partial_{[\mu}L_{\nu]}\;,\\\\[5.69054pt]
\bar{\beta}^{\Phi}&=\beta^{\Phi}+\alpha^{\prime}\,(\nabla\Phi)^{2}+\tfrac{1}{2}\nabla^{\mu}\Phi\,W_{\mu}\;.\end{split}$
(2.51)

This gives the $Q$ contributions as the terms involving $W$ and $L$.

Let us discuss this result in more detail. First of all, the renormalization
of the dilaton operator $A_{\Phi}$ does not produce total derivatives. This
can be understood by observing that the scalar curvature operator
$\bar{R}\sim\partial\partial\gamma$ cannot mix with total derivative
operators, which contain $\partial X$, under renormalization. A similar
argument, which we will review in the next section, shows that the
$\beta-$functions $\beta^{g}_{\mu\nu}$ and $\beta^{B}_{\mu\nu}$ do not depend
on the dilaton, and $\beta^{\Phi}$ itself is only linear in $\Phi$, i.e.
$\beta^{\Phi}=\Delta(g,B)\,\Phi+\omega(g,B)$. We shall thus stress that the
dilaton dependence displayed in (2.51) is exact to all orders in
$\alpha^{\prime}$ and, in particular, $W_{\mu}(g,B)$ and $L_{\mu}(g,B)$ do not
depend on $\Phi$.

We conclude this section by discussing the issue with the $B-$field gauge
invariance. As we have pointed out at the beginning of this subsection, the
definition of the $\varepsilon^{\alpha\beta}$ density (2.35) in $n$ dimensions
breaks gauge invariance, in that the vectors $W_{\mu}$ and $L_{\mu}$ have in
principle a general dependence on $B$, rather than on $H=dB$. Another way to
circumvent this problem, compared to the proposal of [22], is to derive a
manifestly gauge invariant perturbative expansion in two dimensions, and to
extend the gauge invariant interactions to $n=2+\epsilon$ dimensions only
afterwards. Finally, one can try to fix $W$ and $L$ by more indirect means
(see _e.g._ [26]). For instance, by expanding in powers of $\alpha^{\prime}$
and imposing covariance, the most general form of $W$ is
$W_{\mu}(g,B)=\sum_{L=1}^{\infty}(\alpha^{\prime})^{L}W_{\mu}^{(L)}(R,\nabla,H)$,
where $W_{\mu}^{(L)}$ contains $2L-1$ derivatives of the metric and $B-$field.
This already implies that at one-loop $W_{\mu}^{(1)}=0$ and

$W_{\mu}(R,\nabla,H)=(\alpha^{\prime})^{2}\,\Big{(}a_{1}\,\nabla_{\mu}R+a_{2}\,\nabla_{\mu}H^{2}+a_{3}\,H_{\mu\nu\lambda}\nabla_{\rho}H^{\rho\nu\lambda}\Big{)}+{\cal
O}(\alpha^{\prime})^{3}\;,$ (2.52)

and similarly for $L_{\mu}$, respectively. For our subsequent applications to
the duality invariant sigma model in sec. 4 we will also find that at one-loop
level these vectors vanish.

##  3  Background Field Method

In this section we revisit the techniques for computing the
$\beta-$functionals of the string nonlinear sigma model defined by

$S=\frac{1}{2\lambda}\int
d^{2}x\,\sqrt{\gamma}\Big{[}\gamma^{\alpha\beta}g_{\mu\nu}(X)\,\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}+\alpha^{\prime}\,R^{(2)}\,\Phi(X)\Big{]}\;,\quad\lambda=2\pi\alpha^{\prime}\;,$
(3.1)

on a curved _Euclidean_ worldsheet of spherical topology. For now we discard
the coupling to the Kalb-Ramond field $B_{\mu\nu}$. We will compute the
quantum effective action $\Gamma$ using the background-field method and deduce
the corresponding $\beta-$functions by renormalization of the couplings.

###  3.1 Effective Action

We start by considering a generic field theory with Euclidean action
$S[\phi]\,$. The generating functional of all correlators is given by

$Z[J]=\int\mathcal{D}\phi\,e^{-S[\phi]+J\cdot\phi}=Z[0]\,\Big{(}1+\sum_{n=1}^{\infty}\tfrac{1}{n!}\,J_{1}...J_{n}\,\langle\phi_{1}...\phi_{n}\rangle\Big{)}\;,$
(3.2)

where spacetime integrals and positions are condensed in the notation

$\begin{split}&J\cdot\phi:=\int d^{n}x\,J(x)\,\phi(x)\;,\\\
&J_{1}...J_{n}\,\langle\phi_{1}...\phi_{n}\rangle:=\int
d^{n}x_{1}...d^{n}x_{n}\,J(x_{1})...J(x_{n})\,\langle\phi(x_{1})...\phi(x_{n})\rangle\;,\end{split}$
(3.3)

etc. The factor $Z:=Z[0]$ ensures that the correlators in (3.2) are
normalized, $\langle 1\rangle=1\,$. The generating functional of the connected
correlators, $W[J]$, is defined by

$W[J]:=\log\frac{Z[J]}{Z[0]}\;,$ (3.4)

which ensures that $W[0]=0$, meaning that all the vacuum bubbles are
subtracted:

$W[J]=\sum_{n=1}^{\infty}\frac{1}{n!}\,J_{1}...J_{n}\,\langle\phi_{1}...\phi_{n}\rangle_{\rm
connected}\;.$ (3.5)

Following standard terminology we define the field $\varphi$, which is a
function of the source $J$, as the quantum expectation value:

$\varphi(J):=\frac{\delta W}{\delta J}=\langle\phi\rangle_{J}\;.$ (3.6)

Assuming tadpole cancellation one has $\varphi(0)=\langle\phi\rangle=0\,$. If
$\varphi(J)$ is invertible, the generating functional of one-particle-
irreducible (1PI) diagrams, _i.e._ the quantum effective action
$\Gamma[\varphi]$, is given by the Legendre transform of $W[J]\,$:

$\Gamma[\varphi]=J\cdot\varphi-W[J]\;.$ (3.7)

In order to see that $\Gamma$ is genuinely a functional of $\varphi$ only, one
can compute the variation

$\delta\Gamma=\delta J\cdot\varphi+J\cdot\delta\varphi-\frac{\delta W}{\delta
J}\cdot\delta J=J\cdot\delta\varphi\;,$ (3.8)

where we have used (3.6). This also gives the quantum equation of motion in
the form

$\frac{\delta\Gamma}{\delta\varphi}=J\,,$ (3.9)

that determines $J$ as a function of $\varphi\,$.

We can now use these definitions to manipulate the path integral:

$\begin{split}e^{W[J]}&=e^{-\Gamma[\varphi]+J\cdot\varphi}=\frac{1}{Z}\int\mathcal{D}\phi\,e^{-S[\phi]+J\cdot\phi}\;\rightarrow\\\
e^{-\Gamma[\varphi]}&=\frac{1}{Z}\int\mathcal{D}\phi\,e^{-S[\phi]+J\cdot(\phi-\varphi)}\;.\end{split}$
(3.10)

Using the definition (3.9) and shifting the integration variable by
$\phi=\varphi+\pi$ we obtain the most convenient form for the background field
method:

$e^{-\frac{1}{\hbar}\Gamma[\varphi]}=\frac{1}{Z}\int\mathcal{D}\pi\,\exp\Big{\\{}-\frac{1}{\hbar}\,\Big{(}S[\varphi+\pi]-\frac{\delta\Gamma}{\delta\varphi}\cdot\pi\Big{)}\Big{\\}}\;,$
(3.11)

where we have reinstated Planck’s constant $\hbar\,$. Here $\varphi$ is viewed
as the classical background and $\pi$ as the quantum fluctuation.

Let us mention that there is an all-order subtraction in (3.11) given
implicitly by the term $-\frac{\delta\Gamma}{\delta\varphi}\cdot\pi$, which
removes non-(1PI) contributions from $\Gamma\,$. In fact, $\Gamma[\varphi]$
appears on both sides of (3.11), which is therefore an integral equation. As
usual, one can solve it perturbatively order by order in loops. More
precisely, one starts by writing $\Gamma$ as a power series in $\hbar\,$:

$\Gamma[\varphi]=S[\varphi]+\hbar\,\Gamma_{1l}[\varphi]+{\cal
O}(\hbar^{2})\;,$ (3.12)

and expands the shifted action $S[\varphi+\pi]$ in powers of the fluctuation
$\pi$ as

$S[\varphi+\pi]=S[\varphi]+\frac{\delta
S}{\delta\varphi}\cdot\pi+\frac{1}{2}\,\pi\cdot\frac{\delta^{2}S}{\delta\varphi\delta\varphi}\cdot\pi+{\cal
O}(\pi^{3})\;.$ (3.13)

The linear term in $\pi$ above is canceled by the subtraction in (3.11) to
lowest order and no higher subtractions are needed at one-loop, as one can see
by counting $\hbar$ powers. Similarly, one-loop diagrams require terms at most
quadratic in $\pi$ from the above expansion. As the last ingredient, we shall
extract the pure kinetic term for the fluctuations:

$S_{2\pi}[\pi;\varphi]:=\frac{1}{2}\,\pi\cdot\frac{\delta^{2}S}{\delta\varphi\delta\varphi}\cdot\pi=S_{0}[\pi]+S_{\rm
int}[\pi;\varphi]=S_{0}[\pi]+{\cal O}(\varphi)\,,$ (3.14)

and define the partition function of the free theory

$Z_{0}:=\int\mathcal{D}\pi\,e^{-\frac{1}{\hbar}S_{0}[\pi]}\;.$ (3.15)

This gives the expression for the one-loop contribution to the effective
action:

$e^{-\Gamma_{1l}[\varphi]}=\frac{1}{Z_{0}}\int\mathcal{D}\pi\,e^{-\frac{1}{\hbar}\,S_{2\pi}[\pi;\varphi]}=\left\langle
e^{-\frac{1}{\hbar}\,S_{\rm int}[\pi;\varphi]}\right\rangle_{0}\;,$ (3.16)

where the subscript $0$ denotes the normalized average with respect to the
free action $S_{0}[\pi]$.

###  3.2 Nonlinear Sigma Model

It is possible, in principle, to apply the background-field method in the form
just described to the nonlinear sigma model

$S=\frac{1}{2\lambda}\int
d^{2}x\,g_{\mu\nu}(X)\,\partial_{\alpha}X^{\mu}\partial^{\alpha}X^{\nu}\;,$
(3.17)

where, for the time being, we are considering a flat worldsheet. However,
applying the linear background-quantum split by defining
$X^{\mu}=\varphi^{\mu}+\pi^{\mu}$, leads to a perturbative expansion lacking
manifest target space covariance, since the fluctuation field $\pi^{\mu}$ is a
coordinate difference and thus has no geometric meaning. To remedy this we
employ a field redefinition of $\pi^{\mu}$ as follows [25, 27]: We consider a
geodesic $X^{\mu}(t)$, where $t$ is the affine parameter (and we suppress the
dependence on the worldsheet coordinates $x^{\alpha}$), such that
$X^{\mu}(0)=\varphi^{\mu}$ and $X^{\mu}(1)=\varphi^{\mu}+\pi^{\mu}$, with
tangent vector $\xi^{\mu}(t)=\frac{dX^{\mu}}{dt}\,$. We then use the tangent
vector at $t=0$, $\xi^{\mu}:=\xi^{\mu}(0)$, as the quantum field for the
expansion. Since it is a genuine vector, this ensures manifest target space
covariance.

It is possible to derive the exact nonlinear relation
$\pi^{\mu}(\xi)=\xi^{\mu}-\frac{1}{2}\,\Gamma^{\mu}_{\nu\lambda}\,\xi^{\nu}\xi^{\lambda}+{\cal
O}(\xi^{3})$ implementing the field redefinition and use it to write down the
covariant expansion. This procedure, however, becomes very cumbersome after a
few orders, even when resorting to a normal coordinate system. A considerable
simplification was found in [27] by noting that one usually needs to expand
only the Lagrangian, which is just a scalar. We thus consider a scalar field
evaluated along the geodesic: $\Phi(t)\equiv\Phi(X(t))\,$. We can expand it
around $t=0$, yielding

$\Phi(t)=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\,\left.\frac{d^{n}\Phi}{dt^{n}}\right\rvert_{t=0}\;.$
(3.18)

Since $\Phi$ is a scalar field, the derivative along the geodesic
$\frac{d}{dt}\equiv\xi^{\mu}\partial_{\mu}$ is already manifestly covariant,
in that

$\frac{D\Phi}{Dt}:=\xi^{\mu}\nabla_{\mu}\Phi=\xi^{\mu}\partial_{\mu}\Phi=\frac{d\Phi}{dt}\;,$
(3.19)

which defines the covariant derivative $\frac{D}{Dt}$ along the geodesic. One
can of course also define $\frac{D}{Dt}=\xi^{\mu}\nabla_{\mu}$ on general
spacetime tensors evaluated on the curve, and

$\frac{D}{Dt}T^{\mu\dots}_{\nu\dots}(t)=\frac{d}{dt}T^{\mu\dots}_{\nu\dots}+\xi^{\lambda}\,\Gamma^{\mu}_{\lambda\rho}\,T^{\rho\dots}_{\nu\dots}+\cdots$
(3.20)

for objects defined only on the curve. Given that any application of
$\frac{d}{dt}$ maps the scalar into a scalar, it immediately follows by
induction that

$\frac{d^{n}\Phi}{dt^{n}}=\frac{d^{n-1}}{dt^{n-1}}\left(\xi\cdot\nabla\Phi\right)=(\xi\cdot\nabla)^{n}\Phi=\frac{D^{n}\Phi}{Dt^{n}}\;.$
(3.21)

This yields the covariant expansion

$\Phi(t)=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\,\left.\frac{D^{n}\Phi}{Dt^{n}}\right\rvert_{t=0}\;.$
(3.22)

Let us also mention that, thanks to the geodesic equation
$\frac{D\xi^{\mu}}{Dt}=0$, one can freely rearrange the tangent vectors as

$\frac{D^{n}\Phi}{Dt^{n}}=(\xi\cdot\nabla)^{n}\Phi=\xi^{\mu_{1}}\cdots\xi^{\mu_{n}}\,\nabla_{\mu_{1}}\cdots\nabla_{\mu_{n}}\Phi\;.$
(3.23)

We now apply the above result to the Lagrangian, for which we recall that
$\varphi^{\mu}=X^{\mu}(0)$ and $\varphi^{\mu}+\pi^{\mu}=X^{\mu}(1)$ so that
(3.22) should be evaluated at $t=1$. This gives [28]

${\cal L}(\varphi+\pi(\xi))=\exp\left(\frac{D}{Dt}\right){\cal L}(\varphi)\;.$
(3.24)

In order to apply this to the sigma model we need to consider the pullback to
the worldsheet. To this end we use the worldsheet-dependent geodesic
$X^{\mu}(x;t)$ and $\xi^{\mu}(x;t)$ to compute

$\begin{split}\frac{D}{Dt}\partial_{\alpha}X^{\mu}&=\frac{d}{dt}\partial_{\alpha}X^{\mu}+\xi^{\nu}\Gamma^{\mu}_{\nu\lambda}\partial_{\alpha}X^{\lambda}=\partial_{\alpha}\xi^{\mu}+\partial_{\alpha}X^{\nu}\Gamma^{\mu}_{\nu\lambda}\xi^{\lambda}=:D_{\alpha}\xi^{\mu}\;,\end{split}$
(3.25)

where we have defined the covariant derivative $D_{\alpha}$ on the worldsheet.
This covariant derivative acts as
$D_{\alpha}=\partial_{\alpha}X^{\mu}\nabla_{\mu}$ on the pullback of target
space tensors. To act further with $\frac{D}{Dt}$ we need the commutator

$\begin{split}\Big{[}\frac{D}{Dt},D_{\alpha}\Big{]}&=\big{[}\xi^{\mu}\nabla_{\mu},\partial_{\alpha}X^{\nu}\nabla_{\nu}\big{]}\\\
&=\xi^{\mu}\partial_{\alpha}X^{\nu}\,\big{[}\nabla_{\mu},\nabla_{\nu}\big{]}+\Big{(}\frac{D}{Dt}\partial_{\alpha}X^{\nu}\Big{)}\nabla_{\nu}-(D_{\alpha}\xi^{\mu})\nabla_{\mu}\\\
&=\xi^{\mu}\partial_{\alpha}X^{\nu}\,R_{\mu\nu}^{\\#}\;,\end{split}$ (3.26)

where $R_{\mu\nu}^{\\#}$ is the Riemann tensor acting as an operator, _e.g._
$R_{\mu\nu}^{\\#}V^{\rho}=R_{\mu\nu}{}^{\rho}{}_{\lambda}\,V^{\lambda}\,$. One
can thus determine, for instance,

$\Big{(}\frac{D}{Dt}\Big{)}^{2}\partial_{\alpha}X^{\mu}=\frac{D}{Dt}D_{\alpha}\xi^{\mu}=\Big{[}\frac{D}{Dt},D_{\alpha}\Big{]}\xi^{\mu}=\xi^{\nu}\partial_{\alpha}X^{\lambda}\,R_{\nu\lambda}{}^{\mu}{}_{\rho}\,\xi^{\rho}\;.$
(3.27)

These tools allow one to systematically expand the sigma model action in a
simple and recursive manner. Using (3.24) the first few orders of the
expansion of the worldsheet sigma model (3.17) are easily obtained as

$\begin{split}S_{0\xi}&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}g_{\mu\nu}(\varphi)\,\partial_{\alpha}\varphi^{\mu}\partial^{\alpha}\varphi^{\nu}\Big{]}\;,\\\\[5.69054pt]
S_{1\xi}&=\frac{1}{2\lambda}\int
d^{2}x\,\frac{D}{Dt}\Big{[}g_{\mu\nu}(X)\,\partial_{\alpha}X^{\mu}\partial^{\alpha}X^{\nu}\Big{]}\Big{\rvert}_{t=0}=\frac{1}{\lambda}\int
d^{2}x\,\Big{[}g_{\mu\nu}(\varphi)\,D_{\alpha}\xi^{\mu}\partial^{\alpha}\varphi^{\nu}\Big{]}\;,\\\\[5.69054pt]
S_{2\xi}&=\frac{1}{\lambda}\int
d^{2}x\,\frac{1}{2}\,\frac{D}{Dt}\Big{[}g_{\mu\nu}(X)\,D_{\alpha}\xi^{\mu}\partial^{\alpha}X^{\nu}\Big{]}\Big{\rvert}_{t=0}\\\
&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}g_{\mu\nu}(\varphi)\,D^{\alpha}\xi^{\mu}D_{\alpha}\xi^{\nu}+R_{\mu\nu\lambda\rho}(\varphi)\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\rho}\,\xi^{\nu}\xi^{\lambda}\Big{]}\;,\end{split}$
(3.28)

where we recall that, after taking the derivatives, evaluating at $t=0$
amounts to replacing $X^{\mu}$ by $\varphi^{\mu}$. This already exhausts the
terms needed at one-loop.

###  3.3 One-loop Effective Action

Having found the covariant background-field expansion $S_{2\xi}$ in (3.28), we
can write the one-loop effective action in terms of the path integral

$e^{-\Gamma_{1l}[\varphi]}=\frac{1}{Z_{0}}\int\mathcal{D}\xi\,e^{-S_{2\xi}[\xi;\varphi]}\,,$
(3.29)

with

$S_{2\xi}[\xi;\varphi]=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}g_{\mu\nu}\,D^{\alpha}\xi^{\mu}D_{\alpha}\xi^{\nu}+R_{\mu\nu\lambda\rho}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\rho}\,\xi^{\nu}\xi^{\lambda}\Big{]}\;,$
(3.30)

where all target space tensors are evaluated at $\varphi^{\mu}\,$. Comparing
(3.29) with the general formulas (3.11) and (3.16), one can see that the
linear subtraction $S[\varphi+\pi]-\frac{\delta S}{\delta\varphi}\cdot\pi$ has
not been performed in the usual way, since we rather subtracted $\frac{\delta
S}{\delta\varphi}\cdot\xi$ from $S[\varphi+\pi(\xi)]\,$. This choice leads to
a covariant one-loop effective action even for off-shell backgrounds
$\varphi^{\mu}$, while it coincides with any other choice for on-shell
backgrounds [29]. Let us note that the general form of the effective action is

$\Gamma[\varphi]=\frac{1}{2\lambda}\int d^{2}x\,\Big{[}{\cal
T}_{\mu\nu}(g)\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}+\cdots\Big{]}\;,$
(3.31)

where ${\cal T}_{\mu\nu}$ is some target space tensor, which prior to
renormalization contains divergent coefficients. The divergent part of ${\cal
T}_{\mu\nu}$ determines the renormalization of the metric and hence determines
the $\beta-$function. For this reason, it is sufficient for our purposes to
consider contributions to $\Gamma$ with only two factors of
$\partial_{\alpha}\varphi^{\mu}$, while the ellipsis denote terms with more
than two derivatives of $\varphi$.

Let us now examine the perturbative expansion of the action. The kinetic term
of (3.30),
$g_{\mu\nu}(\varphi)\partial^{\alpha}\xi^{\mu}\partial_{\alpha}\xi^{\nu}$, has
a non-standard form, since $g_{\mu\nu}(\varphi)$ is not constant on the
worldsheet. To overcome this difficulty it is customary to introduce vielbeins
$e_{\mu}^{a}(\varphi)$ and flatten the fluctuation by introducing
$\xi^{a}=e_{\mu}^{a}\,\xi^{\mu}\,$. The covariant derivative
$D_{\alpha}\xi^{a}=\partial_{\alpha}\xi^{a}+\partial_{\alpha}\varphi^{\mu}\,\omega_{\mu}{}^{a}{}_{b}\,\xi^{b}$
now involves the spin connection, and the action takes the form

$\begin{split}S_{2\xi}[\xi;\varphi]&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}D^{\alpha}\xi^{a}D_{\alpha}\xi_{a}+R_{\mu
ab\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\,\xi^{a}\xi^{b}\Big{]}\\\
&=\frac{1}{2\lambda}\int
d^{2}x\,\partial^{\alpha}\xi^{a}\partial_{\alpha}\xi_{a}+S_{\rm
int}[\xi;\varphi]\;,\end{split}$ (3.32)

which has a standard kinetic term. One can see that the worldsheet coupling
constant $\lambda=2\pi\alpha^{\prime}$ is the loop counting parameter,
implying that the effective action at $L$ loops is of order
$(\alpha^{\prime})^{L-1}$. Writing out the interaction part of (3.32) we
obtain

$\begin{split}S_{\rm int}[\xi;\varphi]&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}2\,\partial^{\alpha}\varphi^{\mu}\,\omega_{\mu\,ab}\,\xi^{b}\partial_{\alpha}\xi^{a}+\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\,\omega_{\mu\,ca}\,\omega_{\nu}{}^{c}{}_{b}\,\xi^{a}\xi^{b}\\\
&\qquad\qquad\qquad+R_{\mu
ab\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\,\xi^{a}\xi^{b}\Big{]}\\\
&=:S_{\omega}+S_{\omega\omega}+S_{R}\;.\end{split}$ (3.33)

We are now ready to evaluate the one-loop effective action, which is given by

$e^{-\Gamma_{1l}[\varphi]}=\left\langle e^{-S_{\rm
int}[\xi;\varphi]}\right\rangle\,,$ (3.34)

where angle brackets denote normalized free expectation values. The only terms
contributing to the renormalization of the metric are

$\Gamma_{1l}=\langle S_{R}\rangle+\langle
S_{\omega\omega}\rangle-\tfrac{1}{2}\,\langle S_{\omega}^{2}\rangle_{1{\rm
PI}}+\cdots\;,$ (3.35)

where dots stand for terms with more than two factors of
$\partial_{\alpha}\varphi^{\mu}$. The propagator can be derived from the free
part of (3.32) and reads

$\langle\xi^{a}(x)\,\xi^{b}(y)\rangle=\lambda\delta^{ab}G(x-y)\;,\quad
G(x)=\int\frac{d^{2}p}{(2\pi)^{2}}\,\frac{e^{ip\cdot x}}{p^{2}}\;.$ (3.36)

On dimensional grounds, and using gauge invariance, it follows that the terms
$S_{\omega}$ and $S_{\omega\omega}$ involving the spin connection cannot
contribute to UV divergences. For the sake of completeness, we shall compute
their contribution to (3.35) nonetheless and show that it is UV finite. Taking
the expectation values and performing the Wick contractions we have

$\begin{split}\langle S_{2\omega}\rangle&-\tfrac{1}{2}\,\langle
S_{1\omega}^{2}\rangle=\tfrac{1}{2}\int
d^{2}x\,\omega_{\alpha}{}^{ab}(x)\,\omega^{\alpha}{}_{ab}(x)\,G(0)\\\
&+\tfrac{1}{2}\int
d^{2}xd^{2}y\,\omega_{\alpha}{}^{ab}(x)\,\omega_{\beta\,ab}(y)\,\Big{[}G(x-y)\partial^{\alpha}\partial^{\beta}G(x-y)-\partial^{\alpha}G(x-y)\partial^{\beta}G(x-y)\Big{]}\;,\end{split}$
(3.37)

where we introduced the pullback
$\omega_{\alpha}{}^{ab}=\partial_{\alpha}\varphi^{\mu}\,\omega_{\mu}{}^{ab}$,
that behaves as an $SO(D)$ gauge field in two dimensions. Going to momentum
space by defining

$\omega_{\alpha\,ab}(x)=\int\frac{d^{2}p}{(2\pi)^{2}}\,\omega_{\alpha\,ab}(p)\,e^{ip\cdot
x}$ (3.38)

one obtains

$\begin{split}\langle S_{2\omega}\rangle-\tfrac{1}{2}\,\langle
S_{1\omega}^{2}\rangle&=\tfrac{1}{2}\int\frac{d^{2}p}{(2\pi)^{2}}\,\omega_{\alpha}{}^{ab}(p)\,\Pi^{\alpha\beta}(p)\,\omega_{\beta\,ab}(-p)\;,\\\
\Pi^{\alpha\beta}(p)&=-\tfrac{1}{2}\int\frac{d^{2}k}{(2\pi)^{2}}\,\frac{(2k^{\alpha}+p^{\alpha})(2k^{\beta}+p^{\beta})-2\delta^{\alpha\beta}(p+k)^{2}}{k^{2}(p+k)^{2}}\;.\end{split}$
(3.39)

In order to regularize the polarization tensor $\Pi^{\alpha\beta}$ we continue
to $n=2+\epsilon$ dimensions and introduce an infrared mass regulator $m^{2}$
by changing the propagator222This amounts to adding the mass term
$\frac{m^{2}}{2\lambda}\int d^{2}x\,\xi^{a}\xi_{a}$ to the action, that
suffices to regularize infrared divergences at one-loop. as
$\frac{1}{p^{2}}\to\frac{1}{p^{2}+m^{2}}\,$. This yields

$\Pi_{\rm
reg}^{\alpha\beta}(p)=-\tfrac{1}{2}\,\mu^{2-n}\int\frac{d^{n}k}{(2\pi)^{n}}\,\frac{(2k^{\alpha}+p^{\alpha})(2k^{\beta}+p^{\beta})-2\delta^{\alpha\beta}[(p+k)^{2}+m^{2}]}{(k^{2}+m^{2})[(p+k)^{2}+m^{2}]}\;,$
(3.40)

where we introduced the arbitrary mass parameter $\mu$ to keep
$\Pi^{\alpha\beta}_{\rm reg}$ dimensionless. After some standard manipulations
for one-loop integrals (see appendix A.1 for details) one obtains

$\begin{split}\Pi_{\rm
reg}^{\alpha\beta}(p)&=(\delta^{\alpha\beta}p^{2}-p^{\alpha}p^{\beta})\,\Pi_{\rm
reg}(p^{2})\;,\\\ \Pi_{\rm
reg}(p^{2})&=\frac{2\,\mu^{2-n}}{(4\pi)^{n/2}}\Gamma(2-\tfrac{n}{2})\int_{-1/2}^{+1/2}dy\,y^{2}\,\Big{[}(\tfrac{1}{4}-y^{2})p^{2}+m^{2}\Big{]}^{\frac{n}{2}-2}\;,\end{split}$
(3.41)

which is perfectly finite for $n=2$, provided one keeps $m^{2}$ fixed. This
shows that the $\omega-$terms are UV finite and do not contribute to
renormalization. Note that for this computation it was necessary to introduce
the infrared regulator, because the integral is IR divergent in $n=2\,$:

$\Pi_{\rm reg}(p^{2})\;\stackrel{{\scriptstyle
n=2}}{{\longrightarrow}}\;-\frac{1}{2\pi
p^{2}}\Big{[}\frac{R}{2}\log\frac{R-1}{R+1}+1\Big{]}\;,\quad
R=\Big{(}1+\frac{4m^{2}}{p^{2}}\Big{)}^{1/2}\;.$ (3.42)

While at this point it is clear that the logarithmic divergence appearing as
$m^{2}\to 0$ is of infrared nature, it would have been dangerous to set
$m^{2}=0$ beforehand, since the IR divergence would have reincarnated in a
$\frac{1}{\epsilon}$ pole.

From the expansion (3.35) we are left with the single divergent contribution

$\Gamma_{1l}^{\rm div}=\langle S_{R}\rangle=-\frac{1}{2}\,G(0)\int
d^{2}x\,R_{\mu\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\;.$
(3.43)

As before, we regularize the propagator at coinciding points
$G(0)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{p^{2}}$ by continuing to
$n=2+\epsilon$ dimensions and introducing the IR mass regulator:

$G(0)_{\rm
reg}=\mu^{2-n}\int\frac{d^{n}p}{(2\pi)^{n}}\,\frac{1}{p^{2}+m^{2}}=\frac{1}{4\pi}\,\left(\frac{m^{2}}{4\pi\mu^{2}}\right)^{\frac{\epsilon}{2}}\Gamma(-\epsilon/2)\;.$
(3.44)

In order to extract the pole, we expand the gamma function
$\Gamma(x)=\frac{1}{x}-\gamma+{\cal O}(x)$, where $\gamma$ is the Euler-
Mascheroni constant, and obtain

$\Gamma_{1l}^{\rm
div}=\frac{1}{4\pi}\,\left(\frac{1}{\epsilon}+\log\frac{m}{\mu}\right)\,\int
d^{2}x\,R_{\mu\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\;.$
(3.45)

Here we redefined the renormalization scale as $4\pi
e^{-\gamma}\mu^{2}\rightarrow\mu^{2}$, as it is customary in the
$\overline{\rm MS}$ scheme. At this point we can fix the one-loop counterterm
by demanding that it cancels the divergence:

$S_{\rm c.t.}=-\frac{1}{4\pi\epsilon}\int
d^{2}x\,R_{\mu\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\;,$
(3.46)

which yields the renormalized coupling at one-loop order:

$\Gamma_{\rm ren}=\frac{1}{2\lambda}\int
d^{2}x\,\Big{(}g_{\mu\nu}+\frac{\lambda}{2\pi}\,\log\frac{m}{\mu}\,R_{\mu\nu}\Big{)}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\;.$
(3.47)

In order to extract the $\beta-$function, we apply the method outlined in the
previous section. Following eq. (2.6) we write the bare action as

$S_{0}=S+S_{\rm c.t.}=\frac{1}{2\lambda}\int
d^{n}x\,g_{\mu\nu}^{0}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}=\frac{1}{2\lambda}\int
d^{n}x\,\mu^{\epsilon}\,\big{(}g_{\mu\nu}+T_{\mu\nu}(g)\big{)}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\;,$
(3.48)

where, using (3.46), $T_{\mu\nu}=-\frac{\lambda}{2\pi\epsilon}R_{\mu\nu}$.
This determines the bare metric to be

$g_{\mu\nu}^{0}=\mu^{\epsilon}\,\Big{(}g_{\mu\nu}-\frac{\lambda}{2\pi\epsilon}\,R_{\mu\nu}\Big{)}\equiv\mu^{\epsilon}\,\Big{(}g_{\mu\nu}-\frac{\alpha^{\prime}}{\epsilon}\,R_{\mu\nu}\Big{)}\;.$
(3.49)

Defining $t:=\log\mu$ and requiring $g_{\mu\nu}^{0}$ to be independent of $t$
one obtains

$0=\frac{dg_{\mu\nu}^{0}}{dt}=e^{\epsilon
t}\Big{(}\epsilon\,g_{\mu\nu}-\alpha^{\prime}\,R_{\mu\nu}+\frac{dg_{\mu\nu}}{dt}-\frac{\alpha^{\prime}}{\epsilon}\,\frac{dg_{\lambda\rho}}{dt}\cdot\frac{\partial}{\partial
g_{\lambda\rho}}\,R_{\mu\nu}\Big{)}\;.$ (3.50)

Matching the order $\epsilon$ and $\epsilon^{0}$ terms yields

$\frac{dg_{\mu\nu}}{dt}=-\epsilon\,g_{\mu\nu}+\beta_{\mu\nu}(g)\;,\quad\beta_{\mu\nu}(g)=\alpha^{\prime}\,\Big{(}1-g\cdot\frac{\partial}{\partial
g}\Big{)}\,R_{\mu\nu}\;.$ (3.51)

We recall that the operator $g\cdot\frac{\partial}{\partial g}$ should be
regarded as the integrated functional derivative as in (2.12), but here it can
be reduced to an ordinary parametric derivative as follows:

$g_{\lambda\rho}\cdot\frac{\partial}{\partial
g_{\lambda\rho}}\,T_{\mu\nu}(g)=\Lambda\frac{\partial}{\partial\Lambda}T_{\mu\nu}(\Lambda
g)\Big{\rvert}_{\Lambda=1}\;.$ (3.52)

This can be verified by computing the right-hand side, viewing $T_{\mu\nu}$ as
a function of $\tilde{g}_{\lambda\rho}=\Lambda\,g_{\lambda\rho}$ and applying
the (functional) chain rule. Note that the operator
$\Lambda\frac{\partial}{\partial\Lambda}$ counts the number of $g_{\mu\nu}$
minus the number of $g^{\mu\nu}$. For the Ricci tensor this operator has zero
eigenvalue, because the Christoffel symbols contain one $g$ and one $g^{-1}$
and there is no further metric needed in defining the Ricci tensor. Using this
back in (3.51) we can finally read off the one-loop beta function [16]

$\beta_{\mu\nu}(g)=\alpha^{\prime}\,R_{\mu\nu}\;.$ (3.53)

###  3.4 The Dilaton

In this section we include the dilaton coupling in the sigma model and compute
its one-loop $\beta-$function. Treating the dilaton term is technically more
involved, since it requires computations on a curved worldsheet. Moreover, the
resulting $\beta-$functions of the metric-dilaton system are such that
$\beta^{g}_{\mu\nu}=0$ and $\beta^{\Phi}=0$ are _not_ the correct target space
equations, which are instead given by $\bar{\beta}^{g}_{\mu\nu}=0$ and
$\bar{\beta}^{\Phi}=0$, as we have discussed in section 2.

We now compute the background field expansion for the sigma model action
including the dilaton:

$S=\frac{1}{2\lambda}\int
d^{2}x\,\sqrt{\gamma}\Big{[}\gamma^{\alpha\beta}\,g_{\mu\nu}(X)\,\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}+\alpha^{\prime}\,R^{(2)}\,\Phi(X)\Big{]}\;.$
(3.54)

Here we need to consider an arbitrary curved worldsheet with metric
$\gamma_{\alpha\beta}$ and spherical topology. The curved two-metric does not
affect our discussion of the background-field expansion, since it does not
involve the field $X^{\mu}=\varphi^{\mu}+\pi^{\mu}(\xi)\,$. This leads
immediately to the quadratic action

$\begin{split}S_{2\xi}&=\frac{1}{2\lambda}\int
d^{2}x\,\sqrt{\gamma}\Big{[}\gamma^{\alpha\beta}D_{\alpha}\xi^{a}D_{\beta}\xi_{a}+\gamma^{\alpha\beta}R_{\mu
ab\nu}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}\,\xi^{a}\xi^{b}\Big{]}\\\
&+\frac{1}{8\pi}\int
d^{2}x\sqrt{\gamma}R^{(2)}\,\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}\Phi+\frac{m^{2}}{2\lambda}\int
d^{2}x\sqrt{\gamma}\,\xi^{a}\xi_{a}\;,\end{split}$ (3.55)

where we have already included the mass term to regulate infrared divergences,
and we denoted
$\nabla_{a}\nabla_{b}\Phi:=e_{a}^{\mu}e_{b}^{\nu}\nabla_{\mu}\nabla_{\nu}\Phi\,$.
Following conventional perturbation theory, we shall expand
$\gamma_{\alpha\beta}$ around flat space:

$\gamma_{\alpha\beta}=\delta_{\alpha\beta}+h_{\alpha\beta}\;,$ (3.56)

and consider the one-loop effective action perturbatively in powers of
$h_{\alpha\beta}\,$. In particular, the propagators are still extracted from
the flat-space free theory

$S_{0}=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}\partial^{\alpha}\xi^{a}\partial_{\alpha}\xi_{a}+m^{2}\,\xi^{a}\xi_{a}\Big{]}\;,$
(3.57)

and terms with any powers of $h_{\alpha\beta}$ are treated as interactions.

Before starting any computation, there is one immediate consequence that can
be derived from the structure of the action: thanks to the coupling with the
scalar curvature $R^{(2)}$, every term involving the dilaton appears with at
least one factor of $h_{\alpha\beta}\,$. Since the lowest order coupling of
the metric $g_{\mu\nu}$ (and $B-$field if present) only involves the flat
background $\delta_{\alpha\beta}$ one can immediately show that the dilaton
_cannot_ renormalize the metric nor the $B-$field at any order in perturbation
theory. This implies that the $\beta-$functions $\beta_{\mu\nu}^{g}$ and
$\beta_{\mu\nu}^{B}$ do not depend on the dilaton at any order in
$\alpha^{\prime}$, as anticipated in section 2. This fact already shows that
the $\beta-$functions alone cannot provide the correct field equations.

Having shown that the metric $\beta-$function is not affected by the dilaton,
we are left to determine the $\beta-$function of the dilaton itself. To this
end, we have to extract from $S_{\rm int}=S_{2\xi}-S_{0}$ the terms that can
renormalize the coupling $\int d^{2}x\sqrt{\gamma}R^{(2)}\Phi$ in
$e^{-\Gamma_{1l}}=\left\langle e^{-S_{\rm int}}\right\rangle$, which in
particular do not contain $\partial_{\alpha}\varphi^{\mu}$ factors. From
(3.55) one can split the interacting action into several terms, according to
the background fields involved. Since we will work to quadratic order in
$h_{\alpha\beta}$, we have

$\begin{split}S_{\rm int}&=S_{\rm
int}^{h=0}+\big{(}S_{h\omega}+S_{hh\omega}\big{)}+\big{(}S_{h\omega\omega}+S_{hh\omega\omega}\big{)}+\big{(}S_{hR}+S_{hhR}\big{)}\\\
&+\big{(}S_{h\partial\xi\partial\xi}+S_{hh\partial\xi\partial\xi}\big{)}+m^{2}\,\big{(}S_{h\xi\xi}+S_{hh\xi\xi}\big{)}+S_{\Phi}+{\cal
O}(h^{3})\;,\end{split}$ (3.58)

where the subscripts denote the schematic form of the vertices.

In order to perform the perturbative expansion in $h_{\alpha\beta}$ we shall
need [29]

$\sqrt{\gamma}\,\gamma^{\alpha\beta}=\delta^{\alpha\beta}-\bar{h}^{\alpha\beta}+{\cal
O}(h^{2})\;,$ (3.59)

where we defined the trace-adjusted perturbation in arbitrary dimensions

$\bar{h}_{\alpha\beta}=h_{\alpha\beta}-\tfrac{1}{2}\,\delta_{\alpha\beta}h\;,\quad\bar{h}^{\alpha}{}_{\alpha}=-\tfrac{1}{2}\,(n-2)\,h\;.$
(3.60)

In order to recognize the Ricci scalar we shall also need

$\sqrt{\gamma}R^{(n)}=\partial^{\alpha}\partial^{\beta}\bar{h}_{\alpha\beta}-\tfrac{1}{2}\,\Box
h+\tfrac{1}{4}\,\bar{h}^{\alpha\beta}\Box\bar{h}_{\alpha\beta}+\tfrac{1}{2}\,(\partial\cdot\bar{h}_{\alpha})^{2}-\tfrac{1}{16}\,(n-2)\,h\Box
h+\partial_{\alpha}v^{\alpha}+{\cal O}(h^{3})\;,$ (3.61)

modulo total derivatives.

We are now ready to examine the various contributions from the interacting
action. First of all, the entire first line in (3.58) cannot contribute to the
renormalization of the dilaton coupling, since every term contains at least
one factor of $\partial_{\alpha}\varphi^{\mu}\,$. Taking into account the
terms in the second line we find

$\begin{split}\Gamma^{\Phi}_{1l}&=\langle
S_{h\partial\xi\partial\xi}+m^{2}\,S_{h\xi\xi}\rangle+\langle
S_{hh\partial\xi\partial\xi}+m^{2}\,S_{hh\xi\xi}\rangle\\\ &\hskip
14.22636pt-\tfrac{1}{2}\,\langle(S_{h\partial\xi\partial\xi}+m^{2}\,S_{h\xi\xi})^{2}\rangle_{1\rm
PI}+\langle S_{\Phi}\rangle+{\cal O}(h^{3})\;,\end{split}$ (3.62)

where by the superscript $\Phi$ we mean possible contributions proportional to
$\int d^{2}x\sqrt{\gamma}R^{(2)}\,$.

At this point we notice that the “tadpole” contributions from the first line
have no external momentum flowing in the loop, meaning that they give
(possibly UV divergent) contributions of the form $C\int d^{2}x\,F(h)$, with
no derivatives acting on $h_{\alpha\beta}\,$. These do not contribute to the
Ricci scalar, but rather to the cosmological constant

$\Lambda\int d^{2}x\sqrt{\gamma}\;.$ (3.63)

On dimensional grounds $\Lambda$ must scale as $m^{2}$ and thus vanish upon
removing the IR regulator. Similarly, the contributions from the
$m^{2}S_{h\xi\xi}$ term in
$\langle(S_{h\partial\xi\partial\xi}+m^{2}\,S_{h\xi\xi})^{2}\rangle_{1\rm PI}$
vanish in the massless limit. We are thus left with

$\begin{split}\Gamma_{1l}^{\Phi}&=\langle
S_{\Phi}\rangle-\tfrac{1}{2}\,\langle
S_{h\partial\xi\partial\xi}^{2}\rangle_{1\rm PI}+{\cal O}(h^{3})+{\cal
O}(m)\;,\\\ S_{h\partial\xi\partial\xi}&=-\frac{1}{2\lambda}\int
d^{2}x\,\bar{h}^{\alpha\beta}\partial_{\alpha}\xi^{a}\partial_{\beta}\xi_{a}\;,\quad
S_{\Phi}=\frac{1}{8\pi}\int
d^{2}x\sqrt{\gamma}R^{(2)}\,\xi^{a}\xi^{b}\nabla_{a}\nabla_{b}\Phi\;.\end{split}$
(3.64)

We start from the bubble diagram given by the double contraction term

$\begin{split}-\tfrac{1}{2}\,\langle
S_{h\partial\xi\partial\xi}^{2}\rangle_{\rm 1\rm
PI}&=-\frac{1}{8\lambda^{2}}\int
d^{2}xd^{2}y\,\bar{h}^{\alpha\beta}(x)\,\bar{h}^{\gamma\delta}(y)\,\langle\partial_{\alpha}\xi^{a}(x)\partial_{\beta}\xi_{a}(x)\,\partial_{\gamma}\xi^{b}(y)\partial_{\delta}\xi_{b}(y)\rangle_{1\rm
PI}\\\ &=-\frac{D}{4}\int
d^{2}xd^{2}y\,\bar{h}^{\alpha\beta}(x)\,\bar{h}^{\gamma\delta}(y)\,\partial_{\alpha}\partial_{\gamma}G(x-y)\,\partial_{\beta}\partial_{\delta}G(x-y)\\\
&=-\frac{D}{4}\int\frac{d^{2}p}{(2\pi)^{2}}\,\bar{h}^{\alpha\beta}(p)\,\Pi_{\alpha\beta,\gamma\delta}(p)\,\bar{h}^{\gamma\delta}(-p)\;,\end{split}$
(3.65)

where the regularized polarization tensor is given by

$\Pi^{\rm
reg.}_{\alpha\beta,\gamma\delta}(p)=\mu^{2-n}\int\frac{d^{n}k}{(2\pi)^{n}}\,\frac{k_{(\alpha}(p+k)_{\beta)}k_{(\gamma}(p+k)_{\delta)}}{(k^{2}+m^{2})((p+k)^{2}+m^{2})}\;.$
(3.66)

The diagram is IR finite in the massless limit, allowing one to compute
$\Pi^{\rm reg.}_{\alpha\beta,\gamma\delta}(p)$ at $m^{2}=0\,$. This gives [29]

$\begin{split}-\tfrac{1}{2}\,\langle&S_{h\partial\xi\partial\xi}^{2}\rangle_{1\rm
PI}=-\frac{D}{16\pi}\,\Gamma(1-\tfrac{\epsilon}{2})\,B(2+\tfrac{\epsilon}{2},2+\tfrac{\epsilon}{2})\,\int\frac{d^{2}p}{(2\pi)^{2}}\,\left(\frac{p^{2}}{4\pi\mu^{2}}\right)^{\frac{\epsilon}{2}}\\\
&\times\Big{\\{}\Big{(}p_{\alpha}p_{\beta}\bar{h}^{\alpha\beta}(p)-\tfrac{1}{2}\,p^{2}h(p)\Big{)}\,\frac{1}{p^{2}}\,\Big{(}p_{\gamma}p_{\delta}\bar{h}^{\gamma\delta}(-p)-\tfrac{1}{2}\,p^{2}h(-p)\Big{)}\\\
&\hskip
14.22636pt-\frac{2}{\epsilon(1+\tfrac{1}{2}\epsilon)}\,\Big{(}p_{\alpha}\bar{h}^{\alpha\beta}(p)\,p_{\gamma}\bar{h}^{\gamma}{}_{\beta}(-p)-\tfrac{1}{2}\,\bar{h}^{\alpha\beta}(p)\,p^{2}\bar{h}_{\alpha\beta}(-p)+\tfrac{1}{8}\,\epsilon\,h(p)\,p^{2}h(-p)\Big{)}\Big{\\}}\,,\end{split}$
(3.67)

where $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ is the Euler beta
function. The first term above is UV finite and non-local, corresponding to
the covariant term

$\int d^{2}x\sqrt{\gamma}\,R^{(2)}\frac{1}{\Box}\,R^{(2)}\;.$ (3.68)

The second term, which is the one we are after, is divergent, yielding

$\begin{split}\Gamma^{\Phi}_{\rm
div.}&\supset\frac{D}{48\pi}\,\frac{1}{\epsilon}\int\frac{d^{2}p}{(2\pi)^{2}}\,\Big{(}p_{\alpha}\bar{h}^{\alpha\beta}(p)\,p_{\gamma}\bar{h}^{\gamma}{}_{\beta}(-p)-\tfrac{1}{2}\,\bar{h}^{\alpha\beta}(p)\,p^{2}\bar{h}_{\alpha\beta}(-p)+\tfrac{1}{8}\,\epsilon\,h(p)\,p^{2}h(-p)\Big{)}\\\
&=\frac{D}{48\pi}\,\frac{1}{\epsilon}\int
d^{2}x\,\Big{[}(\partial\cdot\bar{h}_{\alpha})^{2}+\tfrac{1}{2}\,\bar{h}^{\alpha\beta}\Box\bar{h}_{\alpha\beta}-\tfrac{1}{8}\,\epsilon\,h\Box
h\Big{]}\\\ &=\frac{D}{24\pi}\,\frac{1}{\epsilon}\int
d^{2}x\sqrt{\gamma}R^{(2)}\;,\end{split}$ (3.69)

where in the last line we recognized the Ricci scalar to order $h^{2}\,$. The
only other contribution to the divergent part of $\Gamma^{\Phi}_{1l}$ is
$\langle S_{\Phi}\rangle$, which is straightforward to evaluate:

$\begin{split}\langle S_{\Phi}\rangle&=\frac{1}{8\pi}\int
d^{2}x\sqrt{\gamma}R^{(2)}\,\langle\xi^{a}\xi^{b}\rangle\nabla_{a}\nabla_{b}\Phi=\frac{\lambda}{8\pi}\,G(0)\,\int
d^{2}x\sqrt{\gamma}R^{(2)}\,\nabla^{2}\Phi\\\
&=-\frac{\lambda}{16\pi^{2}}\frac{1}{\epsilon}\int
d^{2}x\sqrt{\gamma}R^{(2)}\,\nabla^{2}\Phi+{\cal
O}(\epsilon^{0})\;.\end{split}$ (3.70)

In order to cancel the divergences one needs the counterterm

$S_{\rm c.t.}=\frac{1}{\epsilon}\int
d^{2}x\sqrt{\gamma}R^{(2)}\,\Big{[}-\frac{D}{24\pi}+\frac{\alpha^{\prime}}{8\pi}\,\nabla^{2}\Phi\Big{]}\;,$
(3.71)

where we have used $\lambda=2\pi\alpha^{\prime}$. As in (2.7) this allows us
to determine the bare dilaton $\Phi_{0}$ as

$\Phi_{0}=\mu^{\epsilon}\Big{[}\Phi-\frac{1}{\epsilon}\,\frac{D}{6}+\frac{1}{\epsilon}\,\frac{\alpha^{\prime}}{2}\,\nabla^{2}\Phi\Big{]}\;.$
(3.72)

Finally, using $\frac{d\Phi_{0}}{dt}=0$ we compute for the one-loop
$\beta-$function $\beta^{\Phi}=\frac{d\Phi}{dt}+\epsilon\Phi$:

$\beta^{\Phi}=\frac{D}{6}-\frac{\alpha^{\prime}}{2}\,\nabla^{2}\Phi\;.$ (3.73)

Let us make a few comments on this result. Collecting what we have reviewed so
far, we have determined the one-loop $\beta-$functions for the metric-dilaton
system:

$\beta_{\mu\nu}^{g}=\alpha^{\prime}\,R_{\mu\nu}\;,\quad\beta^{\Phi}=\frac{D}{6}-\frac{\alpha^{\prime}}{2}\,\nabla^{2}\Phi\;.$
(3.74)

One needs to include the reparametrization ghost system corresponding to the
worldsheet metric being a dynamical field that needs to be integrated over in
the path integral. As is well-known [30], this shifts the constant term
$D\rightarrow D-26$. Moreover, it is apparent from (3.74) that $\beta^{i}=0$
are not the correct field equations. Including the extra terms as dictated by
(2.51) one rather obtains

$\bar{\beta}^{g}_{\mu\nu}=\alpha^{\prime}\,\Big{(}R_{\mu\nu}+2\,\nabla_{\mu}\nabla_{\nu}\Phi\Big{)}\;,\quad\bar{\beta}^{\Phi}=\frac{D-26}{6}-\frac{\alpha^{\prime}}{2}\,\Big{(}\nabla^{2}\Phi-2\,\nabla^{\mu}\Phi\nabla_{\mu}\Phi\Big{)}\;.$
(3.75)

Setting these functions to zero provides the correct field equations
associated to the target space effective action

$S[g,\Phi]=\frac{1}{2\kappa_{0}^{2}}\int
d^{D}x\sqrt{-g}\,e^{-2\Phi}\Big{[}-\frac{2(D-26)}{3\,\alpha^{\prime}}+R+4\,\nabla^{\mu}\Phi\nabla_{\mu}\Phi\Big{]}\;.$
(3.76)

One should note, however, that the one-loop result (3.74) does _not_ give, in
principle, the full ${\cal O}(\alpha^{\prime})$ contribution to
$\beta^{\Phi}\,$. This is because the dilaton coupling in the sigma model
(3.1) appears with one extra order of $\alpha^{\prime}$ as compared to the
other couplings. This implies that the full ${\cal
O}({\alpha^{\prime}}^{\,L})$ dilaton $\beta-$function requires an $(L+1)-$loop
computation. For instance, while the pure metric-dilaton system does not
produce any further ${\cal O}(\alpha^{\prime})$ term in (3.73), including the
$B-$field would result in a further contribution
$-\frac{\alpha^{\prime}}{24}\,H^{2}$ arising at two loops.

For this reason, together with the complication of computing on a curved
worldsheet, it is often preferable to fix the dilaton equation from
consistency [23, 22, 31, 32, 33]. Rather than discussing the general
procedure, the idea is easily understood by giving the details in the simplest
case at hand. Let us suppose that we do not know the dilaton $\beta-$function.
The metric $\beta-$function, together with the general relation (2.51) fixes
the metric field equation to

$R_{\mu\nu}+2\,\nabla_{\mu}\nabla_{\nu}\Phi=0\;\longrightarrow\;R+2\,\nabla^{2}\Phi=0\;.$
(3.77)

Taking the divergence of the first equation one obtains

$\begin{split}\nabla^{\nu}R_{\mu\nu}&=-2\,\nabla^{2}\nabla_{\mu}\Phi=-2\,\big{(}\nabla_{\mu}\nabla^{2}\Phi+R_{\mu\nu}\,\nabla^{\nu}\Phi\big{)}\\\
&=-2\,\nabla_{\mu}\nabla^{2}\Phi+4\,\big{(}\nabla_{\mu}\nabla_{\nu}\Phi\big{)}\nabla^{\nu}\Phi=-2\,\nabla_{\mu}\big{(}\nabla^{2}\Phi-(\nabla\Phi)^{2}\big{)}\;.\end{split}$
(3.78)

Consistency with the Bianchi identity
$\nabla^{\nu}R_{\mu\nu}\equiv\frac{1}{2}\,\nabla_{\mu}R$ gives the constraint

$0=\nabla_{\mu}\big{(}\nabla^{2}\Phi-(\nabla\Phi)^{2}+\tfrac{1}{4}\,R\big{)}=\tfrac{1}{2}\,\nabla_{\mu}\big{(}\nabla^{2}\Phi-2\,(\nabla\Phi)^{2}\big{)}\;,$
(3.79)

that can be integrated to

$\nabla^{2}\Phi-2\,(\nabla\Phi)^{2}=C\;,$ (3.80)

for an undetermined constant $C\,$. This shows that the dilaton equation in
(3.75) is correctly reproduced by (3.80), apart from the constant $C$ that can
be easily fixed by matching with the one-loop result (3.75) to be
$C=\frac{D-26}{3\alpha^{\prime}}\,$. Here we use that, on dimensional grounds,
the constant term in (3.75) cannot receive $\alpha^{\prime}$ corrections.

##  4 Duality-invariant Sigma Model

Having reviewed the necessary tools to extract target space equations from a
given string sigma model, we turn in this section to the main subject of this
paper. We shall compute the one-loop $\beta-$functions of the
$O(d,d)-$invariant sigma model recently discussed in [9]. This, together with
a general analysis of the Weyl anomaly, gives the complete $O(d,d)-$invariant
target space equations to lowest order in $\alpha^{\prime}$, which agree with
the low-energy effective action constructed by Maharana and Schwarz in [34].
This result generalizes the earlier works of [14, 15] by including the
external $B-$field and Kaluza-Klein gauge vectors and provides a positive
check on the viability of the sigma model [9] to determine higher order
$\alpha^{\prime}$ corrections.

We shall start by briefly reviewing the features of the duality-invariant
sigma model that will be used to compute the $\beta-$functions. In order to
find the target space equations, we will derive an operator expression for the
Weyl anomaly, following the general discussion of section 2. We will then turn
to the actual computation of the $\beta-$functions and finally determine the
target space field equations, concluding with an internal consistency check on
the result.

###  4.1 $O(d,d)$-invariant Worldsheet

To set the stage, let us consider a $(D+d)-$dimensional target space manifold
possessing $d$ abelian isometries. We shall choose coordinates split as
$\hat{x}^{\hat{\mu}}=(x^{\mu},y^{i})$, with $\mu=0,\cdots,D-1$ denoting the
“external” directions and $i=1,\cdots,d$ the “internal” directions along the
isometries, such that the target space fields do not depend on the internal
coordinates $y^{i}\,$.

Upon dimensional reduction, the $(D+d)-$dimensional fields give rise to the
$D-$dimensional metric $g_{\mu\nu}$, $B-$field $B_{\mu\nu}$ and dilaton
$\phi$, together with an $O(d,d)$ multiplet of Kaluza-Klein gauge fields
${\cal A}_{\mu}{}^{M}$ and an $O(d,d)-$valued symmetric matrix of scalar
fields ${\cal H}_{MN}$, also known as the generalized metric. From here on we
shall denote by $M=1,\cdots,2d$ indices in the fundamental representation of
$O(d,d)$, which will be raised and lowered by the $O(d,d)-$invariant metric
$\eta_{MN}\,$. The scalar matrix ${\cal H}_{MN}$, being $O(d,d)-$valued, has
to obey the constraint ${\cal H}_{MP}\eta^{PQ}{\cal H}_{QN}=\eta_{MN}\,$.

The construction of [9], which is based on the dimensional reduction of the
Polyakov sigma model, together with the early proposals of [5, 35] and [36],
leads to a manifestly $O(d,d)-$invariant worldsheet action that generalizes
the one of [5] and agrees with [7, 8]:

$\begin{split}S=&-\frac{1}{2\lambda}\int
d^{2}\sigma\,\Big{[}\sqrt{-h}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,g_{\mu\nu}(X)+\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,B_{\mu\nu}(X)\\\
&-\epsilon^{\alpha\beta}\partial_{\alpha}Y^{M}\partial_{\beta}X^{\mu}\,{\cal
A}_{\mu M}(X)-D_{0}Y^{M}D_{1}Y_{M}+u\,D_{1}Y^{M}D_{1}Y_{M}+e\,{\cal
H}_{MN}\,D_{1}Y^{M}D_{1}Y^{N}\Big{]}\;,\end{split}$ (4.1)

where we denote by $h_{\alpha\beta}$ the worldsheet metric of Lorentzian
signature. As before, we have set $\lambda=2\pi\alpha^{\prime}\,$. In the
above action the worldsheet scalars $X^{\mu}$ are the coordinate embeddings
corresponding to the “external” directions $x^{\mu}$, while the $O(d,d)$
fields $Y^{M}=(Y^{i},\widetilde{Y}_{i})$ correspond to the doubled “internal”
sector [34, 37]. The gauge-covariant derivatives are defined as [34]

$D_{\alpha}Y^{M}:=\partial_{\alpha}Y^{M}+\partial_{\alpha}X^{\mu}{\cal
A}_{\mu}{}^{M}(X)\equiv\partial_{\alpha}Y^{M}+{\cal A}_{\alpha}{}^{M}\;,$
(4.2)

and the Weyl-invariant metric combinations $e$ and $u$ are given by

$e=\frac{\big{(}(h_{01})^{2}-h_{00}h_{11}\big{)}^{1/2}}{h_{11}}\;,\quad
u=\frac{h_{01}}{h_{11}}\;.$ (4.3)

It is clear from (4.1) that the price for having a manifestly
$O(d,d)-$invariant action that does not need any subsidiary constraints333The
proposal of [38] does provide manifestly Lorentz-invariant actions for chiral
forms, but they are either non-local or non-polynomial. is the lack of
manifest worldsheet diffeomorphism invariance. The action (4.1) is
diffeomorphism invariant nonetheless, albeit with the non-standard
transformation rules

$\begin{split}\delta_{\xi}X^{\mu}&=\xi^{\alpha}\partial_{\alpha}X^{\mu}\;,\qquad\qquad\;\;\;\;\;\,\delta_{\xi}h_{\alpha\beta}=\nabla_{\alpha}\xi_{\beta}+\nabla_{\beta}\xi_{\alpha}\;,\\\
\delta_{\xi}Y^{M}&=\xi^{\alpha}\partial_{\alpha}Y^{M}-\xi^{0}\,\mathfrak{D}^{M}\;,\qquad\mathfrak{D}^{M}=D_{0}Y^{M}-u\,D_{1}Y^{M}-e\,{\cal
H}^{MN}D_{1}Y_{N}\;.\end{split}$ (4.4)

It should be emphasized that this diffeomorphism invariance requires ${\cal
H}_{MN}$ to obey the $O(d,d)$ constraint ${\cal H}_{MP}{\cal
H}^{P}{}_{N}=\eta_{MN}\,$. Let us also mention that the action (4.1) is
invariant under the “time-local” shifts
$\delta_{\Xi}Y^{M}=\Xi^{M}(\sigma^{0})\,$. This symmetry allows one to
integrate the second-order $Y-$equations, which are given by
$\partial_{1}\mathfrak{D}^{M}=0$, to the first-order self-duality relations
$\mathfrak{D}^{M}=0\,$. These in turn render the equations of motion
equivalent to the ones of [34].

As for the symmetry under gauge transformations of the background fields, the
discussion of target space diffeomorphisms and $B-$field gauge transformations
is standard. The $U(1)$ Kaluza-Klein transformation $\delta_{\lambda}{\cal
A}_{\mu}{}^{M}=\partial_{\mu}\lambda^{M}$ instead has to be accompanied by

$\delta_{\lambda}Y^{M}=-\lambda^{M}(X)\;,\quad\delta_{\lambda}B_{\mu\nu}=\frac{1}{2}\,{\cal
F}_{\mu\nu}{}^{M}\lambda_{M}\;,$ (4.5)

where ${\cal F}_{\mu\nu}{}^{M}=2\,\partial_{[\mu}{\cal A}_{\nu]}{}^{M}$ is the
abelian curvature two-form. The non-standard transformation of the $B-$field
under the vector gauge symmetries [34] requires a modification of the naive
field strength by an abelian Chern-Simons term

$H_{\mu\nu\lambda}=3\,\partial_{[\mu}B_{\nu\lambda]}-3\,{\cal
A}_{[\mu}{}^{M}\partial_{\nu}{\cal A}_{\lambda]M}\;,$ (4.6)

to ensure gauge invariance of $H_{\mu\nu\lambda}\,$. Correspondingly, the
standard Bianchi identity $dH=0$ is replaced by

$\partial_{[\mu}H_{\nu\lambda\rho]}+\frac{3}{4}\,{\cal F}_{[\mu\nu}{}^{M}{\cal
F}_{\lambda\rho]M}=0\;.$ (4.7)

In [9], the worldsheet action (4.1) was derived without coupling to the
dilaton. We now include the dilaton coupling by a standard Fradkin-Tseytlin
term:

$\begin{split}S=&-\frac{1}{2\lambda}\int
d^{2}\sigma\,\Big{[}\sqrt{-h}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,g_{\mu\nu}(X)+\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,B_{\mu\nu}(X)\\\
&\hskip
65.44133pt-\epsilon^{\alpha\beta}\partial_{\alpha}Y^{M}\partial_{\beta}X^{\mu}\,{\cal
A}_{\mu\,M}(X)+\alpha^{\prime}\sqrt{-h}R^{(2)}\phi(X)\Big{]}\\\
&+\frac{1}{2\lambda}\int
d^{2}\sigma\,\Big{[}D_{0}Y^{M}D_{1}Y_{M}-u\,D_{1}Y^{M}D_{1}Y_{M}-e\,{\cal
H}_{MN}\,D_{1}Y^{M}D_{1}Y^{N}\Big{]}\;,\end{split}$ (4.8)

where $\phi$ is the $D-$dimensional $O(d,d)-$invariant dilaton [15], which is
related to the higher-dimensional standard dilaton $\Phi$ by

$\phi=\Phi-\frac{1}{4}\,\log{\rm det\,}G_{ij}\;,$ (4.9)

where $G_{ij}(X)$ is the internal metric.

We can give a heuristic justification for the appearance of the
$O(d,d)-$invariant dilaton in the sigma model (4.8). In the Buscher procedure
[39] to derive the action of $T-$duality at the worldsheet level, one trades
the derivatives of the original coordinates $\partial_{\alpha}Y^{i}$ for a
curl-free vector field $V_{\alpha}^{i}\,$. The curl-free condition
$\epsilon^{\alpha\beta}\partial_{\alpha}V_{\beta}^{i}=0$ is imposed via
Lagrange multipliers $\widetilde{Y}_{i}$, that play the role of dual
coordinates. Integrating out the vector fields $V_{\alpha}^{i}$ gives rise to
the $T-$dual action for the $\widetilde{Y}_{i}\,$. It is well-known that
integrating out the auxiliary fields $V_{\alpha}^{i}$ at the quantum level
produces a one-loop shift of the action proportional to $\int
d^{2}\sigma\sqrt{-h}R^{(2)}\,\log{\rm det\,}G_{ij}\,$ [40, 41, 42]. This
results in the original dilaton $\Phi$ being shifted to the T-dual dilaton

$\widetilde{\Phi}=\Phi-\frac{1}{2}\,\log{\rm det\,}G_{ij}\;.$ (4.10)

Now consider the above action and integrate out the dual coordinates
$\widetilde{Y}_{i}$ in the $O(d,d)$ multiplet
$Y^{M}=(Y^{i},\widetilde{Y}_{i})\,$. Similarly to the Buscher procedure, this
leads back to the original Polyakov sigma model at the classical level.
Including the one-loop effect should shift the dilaton coupling $\phi$ (that
we now suppose to be unknown) by

$\phi\rightarrow\phi+k\,\log{\rm det\,}G_{ij}\;,$ (4.11)

for some constant $k$. If this is to give the original theory, one must have
$\phi+k\,\log{\rm det\,}G_{ij}=\Phi$. If one integrates out the original
coordinates $Y^{i}$ instead, the shift has to be

$\phi\rightarrow\phi-k\,\log{\rm det\,}G_{ij}\;,$ (4.12)

since $Y^{i}$ couples to the metric $G_{ij}$ as $\widetilde{Y}_{i}$ does to
the inverse $G^{ij}\,$. This time, integrating out the original coordinates
produces the T-dual sigma model at tree-level, but to match at one-loop the
shift has to produce the T-dual dilaton, _i.e._ $\phi-k\,\log{\rm
det\,}G_{ij}=\widetilde{\Phi}$. Using (4.10) together with $\phi+k\,\log{\rm
det\,}G_{ij}=\Phi$ established above then fixes $k=\frac{1}{4}$. This
reproduces (4.9).

###  4.2 Weyl Anomaly Equations

Before attempting to compute the $\beta-$functions of the sigma model (4.8),
we shall derive the analogue of the Weyl anomaly equations (2.51) reviewed in
section 2. This is necessary in order to establish the correct field equations
$\bar{\beta}^{i}=0$ in terms of the RG $\beta-$functions.

We begin by classifying the possible diffeomorphism invariant dimension two
operators. It is important to note that, besides being background field
dependent, the transformations (4.4) do not leave individually invariant any
terms in the action (4.8) that contain $Y^{M}$. This implies that the only
classically diffeomorphism invariant operator involving the $Y^{M}$ fields is
the $Y-$action itself.

We now come to the equations of motion, that have to be used to derive the
Weyl anomaly, as we have seen in section 2. The $Y-$equation simply reads

$\partial_{1}\mathfrak{D}^{M}=0\;.$ (4.13)

Upon exploiting the time-dependent $\Xi^{M}$ shift symmetry this is equivalent
to the twisted self-duality relation $\mathfrak{D}^{M}=0\,$. The $X-$equation
is considerably more involved and is given by

$\begin{split}\sqrt{-h}\,g_{\mu\nu}\,\mathcal{D}^{\alpha}\partial_{\alpha}X^{\nu}&=\frac{1}{2}\,\epsilon^{\alpha\beta}\partial_{\alpha}X^{\nu}\partial_{\beta}X^{\lambda}\,H_{\mu\nu\lambda}+\frac{\alpha^{\prime}}{2}\,\sqrt{-h}\,R^{(2)}\,\partial_{\mu}\phi+\epsilon^{\alpha\beta}\partial_{\alpha}X^{\nu}D_{\beta}Y_{M}\,{\cal
F}_{\mu\nu}{}^{M}\\\ &\;\;+\frac{e}{2}\,\partial_{\mu}{\cal
H}_{MN}\,D_{1}Y^{M}D_{1}Y^{N}-\partial_{1}X^{\nu}\,{\cal
F}_{\mu\nu}{}^{M}\,\mathfrak{D}_{M}\;,\end{split}$ (4.14)

where we recall that the field strength $H_{\mu\nu\lambda}$ includes the
Chern-Simons correction (4.6).

In order to write down the action as a sum of dimension two operators
associated to different couplings as in (2.4), one has to write out the
covariant derivatives since the Kaluza-Klein vectors ${\cal A}_{\mu}{}^{M}$
give rise to quadratic and cubic composite couplings:

$\begin{split}S&=\int d^{2}\sigma\,\Big{[}A^{\mu\nu}_{g}\cdot
g_{\mu\nu}+A^{\mu\nu}_{\tilde{g}}\cdot\big{(}g_{\mu\nu}+{\cal
A}_{\mu}{}^{M}{\cal H}_{MN}{\cal A}_{\nu}{}^{N}\big{)}+A^{\mu\nu}_{{\cal
A}^{2}}\cdot\big{(}{\cal A}_{\mu}{}^{M}{\cal A}_{\nu M}\big{)}\\\
&+A^{\mu\nu}_{B}\cdot B_{\mu\nu}+A^{\mu M}_{\cal A}\cdot{\cal A}_{\mu
M}+A^{\mu M}_{{\cal A}{\cal H}}\cdot\big{(}{\cal A}_{\mu}{}^{N}{\cal
H}_{MN}\big{)}+A^{MN}_{\eta}\cdot\eta_{MN}+A^{MN}_{\cal H}\cdot{\cal
H}_{MN}+A_{\phi}\cdot\phi\Big{]}\;,\end{split}$ (4.15)

where we defined

$\begin{split}A^{\mu\nu}_{g}&=\frac{1}{2\lambda
e}\,d_{0}X^{\mu}d_{0}X^{\nu}\;,\qquad\hskip
28.45274ptA^{\mu\nu}_{\tilde{g}}=-\frac{e}{2\lambda}\,\partial_{1}X^{\mu}\partial_{1}X^{\nu}\;,\quad\hskip
2.84526ptA^{\mu\nu}_{{\cal
A}^{2}}=\frac{1}{2\lambda}\,\partial_{1}X^{(\mu}d_{0}X^{\nu)}\;,\\\\[5.69054pt]
A^{\mu\nu}_{B}&=-\frac{1}{2\lambda}\,\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\;,\qquad
A^{\mu\,M}_{\cal A}=\frac{1}{\lambda}\,d_{0}X^{\mu}\partial_{1}Y^{M}\;,\qquad
A^{\mu\,M}_{{\cal A}{\cal
H}}=-\frac{e}{\lambda}\,\partial_{1}X^{\mu}\partial_{1}Y^{M}\;,\\\\[5.69054pt]
A^{MN}_{\eta}&=\frac{1}{2\lambda}\,\partial_{1}Y^{(M}d_{0}Y^{N)}\;,\qquad\hskip
14.22636ptA^{MN}_{\cal
H}=-\frac{e}{2\lambda}\,\partial_{1}Y^{M}\partial_{1}Y^{N}\;,\quad\hskip
5.69054ptA_{\phi}=-\frac{\alpha^{\prime}}{2\lambda}\,\sqrt{-h}\,R^{(2)}\;,\end{split}$
(4.16)

with $d_{0}:=\partial_{0}-u\partial_{1}$, and we suppressed the factor
$\delta^{D}(x-X(\sigma))\,$ in every $A_{i}$.

Since the linear couplings $\varphi^{i}_{\rm lin}=(g_{\mu\nu},B_{\mu\nu},{\cal
A}_{\mu\,M},{\cal H}_{MN},\phi)$ all multiply different operators, their
$\beta-$functions are extracted as usual from the corresponding expressions
for the bare couplings:

$\begin{split}g_{0\mu\nu}&=\mu^{\epsilon}\Big{[}g_{\mu\nu}+\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,T_{n\mu\nu}^{g}\Big{]}\;,\quad
B_{0\mu\nu}=\mu^{\epsilon}\Big{[}B_{\mu\nu}+\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,T_{n\mu\nu}^{B}\Big{]}\;,\quad\phi_{0}=\mu^{\epsilon}\Big{[}\phi+\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,T_{n}^{\phi}\Big{]}\;,\\\
{\cal A}_{0\mu\,M}&=\mu^{\epsilon}\Big{[}{\cal
A}_{\mu\,M}+\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,T_{n\mu\,M}^{\cal
A}\Big{]}\;,\quad{\cal H}_{0MN}=\mu^{\epsilon}\Big{[}{\cal
H}_{MN}+\sum_{n=1}^{\infty}\frac{1}{\epsilon^{n}}\,T_{nMN}^{\cal
H}\Big{]}\;,\quad\eta_{0MN}=\mu^{\epsilon}\,\eta_{MN}\;.\end{split}$ (4.17)

Notice that, in order to keep both $X^{\mu}$ and $Y^{M}$ dimensionless in
$n=2+\epsilon$ dimensions, one has to define the bare $O(d,d)$ metric as444We
shall see in the following that $\eta_{MN}$ does not renormalize.
$\eta_{0MN}=\mu^{\epsilon}\,\eta_{MN}\,$. Due to this one has to be careful
with the position of the $O(d,d)$ indices. We take $A_{0\mu M}$ to have
dimension $\epsilon$, which implies that ${\cal
A}_{0\mu}{}^{M}=\eta_{0}^{MN}{\cal A}_{0\mu\,N}={\cal A}_{\mu}{}^{M}+\cdots$
is dimensionless and that ${\cal A}_{0\alpha}{}^{M}$ has dimension one (as it
should in order to define the bare covariant derivative
$D^{0}_{\alpha}Y^{M}$). Knowing the $\beta-$functions for the linear
couplings, one can easily derive the ones for the composite couplings. For
instance,

$\begin{split}&\frac{d}{dt}\,\big{(}{\cal A}_{\mu}{}^{N}\,{\cal
H}_{MN}\big{)}=\frac{d{\cal A}_{\mu\,P}}{dt}\,\eta^{PN}\,{\cal H}_{MN}+{\cal
A}_{\mu\,P}\,\frac{d\eta^{PN}}{dt}\,{\cal H}_{MN}+{\cal
A}_{\mu\,P}\,\eta^{PN}\,\frac{d{\cal H}_{MN}}{dt}\\\ &=\big{(}-\epsilon\,{\cal
A}_{\mu\,P}+\beta^{\cal A}_{\mu\,P}\big{)}\,\eta^{PN}\,{\cal H}_{MN}+{\cal
A}_{\mu\,P}\,\big{(}\epsilon\,\eta^{PN}\big{)}\,{\cal H}_{MN}+{\cal
A}_{\mu\,P}\,\eta^{PN}\,\big{(}-\epsilon\,{\cal H}_{MN}+\beta^{\cal
H}_{MN}\big{)}\\\ &=-\epsilon\,\big{(}{\cal A}_{\mu}{}^{N}{\cal
H}_{MN}\big{)}+\beta^{\cal A}_{\mu\,N}{\cal H}_{M}{}^{N}+{\cal
A}_{\mu}{}^{N}\beta^{\cal H}_{MN}\;.\end{split}$ (4.18)

In order to compute the Weyl anomaly as in (2.24) (albeit now in Lorentzian
signature), we need to define the bare action in $n=2+\epsilon$ dimensions.
Similar to the problem of defining the antisymmetric density
$\varepsilon^{\alpha\beta}$ in $n$ dimensions, we now encounter the difficulty
of defining $e$ and $u$ in arbitrary dimensions. If we keep the definitions
(4.3) in arbitrary dimensions, $e$ and $u$ remain Weyl invariant, but in order
to also keep $Y^{M}$ invariant one has to define the bare action for $Y$
as555To see this one can repeat the steps [9] that lead to the action (4.8)
starting from the Polyakov sigma model directly in $n=2+\epsilon$ dimensions.

$S_{0Y}=\frac{1}{2\lambda}\int
d^{n}\sigma\,e^{\epsilon\rho}\,\big{(}D_{1}Y_{M}\big{)}_{0}\,\big{(}D_{0}Y^{M}-u\,D_{1}Y^{M}-e\,{\cal
H}^{MN}D_{1}Y_{N}\big{)}_{0}\;,$ (4.19)

where the subscript $0$ denotes that all the couplings, including the vectors
${\cal A}_{\mu}{}^{M}$ hidden in $D_{\alpha}Y^{M}$, are the bare ones, and
$\rho$ is defined by $e^{(2+\epsilon)\rho}=\sqrt{-h}\,$.

Taking the Weyl variation now leads to

$\sqrt{-h}\,T^{\alpha}{}_{\alpha}=A_{i0}\cdot(-\epsilon\,\varphi^{i}_{0})-\frac{1}{2\pi}\,\partial_{\alpha}\big{[}\sqrt{-h}h^{\alpha\beta}\partial_{\beta}X^{\mu}\partial_{\mu}\phi_{0}\big{]}\;,$
(4.20)

where all couplings $\varphi^{i}$ and operators $A_{i}$ are listed in (4.15)
and (4.16). By repeating the general procedure reviewed in section 2 and using
the equations of motion (4.14) to decompose the total derivative one finds

$\begin{split}\sqrt{-h}\,T^{\alpha}{}_{\alpha}&=[A_{i}]\cdot\hat{\beta}^{i}-\frac{1}{2\lambda}\,\Big{\\{}\sqrt{-h}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,\big{(}2\alpha^{\prime}\,\nabla_{0\mu}\partial_{\nu}\phi_{0}\big{)}+\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\,\big{(}\alpha^{\prime}\,\partial_{\lambda}\phi_{0}\,H_{0}^{\lambda}{}_{\mu\nu}\big{)}\\\
&+\alpha^{\prime}\,\sqrt{-h}\,\bar{R}\,\big{(}\alpha^{\prime}\,\partial_{\mu}\phi_{0}\nabla_{0}^{\mu}\phi_{0}\big{)}+e\,e^{\epsilon\rho}\,D_{1}^{0}Y^{M}D_{1}^{0}Y^{N}\,\big{(}\alpha^{\prime}\,\partial_{\mu}\phi_{0}\nabla_{0}^{\mu}{\cal
H}_{0MN}\big{)}\\\
&+2\,\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}D_{\beta}^{0}Y^{M}\,\big{(}\alpha^{\prime}\,\nabla_{0}^{\lambda}\phi_{0}\,{\cal
F}_{0\lambda\mu\,M}\big{)}-2\,e^{\epsilon\rho}\,\partial_{1}X^{\mu}\,\mathfrak{D}_{0}^{M}\,\big{(}\alpha^{\prime}\,\nabla_{0}^{\lambda}\phi_{0}\,{\cal
F}_{0\lambda\mu\,M}\big{)}\Big{\\}}\\\
&=[A_{i}]\cdot\hat{\beta}^{i}+\alpha^{\prime}\,A_{i0}\cdot\big{(}{\cal
L}_{\nabla_{0}\phi_{0}}\varphi^{i}_{0}\big{)}\;,\end{split}$ (4.21)

where, for the moment, we have discarded the $Q-$type total derivative
contributions coming from $[A_{i}]$ (see (2.31)). The Lie derivative displayed
in the last line acts on $g_{\mu\nu}$, $\phi$ and ${\cal H}_{MN}$ as usual,
while it differs in the case of ${\cal A}_{\mu}{}^{M}$ and $B_{\mu\nu}$ by
gauge transformations:

${\cal L}_{\xi}{\cal A}_{\mu}{}^{M}:=\xi^{\lambda}{\cal
F}_{\lambda\mu}{}^{M}\;,\quad{\cal
L}_{\xi}B_{\mu\nu}:=\xi^{\lambda}\,\big{(}H_{\lambda\mu\nu}+{\cal
A}_{[\mu}{}^{M}{\cal F}_{\nu]\lambda\,M}\big{)}\;.$ (4.22)

The main difference from the standard theory is in the transformation ${\cal
L}_{\xi}B_{\mu\nu}$, that is _not_ covariant, due to the mixing with the
Kaluza-Klein vector fields. Such non-covariant term, however, is canceled in
the sum $A_{i0}\cdot\big{(}{\cal L}_{\xi}\varphi^{i}_{0}\big{)}$, as can be
seen directly from the expanded form of (4.21), which is manifestly gauge-
invariant. Since both the $\beta-$functions of composite couplings (see e.g.
(4.18)) and the Lie derivative obey the Leibniz rule, one can see that the
vanishing of the Weyl anomaly (4.21), which involves _all_ the couplings
(4.15), is ensured by the vanishing of the $\bar{\beta}^{i}$ functions
associated to the independent linear couplings, namely

$\begin{split}\bar{\beta}^{i}&=\beta^{i}+\alpha^{\prime}\,{\cal
L}_{\nabla\phi}\varphi^{i}+\cdots=0\quad{\rm for}\quad i=g,\phi,{\cal A},{\cal
H}\;,\\\
\bar{\beta}_{\mu\nu}^{B}&=\beta_{\mu\nu}^{B}+\alpha^{\prime}\,\nabla^{\lambda}\phi\,H_{\lambda\mu\nu}+\cdots=0\;,\end{split}$
(4.23)

where the dots stand for the possible total derivative terms arising from
$[A_{i}]$.

Let us now study the total derivative terms related to $[A_{i}]$. These terms
come from renormalization of the composite operators $A_{i}$ in flat space,
and so we can take $h_{\alpha\beta}=\eta_{\alpha\beta}$ and thus $\rho=u=0$
and $e=1\,$. The possible dimension two total derivatives arising from
$\partial_{\alpha}X^{\mu}$ are $d^{\mu}_{h}\cdot N_{\mu}$ and
$d^{\mu}_{\varepsilon}\cdot K_{\mu}$ as in (2.37). Coming now to possible
dimension two total derivatives constructed from $\partial_{\alpha}Y^{M}$, the
situation is much more subtle. In flat space, the Lorentz transformations of
$Y^{M}$ (that can be found by specializing the diffeomorphism transformations
(4.4)), are highly non-linear and background field dependent. In order to
determine possible total derivatives including $\partial_{\alpha}Y^{M}$, we
notice that time derivatives $\partial_{0}Y^{M}$ appear in the action (4.8)
only as $\mathfrak{D}^{M}$ or $\epsilon^{\alpha\beta}D_{\beta}Y^{M}$.
Moreover, double time derivatives never appear acting on $Y^{M}$. For these
reasons, one can naively construct two independent operators acting on
different fields: $d^{M}_{\epsilon}\cdot
V_{M}:=-\frac{1}{\lambda}\partial_{\alpha}\big{[}\epsilon^{\alpha\beta}D_{\beta}Y^{M}\,V_{M}(X)\big{]}$
and $d^{M}_{\mathfrak{D}}\cdot
W_{M}:=-\frac{1}{\lambda}\partial_{1}\big{[}\mathfrak{D}^{M}\,W_{M}(X)\big{]}$,
where we recall that $\mathfrak{D}^{M}$ is defined in (4.4). These two
operators, however, cannot appear in an arbitrary way. One can see that the
only (on-shell) Lorentz-invariant combination is given by

$(d_{\epsilon}^{M}-d_{\mathfrak{D}}^{M})\cdot V_{M}=:d^{M}\cdot
V_{M}=-\frac{1}{\lambda}\Big{[}\partial_{\alpha}\big{(}\epsilon^{\alpha\beta}D_{\beta}Y^{M}\,V_{M}(X)\big{)}-\partial_{1}\big{(}\mathfrak{D}^{M}\,V_{M}(X)\big{)}\Big{]}\;,$
(4.24)

and depends on a single target space field $V_{M}$. Decomposing the above
expression gives

$\begin{split}d^{M}\cdot
V_{M}&=A_{i}\cdot\big{(}\delta_{V}\varphi^{i}\big{)}-\frac{1}{2\lambda}\,\partial_{\alpha}\big{[}\epsilon^{\alpha\beta}\partial_{\beta}X^{\mu}\,K_{\mu}^{\prime}(X)\big{]}\;,\\\
\delta_{V}{\cal
A}_{\mu}{}^{M}&=\partial_{\mu}V^{M}\;,\quad\delta_{V}B_{\mu\nu}=\frac{1}{2}\,{\cal
F}_{\mu\nu}{}^{M}V_{M}\;,\end{split}$ (4.25)

where $K^{\prime}_{\mu}:=V_{M}{\cal A}_{\mu}{}^{M}$. This latter contribution
is of the $d^{\mu}_{\varepsilon}$ type and thus does not introduce new
independent structures.

We have thus shown that in the $O(d,d)-$invariant sigma model any total
derivative term giving rise to a $Q-$contribution to (4.23) can be
parametrized by two field-dependent vector fields $W_{\mu}(g,B,{\cal A},{\cal
H})$ and $L_{\mu}(g,B,{\cal A},{\cal H})$ as in section 2, together with an
$O(d,d)$ vector-valued scalar field $V_{M}(g,B,{\cal A},{\cal H})$. The
contribution of $W_{\mu}$ and $L_{\mu}$ to the Weyl anomaly coefficients
(4.23) is the same as in (2.51), including the covariant Lie derivative of
${\cal A}_{\mu}{}^{M}$, while the one from $V_{M}$ can be inferred from the
decomposition (4.25). This finally gives the relation between $\beta$ and
$\bar{\beta}$:

$\begin{split}\bar{\beta}_{\mu\nu}^{g}&=\beta_{\mu\nu}^{g}+2\alpha^{\prime}\,\nabla_{\mu}\nabla_{\nu}\phi+\nabla_{(\mu}W_{\nu)}\;,\\\
\bar{\beta}^{\phi}&=\beta^{\phi}+\alpha^{\prime}\,(\nabla\phi)^{2}+\tfrac{1}{2}\nabla^{\mu}\phi\,W_{\mu}\;,\\\
\bar{\beta}^{B}_{\mu\nu}&=\beta^{B}_{\mu\nu}+\alpha^{\prime}\,H^{\lambda}{}_{\mu\nu}\nabla_{\lambda}\phi+\tfrac{1}{2}\,H^{\lambda}{}_{\mu\nu}W_{\lambda}+\partial_{[\mu}L_{\nu]}+\tfrac{1}{2}\,{\cal
F}_{\mu\nu}{}^{M}V_{M}\;,\\\ \bar{\beta}_{\mu\,M}^{\cal
A}&=\beta_{\mu\,M}^{\cal A}+\alpha^{\prime}\,\nabla^{\lambda}\phi\,{\cal
F}_{\lambda\mu\,M}+\tfrac{1}{2}\,W^{\lambda}\,{\cal
F}_{\lambda\mu\,M}+\nabla_{\mu}V_{M}\;,\\\ \bar{\beta}_{MN}^{\cal
H}&=\beta_{MN}^{\cal H}+\alpha^{\prime}\,\nabla^{\mu}\phi\,\nabla_{\mu}{\cal
H}_{MN}+\tfrac{1}{2}\,W^{\mu}\,\nabla_{\mu}{\cal H}_{MN}\;.\end{split}$ (4.26)

As we have previously discussed, if one tries to compute the vectors
$W_{\mu}$, $L_{\mu}$ and $V_{M}$ directly from the bare action, they will in
general break gauge invariance in both the $B-$field and Kaluza-Klein gauge
sectors, in agreement with the fact that our prescription for extending the
action to $n=2+\epsilon$ dimensions violates said symmetries. One can employ a
manifestly gauge invariant perturbative scheme and try to fix $W$, $L$ and $V$
by other means. Based on covariance, they can only depend on the Riemann
tensor, covariant derivatives and the gauge-invariant field strengths
$H_{\mu\nu\rho}$ and ${\cal F}_{\mu\nu}{}^{M}\,$. One can then see that they
must vanish at one-loop order, since there is no way of constructing a gauge-
invariant vector, nor an $O(d,d)-$vector valued scalar with one spacetime
derivative. We shall thus set all of them to zero and proceed in the following
with the actual computation of the $\beta-$functions.

###  4.3 Beta Functions of Duality Invariant Sigma Model

Having found the dilaton contributions to the field equations, we can now
consider the sigma model on a flat worldsheet, thereby discarding the dilaton
coupling. After deriving the equations of motion for the other fields, the
dilaton equation will be fixed by consistency.

In flat two-dimensional space one can safely perform the Wick rotation. We
will thus study the Euclidean action

$\begin{split}S&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{\\{}g_{\mu\nu}(X)\,\partial^{\alpha}X^{\mu}\partial_{\alpha}X^{\nu}+i\,\epsilon^{\alpha\beta}\,\Big{[}B_{\mu\nu}(X)\,\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}+{\cal
A}_{\mu}{}^{M}(X)\,\partial_{\alpha}X^{\mu}D_{\beta}Y_{M}\Big{]}\\\ &\hskip
65.44133pt-i\,D_{1}Y^{M}D_{2}Y_{M}+{\cal
H}_{MN}(X)\,D_{1}Y^{M}D_{1}Y^{N}\Big{\\}}\;,\end{split}$ (4.27)

where $x^{2}$ is the Euclidean time, $\epsilon^{12}=+1$ and we recall
$D_{\alpha}Y^{M}=\partial_{\alpha}Y^{M}+{\cal
A}_{\mu}{}^{M}(X)\,\partial_{\alpha}X^{\mu}$.

#### Gauge-invariant perturbative expansion

In order to perform the background field expansion we shall employ the
covariant fluctuation $\xi^{\mu}$ and split

$X^{\mu}=\varphi^{\mu}+\pi^{\mu}(\xi)\;,$ (4.28)

as we have extensively discussed. If one now performs a _linear_ split
$Y^{M}=\phi^{M}+\pi^{M}$ in the internal sector, manifest gauge invariance
under the $U(1)$ Kaluza-Klein symmetries will be lost. The reason is that
$Y^{M}$ transforms as $\delta Y^{M}=-\lambda^{M}(X)$. Combining this with the
split of $X^{\mu}\,$, one finds that the background field $\phi^{M}$
transforms with respect to the background gauge parameter:

$\delta\phi^{M}=-\lambda^{M}(\varphi)\;,$ (4.29)

ensuring that the background covariant derivative

$D_{\alpha}\phi^{M}:=\partial_{\alpha}\phi^{M}+{\cal
A}_{\mu}{}^{M}(\varphi)\,\partial_{\alpha}\varphi^{\mu}$ (4.30)

is gauge-invariant. The linear fluctuation $\pi^{M}$, however, transforms to
all orders, as follows from (4.29):

$\delta\pi^{M}=-\big{[}\lambda^{M}(X)-\lambda^{M}(\varphi)\big{]}=-\sum_{n=1}^{\infty}\frac{1}{n!}\,(\xi\cdot\nabla)^{n}\lambda^{M}(\varphi)\;,$
(4.31)

where we used the covariant Taylor expansion in terms of $\xi^{\mu}\,$. We can
rewrite the right-hand side of (4.31) in terms of $\delta{\cal
A}_{\mu}{}^{M}=\nabla_{\mu}\lambda^{M}$ as

$-\sum_{n=1}^{\infty}\frac{1}{n!}\,(\xi\cdot\nabla)^{n}\lambda^{M}(\varphi)=-\sum_{n=0}^{\infty}\frac{1}{(n+1)!}\,(\xi\cdot\nabla)^{n}\Big{(}\xi^{\mu}\delta{\cal
A}_{\mu}{}^{M}(\varphi)\Big{)}\;.$ (4.32)

Since the right-hand side here is a total gauge variation we can define a
gauge invariant fluctuation $\zeta^{M}$ as follows:

$\pi^{M}(\xi,\zeta)=\zeta^{M}-\sum_{n=0}^{\infty}\frac{1}{(n+1)!}\,(\xi\cdot\nabla)^{n}\,\Big{(}\xi^{\mu}{\cal
A}_{\mu}{}^{M}(\varphi)\Big{)}\;.$ (4.33)

This ensures that the perturbation theory is manifestly gauge invariant.

Even though it is guaranteed that the expansion in powers of $\xi^{\mu}$ and
$\zeta^{M}$ will be gauge-invariant, we will not use (4.33) directly, just as
one does not use $\pi^{\mu}(\xi)$ to derive the usual covariant expansion. We
shall rather use (4.33) in order to find a set of covariant rules to perform
the expansion. Let us recall the strategy for the covariant background field
method: given any target space scalar $\Phi(X)$ its covariant $\xi-$expansion
is given by

$\Phi(X)=\Phi(\varphi+\pi(\xi))=e^{\mathcal{D}}\Phi(\varphi)\;.$ (4.34)

The operator $\mathcal{D}$ is defined in terms of an auxiliary geodesic
$X^{\mu}(t)$ as discussed in sec. 3:

$\mathcal{D}\Phi:=\left(\frac{D\Phi(t)}{Dt}\right)_{t=0}\;.$ (4.35)

This yields the effective rules

$\begin{split}\mathcal{D}T&=\xi^{\mu}\,\nabla_{\mu}T\;,\quad
D_{\alpha}T=\partial_{\alpha}\varphi^{\mu}\,\nabla_{\mu}T\quad\text{for target
space tensors}\;,\\\
\mathcal{D}\xi^{\mu}&=0\;,\quad\mathcal{D}\partial_{\alpha}\varphi^{\mu}=\partial_{\alpha}\xi^{\mu}+\partial_{\alpha}\varphi^{\nu}\,\Gamma^{\mu}_{\nu\lambda}(\varphi)\,\xi^{\lambda}=D_{\alpha}\xi^{\mu}\;,\\\
[\mathcal{D},D_{\alpha}]&=\xi^{\mu}\partial_{\alpha}\varphi^{\nu}\,R_{\mu\nu}^{\\#}\quad\text{acting
on target space tensors and}\,\xi^{\mu}\;\end{split}$ (4.36)

that ensure manifest covariance under external target space diffeomorphisms.
Note that one should distinguish
$D_{\alpha}V^{\mu}=\partial_{\alpha}\varphi^{\nu}\nabla_{\nu}V^{\mu}$ from
$D_{\alpha}\phi^{M}=\partial_{\alpha}\phi^{M}+\partial_{\alpha}\varphi^{\mu}{\cal
A}_{\mu}{}^{M}\,$. Next, we derive effective rules for fields transforming
under $U(1)$. Using that $D_{\alpha}Y^{M}$ is a scalar with respect to
external (target space) diffeomorphisms and using (4.33) we may now derive the
gauge-invariant expansion of $D_{\alpha}Y^{M}$:

$\begin{split}D_{\alpha}Y^{M}&=\partial_{\alpha}\phi^{M}+\partial_{\alpha}\zeta^{M}+\partial_{\alpha}X^{\mu}\,{\cal
A}_{\mu}{}^{M}(X)-\partial_{\alpha}\sum_{n=0}^{\infty}\frac{1}{(n+1)!}\,(\xi\cdot\nabla)^{n}\,\Big{(}\xi^{\mu}{\cal
A}_{\mu}{}^{M}(\varphi)\Big{)}\\\
&=\partial_{\alpha}\phi^{M}+\partial_{\alpha}\zeta^{M}+e^{\mathcal{D}}\left[\partial_{\alpha}\varphi^{\mu}\,{\cal
A}_{\mu}{}^{M}(\varphi)\right]-\partial_{\alpha}\sum_{n=0}^{\infty}\frac{1}{(n+1)!}\,(\xi\cdot\nabla)^{n}\,\Big{(}\xi^{\mu}{\cal
A}_{\mu}{}^{M}(\varphi)\Big{)}\\\
&=D_{\alpha}\phi^{M}+\partial_{\alpha}\zeta^{M}+\sum_{n=1}^{\infty}\frac{1}{n!}\,\mathcal{D}^{n}\left[\partial_{\alpha}\varphi^{\mu}\,{\cal
A}_{\mu}{}^{M}(\varphi)\right]-\partial_{\alpha}\sum_{n=0}^{\infty}\frac{1}{(n+1)!}\,(\xi\cdot\nabla)^{n}\,\Big{(}\xi^{\mu}{\cal
A}_{\mu}{}^{M}(\varphi)\Big{)}\;,\end{split}$ (4.37)

where we have extracted the zeroth order part from $e^{\mathcal{D}}$ in order
to recover $D_{\alpha}\phi^{M}$. From now on we will omit the explicit
evaluation of spacetime fields at the base point $\varphi^{\mu}$, since it is
always implied in the expansion. The final step to prove gauge invariance of
the above expansion requires to evaluate the first $\mathcal{D}$ on
$\partial_{\alpha}\varphi^{\mu}{\cal A}_{\mu}{}^{M}\,$:

$\begin{split}\sum_{n=1}^{\infty}\frac{1}{n!}\,\mathcal{D}^{n}\left[\partial_{\alpha}\varphi^{\mu}{\cal
A}_{\mu}{}^{M}\right]&=\sum_{n=1}^{\infty}\frac{1}{n!}\,\mathcal{D}^{n-1}\left[D_{\alpha}\xi^{\mu}{\cal
A}_{\mu}{}^{M}+\partial_{\alpha}\varphi^{\mu}\xi^{\nu}\nabla_{\nu}{\cal
A}_{\mu}{}^{M}\right]\\\
&=\sum_{n=1}^{\infty}\frac{1}{n!}\,\mathcal{D}^{n-1}\left[\xi^{\mu}\partial_{\alpha}\varphi^{\nu}{\cal
F}_{\mu\nu}{}^{M}+\partial_{\alpha}\big{(}\xi^{\mu}{\cal
A}_{\mu}{}^{M}\big{)}\right]\;,\end{split}$ (4.38)

where we used (4.36). Given that $[\mathcal{D},D_{\alpha}]=0$ when acting on
scalars, the non-covariant term can be rewritten as

$\sum_{n=1}^{\infty}\frac{1}{n!}\,\mathcal{D}^{n-1}\partial_{\alpha}\big{(}\xi^{\mu}{\cal
A}_{\mu}{}^{M}\big{)}=\partial_{\alpha}\sum_{n=0}^{\infty}\frac{1}{(n+1)!}\,\mathcal{D}^{n}\left(\xi^{\mu}{\cal
A}_{\mu}{}^{M}\right)\;,$ (4.39)

which cancels the last term in the third line of (4.37). We have thus found
the manifestly gauge-invariant expansion

$D_{\alpha}Y^{M}=D_{\alpha}\phi^{M}+\partial_{\alpha}\zeta^{M}+\sum_{n=0}^{\infty}\frac{1}{(n+1)!}\,\mathcal{D}^{n}\left[\xi^{\mu}\partial_{\alpha}\varphi^{\nu}{\cal
F}_{\mu\nu}{}^{M}\right]\;.$ (4.40)

Finally, by formally declaring

$\mathcal{D}D_{\alpha}\phi^{M}:=\xi^{\mu}\partial_{\alpha}\varphi^{\nu}{\cal
F}_{\mu\nu}{}^{M}\;,\quad\mathcal{D}\partial_{\alpha}\zeta^{M}:=0\;,$ (4.41)

the entire expansion can be recast in the exponential form

$D_{\alpha}Y^{M}=e^{\mathcal{D}}\big{(}D_{\alpha}\phi^{M}+\partial_{\alpha}\zeta^{M}\big{)}\;.$
(4.42)

The set of covariant rules (4.36) and (4.41) allows one to write down the
${\cal O}(\xi^{n})$ term in the action recursively. By expanding in powers of
$\xi$ one obtains the recursive relations

$\begin{split}S[\varphi+\pi(\xi)]&=e^{\mathcal{D}}S[\varphi]=\sum_{n=0}^{\infty}S_{n\xi}\;,\\\
S_{n\xi}&=\frac{1}{n!}\,\mathcal{D}^{n}S_{0\xi}=\frac{1}{n}\,\mathcal{D}S_{(n-1)\xi}\;,\end{split}$
(4.43)

starting from

$\begin{split}S_{0\xi}&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{\\{}g_{\mu\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}+i\,\epsilon^{\alpha\beta}\,\Big{[}B_{\mu\nu}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}+{\cal
A}_{\mu}{}^{M}\,\partial_{\alpha}\varphi^{\mu}(D_{\beta}\phi_{M}+\partial_{\beta}\zeta_{M})\Big{]}\\\
&\hskip
36.98857pt-i\,(D_{1}\phi^{M}+\partial_{1}\zeta^{M})(D_{2}\phi_{M}+\partial_{2}\zeta_{M})+{\cal
H}_{MN}\,(D_{1}\phi^{M}+\partial_{1}\zeta^{M})(D_{1}\phi^{N}+\partial_{1}\zeta^{N})\Big{\\}}\;.\end{split}$
(4.44)

This action is not manifestly gauge invariant due to the topological terms
involving $B_{\mu\nu}$ and ${\cal A}_{\mu}{}^{M}$, which transform non
trivially, but the higher-order terms $S_{n\xi}$ will be manifestly gauge
invariant. To see this we compute explicitly $S_{1\xi}$:

$\begin{split}&S_{1\xi}=\mathcal{D}S_{0\xi}=\frac{1}{2\lambda}\int
d^{2}x\,\Big{\\{}2\,g_{\mu\nu}\,D^{\alpha}\xi^{\mu}\partial_{\alpha}\varphi^{\nu}+i\,\epsilon^{\alpha\beta}\,\xi^{\lambda}H^{0}_{\lambda\mu\nu}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}\\\
&+i\,\epsilon^{\alpha\beta}\Big{[}\xi^{\nu}\,\nabla_{\nu}{\cal
A}_{\mu}^{M}\partial_{\alpha}\varphi^{\mu}(D_{\beta}\phi_{M}+\partial_{\beta}\zeta_{M})+{\cal
A}_{\mu}^{M}D_{\alpha}\xi^{\mu}\,(D_{\beta}\phi_{M}+\partial_{\beta}\zeta_{M})+{\cal
A}_{\mu}^{M}\partial_{\alpha}\varphi^{\mu}\,\xi^{\nu}\partial_{\beta}\varphi^{\lambda}{\cal
F}_{\nu\lambda\,M}\Big{]}\\\
&-i\,\Big{[}(D_{1}\phi_{M}+\partial_{1}\zeta_{M})\xi^{\mu}\partial_{2}\varphi^{\nu}{\cal
F}_{\mu\nu}^{M}+(D_{2}\phi_{M}+\partial_{2}\zeta_{M})\xi^{\mu}\partial_{1}\varphi^{\nu}{\cal
F}_{\mu\nu}^{M}\Big{]}\\\ &+\xi^{\mu}\,\nabla_{\mu}{\cal
H}_{MN}\,(D_{1}\phi^{M}+\partial_{1}\zeta^{M})(D_{1}\phi^{N}+\partial_{1}\zeta^{N})+2\,{\cal
H}_{MN}\,(D_{1}\phi^{M}+\partial_{1}\zeta^{M})\,\xi^{\mu}\partial_{1}\varphi^{\nu}{\cal
F}_{\mu\nu}^{N}\Big{\\}}\;,\end{split}$ (4.45)

where we have already integrated by parts the term
$2i\,\epsilon^{\alpha\beta}B_{\mu\nu}D_{\alpha}\xi^{\mu}\partial_{\beta}\varphi^{\nu}$
in order to generate
$H^{0}_{\lambda\mu\nu}=3\,\partial_{[\lambda}B_{\mu\nu]}\,$. Integrating by
parts $D_{\alpha}\xi^{\mu}$ in the term ${\cal
A}_{\mu}{}^{M}D_{\alpha}\xi^{\mu}\,(D_{\beta}\phi_{M}+\partial_{\beta}\zeta_{M})$
we finally obtain

$\begin{split}S_{1\xi}&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{\\{}2\,g_{\mu\nu}\,D^{\alpha}\xi^{\mu}\partial_{\alpha}\varphi^{\nu}+i\,\epsilon^{\alpha\beta}\,\xi^{\lambda}H_{\lambda\mu\nu}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}-2i\,\xi^{\mu}\partial_{2}\varphi^{\nu}{\cal
F}_{\mu\nu}{}^{M}\,(D_{1}\phi_{M}+\partial_{1}\zeta_{M})\\\
&+\xi^{\mu}\,\nabla_{\mu}{\cal
H}_{MN}\,(D_{1}\phi^{M}+\partial_{1}\zeta^{M})(D_{1}\phi^{N}+\partial_{1}\zeta^{N})+2\,{\cal
H}_{MN}\,(D_{1}\phi^{M}+\partial_{1}\zeta^{M})\,\xi^{\mu}\partial_{1}\varphi^{\nu}{\cal
F}_{\mu\nu}{}^{N}\Big{\\}}\;.\end{split}$ (4.46)

This is manifestly gauge-invariant under all symmetries, since it displays the
invariant field strength $H_{\mu\nu\rho}=H^{0}_{\mu\nu\rho}-\frac{3}{2}\,{\cal
A}_{[\mu}{}^{M}{\cal F}_{\nu\rho]\,M}$, which ensures that the perturbative
expansion will be manifestly gauge-invariant at all orders.

For the one-loop computation we only need the action to quadratic order in all
fluctuations, which is schematically given by
$S_{2}=S_{\zeta\zeta}+S_{\zeta\xi}+S_{\xi\xi}\,$. The term $S_{\zeta\zeta}$ is
easily obtained from (4.44):

$S_{\zeta\zeta}=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}-i\,\partial_{1}\zeta^{M}\partial_{2}\zeta_{M}+{\cal
H}_{MN}\,\partial_{1}\zeta^{M}\partial_{1}\zeta^{N}\Big{]}\;,$ (4.47)

while $S_{\zeta\xi}$ is obtained by keeping the ${\cal O}(\zeta)$ terms of
(4.46):

$S_{\zeta\xi}=\frac{1}{\lambda}\int
d^{2}x\,\Big{[}\xi^{\mu}\,\nabla_{\mu}{\cal
H}_{MN}\,D_{1}\phi^{M}\partial_{1}\zeta^{N}+\xi^{\mu}\partial_{1}\zeta^{M}{\cal
F}_{\mu\nu}^{N}\,\big{(}\partial_{1}\varphi^{\nu}{\cal
H}_{MN}-i\,\partial_{2}\varphi^{\nu}\eta_{MN}\big{)}\Big{]}\;.$ (4.48)

$S_{\xi\xi}$ on the other hand requires one to compute
$\frac{1}{2}\,\mathcal{D}S_{1\xi}$ and set $\zeta^{M}=0$ in (4.46), finally
yielding

$\begin{split}S_{\xi\xi}&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{\\{}g_{\mu\nu}\,D^{\alpha}\xi^{\mu}D_{\alpha}\xi^{\nu}+R_{\mu\nu\lambda\rho}\,\xi^{\nu}\xi^{\lambda}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\rho}+\tfrac{i}{2}\,\epsilon^{\alpha\beta}\,\xi^{\lambda}\xi^{\rho}\,\nabla_{\lambda}H_{\rho\mu\nu}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}\\\
&+i\,\epsilon^{\alpha\beta}\,\xi^{\lambda}D_{\alpha}\xi^{\mu}H_{\lambda\mu\nu}\,\partial_{\beta}\varphi^{\nu}+\tfrac{1}{2}\,\xi^{\mu}\xi^{\nu}\,\nabla_{\mu}\nabla_{\nu}{\cal
H}_{MN}\,D_{1}\phi^{M}D_{1}\phi^{N}\\\ &+D_{1}\phi^{M}{\cal
F}_{\mu\nu}^{N}\,\big{(}\xi^{\mu}D_{1}\xi^{\nu}\,{\cal
H}_{MN}-i\,\xi^{\mu}D_{2}\xi^{\nu}\,\eta_{MN}\big{)}+\xi^{\lambda}\xi^{\mu}\,D_{1}\phi^{M}\nabla_{\lambda}{\cal
F}_{\mu\nu}^{N}\,\big{(}\partial_{1}\varphi^{\nu}\,{\cal
H}_{MN}-i\,\partial_{2}\varphi^{\nu}\,\eta_{MN}\big{)}\\\
&+\xi^{\lambda}\xi^{\mu}\partial_{1}\varphi^{\rho}{\cal F}_{\mu\nu}^{M}{\cal
F}_{\lambda\rho}^{N}\,\big{(}\partial_{1}\varphi^{\nu}\,{\cal
H}_{MN}-i\,\partial_{2}\varphi^{\nu}\,\eta_{MN}\big{)}+2\,\xi^{\mu}\xi^{\nu}\partial_{1}\varphi^{\lambda}\,\nabla_{\mu}{\cal
H}_{MN}\,{\cal F}_{\nu\lambda}^{M}\,D_{1}\phi^{N}\Big{\\}}\;.\end{split}$
(4.49)

In order to have standard kinetic terms we introduce background vielbeins
$e_{\mu}^{a}(\varphi)$ and consider the flattened fluctuation
$\xi^{a}=e_{\mu}^{a}\,\xi^{\mu}\,$. This transforms the $\xi-$kinetic term
into $D^{\alpha}\xi^{a}D_{\alpha}\xi_{a}$, with

$D_{\alpha}\xi^{a}=\partial_{\alpha}\xi^{a}+\partial_{\alpha}\varphi^{\mu}\,\omega_{\mu}{}^{a}{}_{b}\,\xi^{b}=\partial_{\alpha}\xi^{a}+\omega_{\alpha}{}^{a}{}_{b}\,\xi^{b}\;,$
(4.50)

that allows us to extract a standard propagator from $\frac{1}{2\lambda}\int
d^{2}x\,\partial^{\alpha}\xi^{a}\partial_{\alpha}\xi_{a}\,$. As recalled in
sec. 3, dimensional arguments using local Lorentz symmetry ensure that the
spin connection cannot contribute to UV divergences at one loop. This allows
one to effectively treat all $D_{\alpha}\xi^{a}$ as $\partial_{\alpha}\xi^{a}$
in the interaction vertices.

A similar treatment is needed for the $\zeta-$kinetic term $S_{\zeta\zeta}$,
given the non-constant background ${\cal H}_{MN}(\varphi)\,$. We shall thus
introduce $O(d,d)-$valued frames $E_{M}{}^{A}(\varphi)$, such that

${\cal
H}_{MN}=E_{M}{}^{A}\,h_{AB}\,E_{N}{}^{B}\;,\quad\eta_{MN}=E_{M}{}^{A}\,\eta_{AB}\,E_{N}{}^{B}\;,$
(4.51)

where $\eta_{AB}$ has the same form as $\eta_{MN}$, while $h_{AB}$ is a
_constant_ matrix that we choose to be a metric of $SO(d)\times SO(d)$
embedded in $O(d,d)\,$. The frames are then subject to _local_ $SO(d)\times
SO(d)$ transformations that rotate the flat index $A\,$. We then flatten the
fluctuations by defining $\zeta^{A}=E_{M}{}^{A}(\varphi)\,\zeta^{M}\,$. The
derivatives, accordingly, are modified as

$\partial_{\alpha}\zeta^{M}=E^{M}{}_{A}\,D_{\alpha}\zeta^{A}\;,\quad
D_{\alpha}\zeta^{A}=\partial_{\alpha}\zeta^{A}+\partial_{\alpha}\varphi^{\mu}\,W_{\mu}{}^{A}{}_{B}\,\zeta^{B}=\partial_{\alpha}\zeta^{A}+W_{\alpha}{}^{A}{}_{B}\,\zeta^{B}\;,$
(4.52)

where $W_{\mu}{}^{AB}$ is the Maurer-Cartan flat connection

$W_{\mu}{}^{AB}=E^{MA}\partial_{\mu}E_{M}{}^{B}=Q_{\mu}{}^{AB}+P_{\mu}{}^{AB}$
(4.53)

that splits into an $SO(d)\times SO(d)$ connection $Q$ and an $SO(d)\times
SO(d)$ tensor $P$, as familiar in coset constructions. The kinetic term
$S_{\zeta\zeta}$ is rewritten as

$\begin{split}S_{\zeta\zeta}&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}-i\,D_{1}\zeta^{A}D_{2}\zeta_{A}+h_{AB}\,D_{1}\zeta^{A}D_{1}\zeta^{B}\Big{]}\\\
&=\frac{1}{2\lambda}\int
d^{2}x\,\Big{[}-i\,\partial_{1}\zeta^{A}\partial_{2}\zeta_{A}+h_{AB}\,\partial_{1}\zeta^{A}\partial_{1}\zeta^{B}\Big{]}+S_{W}+S_{WW}\;,\end{split}$
(4.54)

where the free part defines the propagator, while the interaction terms
involving the Maurer-Cartan form are given by

$S_{W}=\frac{1}{\lambda}\int
d^{2}x\,T^{\alpha\beta}_{AB}\,W_{\alpha}{}^{A}{}_{C}\,\partial_{\beta}\zeta^{B}\zeta^{C}\;,\quad
S_{WW}=\frac{1}{2\lambda}\int
d^{2}x\,T^{\alpha\beta}_{AB}\,W_{\alpha}{}^{A}{}_{C}\,W_{\beta}{}^{B}{}_{D}\,\zeta^{C}\zeta^{D}\;,$
(4.55)

where $T^{\alpha\beta}_{AB}$ is defined by

$T^{11}_{AB}=h_{AB}\;,\quad
T^{12}_{AB}=T^{21}_{AB}=-\frac{i}{2}\,\eta_{AB}\;,\quad T^{22}_{AB}=0\;.$
(4.56)

Despite the similarities with the Lorentz connection $\omega_{\alpha}{}^{ab}$,
the Maurer-Cartan form has two important differences: first of all, it
contains an $SO(d)\times SO(d)$ tensor, $P_{\alpha}{}^{AB}$, that _does_
contribute to divergences. Moreover, as it has been recently shown [9], the
local $SO(d)\times SO(d)$ symmetry is anomalous, due to the chiral nature of
the $Y-$sector. The anomaly, that is canceled by a suitable ${\cal
O}(\alpha^{\prime})$ transformation of $B_{\mu\nu}\,$ [10, 9], is a one-loop
effect but is _finite_. Therefore, it does not affect our computation of the
$\beta-$functions at this order, but it should intervene at higher loops,
thereby triggering the Green-Schwarz type deformation of the field strength
$H_{\mu\nu\rho}$ at order $\alpha^{\prime}\,$. For these reasons, the
connection terms arising from both $S_{\zeta\zeta}$ and $S_{\zeta\xi}$
_cannot_ be ignored.

#### One-loop $\beta-$functions

From the free kinetic terms we can derive the propagators, that read

$\begin{split}\langle\xi^{a}(x)\,\xi^{b}(y)\rangle&=\lambda\,\delta^{ab}\,G(x-y)\;,\quad
G(x)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{ip\cdot x}}{p^{2}}\;,\\\
\langle\zeta^{A}(x)\,\zeta^{B}(y)\rangle&=\lambda\,G^{AB}(x-y)\;,\quad
G^{AB}(x)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{ip\cdot
x}}{p^{2}}\,\Big{[}h^{AB}+i\,\frac{p_{2}}{p_{1}}\,\eta^{AB}\Big{]}\;.\end{split}$
(4.57)

The above massless propagators suffer from infrared divergences. The
$\xi-$propagator $G(x)$ can be regularized in the IR by adding a mass term
$\frac{m^{2}}{2\lambda}\int d^{2}x\,\xi^{a}\xi_{a}$ to the action, thereby
modifying $\frac{1}{p^{2}}\to\frac{1}{p^{2}+m^{2}}$ as usual. The IR
regularization of the $\zeta-$propagator is more subtle and is discussed in
Appendix A.2. The interaction vertices extracted from (4.47), (4.48) and
(4.49) can be summarized as

$\begin{split}S_{\zeta\zeta}&\;\longrightarrow\;S_{W}+S_{WW}\;,\\\
S_{\zeta\xi}&\;\longrightarrow\;S_{\nabla{\cal H}}+S_{W\nabla{\cal H}}+S_{\cal
F}+S_{W{\cal F}}\;,\\\ S_{\xi\xi}&\;\longrightarrow\;S_{R}+S_{\nabla
H}+S_{H}+S_{\nabla\nabla{\cal H}}+S_{{\cal F}^{\prime}}+S_{\nabla{\cal
F}}+S_{{\cal F}{\cal F}}+S_{\nabla{\cal H}{\cal F}}\;,\end{split}$ (4.58)

where we schematically distinguished them according to the target space fields
entering the interaction. Let us notice that the new terms $S_{W\nabla{\cal
H}}$ and $S_{W{\cal F}}$ arise from the replacement
$\partial_{\alpha}\zeta^{M}=E^{M}{}_{A}\,(\partial_{\alpha}\zeta^{A}+W_{\alpha}{}^{AB}\zeta_{B})\,$.
Another way to classify the vertices is according to the number of background
fields $\partial_{\alpha}\varphi^{\mu}$ and $D_{\alpha}\phi^{M}$ involved:

$\begin{split}\partial\varphi\partial\varphi&\;\longrightarrow\;S_{WW}+S_{W{\cal
F}}+S_{R}+S_{\nabla H}+S_{{\cal F}{\cal F}}\;,\\\ \partial\varphi
D\phi&\;\longrightarrow\;S_{W\nabla{\cal H}}+S_{\nabla{\cal F}}+S_{\nabla{\cal
H}{\cal F}}\;,\\\ D\phi D\phi&\;\longrightarrow\;S_{\nabla\nabla{\cal
H}}\;,\\\ \partial\varphi&\;\longrightarrow\;S_{W}+S_{\cal F}+S_{H}\;,\\\
D\phi&\;\longrightarrow\;S_{\nabla{\cal H}}+S_{{\cal
F}^{\prime}}\;.\end{split}$ (4.59)

Since the divergences in $e^{-\Gamma_{1l}}=\left\langle e^{-S_{\rm
int}}\right\rangle$ can only renormalize the dimension two operators
$\partial\varphi\partial\varphi$, $\partial\varphi D\phi$ and $D\phi D\phi$,
they can only arise from the divergent part of

$\begin{split}\Gamma_{1l\,\partial\varphi\partial\varphi}&=\langle
S_{R}\rangle+\langle S_{\nabla H}\rangle+\langle S_{WW}\rangle+\langle
S_{{\cal F}{\cal F}}\rangle-\tfrac{1}{2}\,\langle S_{W}^{2}+S_{\cal
F}^{2}+S_{H}^{2}\rangle_{1{\rm PI}}\;,\\\ \Gamma_{1l\,D\phi D\phi}&=\langle
S_{\nabla\nabla{\cal H}}\rangle-\tfrac{1}{2}\,\langle S_{\nabla{\cal
H}}^{2}+S_{{\cal F}^{\prime}}^{2}\rangle_{1{\rm PI}}\;,\\\
\Gamma_{1l\,\partial\varphi D\phi}&=\langle S_{\nabla{\cal F}}\rangle+\langle
S_{\nabla{\cal H}{\cal F}}\rangle-\langle S_{\cal F}S_{\nabla{\cal
H}}\rangle_{1\rm{PI}}-\langle S_{H}S_{{\cal
F}^{\prime}}\rangle_{1\rm{PI}}\;,\end{split}$ (4.60)

where we have set to zero all the contributions that have trivially vanishing
contractions.

We are now ready to compute the divergent parts of (4.60) starting from
$\Gamma_{1l\,\partial\varphi\partial\varphi}$, that gives the
$\beta-$functions of the metric $g_{\mu\nu}$ and $B-$field. The diagrams with
a single contraction, _e.g._ $\langle S_{R}\rangle$, are tadpole diagrams that
give a purely local contribution. For instance,

$\begin{split}\langle S_{R}\rangle&=\frac{1}{2\lambda}\int
d^{2}x\,\langle\xi^{a}\xi^{b}\rangle\,R_{\mu
ab\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}=-\frac{1}{2}\,G(0)\int
d^{2}x\,R_{\mu\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\\\
&=\frac{1}{4\pi\epsilon}\,\int
d^{2}x\,R_{\mu\nu}\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}+{\cal
O}(\epsilon^{0})\;,\end{split}$ (4.61)

and

$\begin{split}\langle S_{\nabla H}\rangle&=\frac{i}{4\lambda}\int
d^{2}x\,\epsilon^{\alpha\beta}\,\langle\xi^{a}\xi^{b}\rangle\,e^{\lambda}_{a}e^{\rho}_{b}\,\nabla_{\lambda}H_{\rho\mu\nu}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}\\\
&=\frac{i}{4}\,G(0)\int
d^{2}x\,\nabla^{\rho}H_{\rho\mu\nu}\,\epsilon^{\alpha\beta}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}\\\
&=-\frac{i}{8\pi\epsilon}\int
d^{2}x\,\nabla^{\rho}H_{\rho\mu\nu}\,\epsilon^{\alpha\beta}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}+{\cal
O}(\epsilon^{0})\;,\end{split}$ (4.62)

where we used the $\frac{1}{\epsilon}$ contribution to $G(0)$ given in (3.44)
(c.f. (A.2)). The bubble diagrams that arise from double contractions, such as
$\langle S_{H}^{2}\rangle_{1{\rm PI}}$, are non-local, but their divergent
contribution only comes from the local leading order, when expanding in
external momenta. As an example, let us consider $\langle
S_{H}^{2}\rangle_{1{\rm PI}}\,$. Its contribution to
$\Gamma_{1l\,\partial\varphi\partial\varphi}$ reads

$\begin{split}-\frac{1}{2}\,\langle S_{H}^{2}\rangle_{1{\rm
PI}}&=\frac{1}{8\lambda^{2}}\int
d^{2}xd^{2}y\,\epsilon^{\alpha\beta}\epsilon^{\gamma\delta}\,\big{(}H_{ab\mu}\,\partial_{\beta}\varphi^{\mu}\big{)}(x)\,\big{(}H_{cd\nu}\,\partial_{\delta}\varphi^{\nu}\big{)}(y)\,\big{\langle}\xi^{a}(x)\,\partial_{\alpha}\xi^{b}(x)\,\xi^{c}(y)\,\partial_{\gamma}\xi^{d}(y)\big{\rangle}\\\
&=\frac{1}{8}\int\frac{d^{2}p}{(2\pi)^{2}}\,\epsilon^{\alpha\beta}\epsilon^{\gamma\delta}\,\big{(}H_{\mu
ab}\,\partial_{\beta}\varphi^{\mu}\big{)}(p)\,\Pi_{\alpha\gamma}(p)\,\big{(}H_{\nu}{}^{ab}\,\partial_{\delta}\varphi^{\nu}\big{)}(-p)\;,\end{split}$
(4.63)

where the polarization tensor is given by

$\Pi_{\alpha\gamma}(p)=\frac{1}{2}\int\frac{d^{2}k}{(2\pi)^{2}}\,\frac{(p+2k)_{\alpha}(p+2k)_{\gamma}}{(k^{2}+m^{2})((p+k)^{2}+m^{2})}\;.$
(4.64)

The divergent part of $\Pi_{\alpha\gamma}(p)$ comes from
$\Pi_{\alpha\gamma}(0)$, which is the only contribution we need to compute.
Upon extending the integral to $n=2+\epsilon$ dimensions we have

$\begin{split}\Pi^{\rm
reg}_{\alpha\gamma}(0)&=\mu^{2-n}\int\frac{d^{n}k}{(2\pi)^{n}}\,\frac{2\,k_{\alpha}k_{\gamma}}{(k^{2}+m^{2})^{2}}=\mu^{2-n}\,\frac{2}{n}\,\delta_{\alpha\gamma}\,\int\frac{d^{n}k}{(2\pi)^{n}}\,\frac{k^{2}}{(k^{2}+m^{2})^{2}}\\\
&=\frac{\delta_{\alpha\gamma}}{4\pi}\,\left(\frac{m^{2}}{4\pi\mu^{2}}\right)^{\frac{\epsilon}{2}}\Gamma(-\epsilon/2)=\delta_{\alpha\gamma}\,G(0)=-\frac{\delta_{\alpha\gamma}}{2\pi\epsilon}+{\cal
O}(\epsilon^{0})\;,\end{split}$ (4.65)

yielding

$\begin{split}-\frac{1}{2}\,\langle S_{H}^{2}\rangle_{1{\rm
PI}}&=-\frac{1}{16\pi\epsilon}\int\frac{d^{2}p}{(2\pi)^{2}}\,\big{(}H_{\mu
ab}\,\partial_{\alpha}\varphi^{\mu}\big{)}(p)\,\big{(}H_{\nu}{}^{ab}\,\partial^{\alpha}\varphi^{\nu}\big{)}(-p)+{\cal
O}(\epsilon^{0})\\\ &=-\frac{1}{16\pi\epsilon}\int
d^{2}x\,H_{\mu\lambda\rho}H_{\nu}{}^{\lambda\rho}\,\partial_{\alpha}\varphi^{\mu}\partial^{\alpha}\varphi^{\nu}+{\cal
O}(\epsilon^{0})\;.\end{split}$ (4.66)

All the remaining integrals in (4.60) can be computed in a similar way, upon
using the results of Appendix A.2 for the $\zeta-$sector, the results of which
we summarize now. Proceeding with the terms involving the Maurer-Cartan form
$W_{\mu}{}^{AB}$, it is interesting to notice that the separate diagrams
$\langle S_{WW}\rangle$ and $\langle S^{2}_{W}\rangle_{1{\rm PI}}$ are neither
2D-Lorentz nor $SO(d)\times SO(d)$ gauge-invariant:

$\begin{split}\langle S_{WW}\rangle&=\frac{1}{2\lambda}\int
d^{2}x\,T^{\alpha\beta}_{AB}\,\partial_{\alpha}\varphi^{\mu}\partial_{\beta}\varphi^{\nu}\,W_{\mu}{}^{A}{}_{C}\,W_{\nu}{}^{B}{}_{D}\,\langle\zeta^{C}\zeta^{D}\rangle\\\
&=-\frac{1}{4\pi\epsilon}\int
d^{2}x\,W_{\mu}{}^{AC}W_{\nu}{}^{BD}\,\Big{[}h_{AB}\,h_{CD}\,\partial_{1}\varphi^{\mu}\partial_{1}\varphi^{\nu}-i\,\eta_{AB}\,h_{CD}\,\partial_{1}\varphi^{\mu}\partial_{2}\varphi^{\nu}\Big{]}+{\cal
O}(\epsilon^{0})\end{split}$ (4.67)

and

$\begin{split}-\frac{1}{2}\,\langle S_{W}^{2}\rangle_{1{\rm
PI}}&=-\frac{1}{2\lambda^{2}}\int
d^{2}xd^{2}y\,T^{\alpha\beta}_{AB}\,\big{(}\partial_{\alpha}\varphi^{\mu}\,W_{\mu}{}^{A}{}_{C}\big{)}(x)\,T^{\gamma\delta}_{DE}\,\big{(}\partial_{\gamma}\varphi^{\nu}\,W_{\nu}{}^{D}{}_{F}\big{)}(y)\\\
&\hskip
56.9055pt\times\big{\langle}\partial_{\beta}\zeta^{B}(x)\,\zeta^{C}(x)\,\partial_{\delta}\zeta^{E}(y)\,\zeta^{F}(y)\big{\rangle}_{1{\rm
PI}}\\\ &=\frac{1}{4\pi\epsilon}\int
d^{2}x\,W_{\mu\,AB}\,W_{\nu\,CD}\,\Big{[}\partial_{1}\varphi^{\mu}\partial_{1}\varphi^{\nu}\,\big{(}\tfrac{3}{4}\,h^{AC}\,h^{BD}+\tfrac{1}{4}\,\eta^{AC}\,\eta^{BD}\big{)}\\\
&\hskip
42.67912pt+\partial_{2}\varphi^{\mu}\partial_{2}\varphi^{\nu}\,\big{(}\tfrac{1}{4}\,h^{AC}\,h^{BD}-\tfrac{1}{4}\,\eta^{AC}\,\eta^{BD}\big{)}-i\,\partial_{1}\varphi^{\mu}\partial_{2}\varphi^{\nu}\,\eta^{AC}\,h^{BD}\Big{]}+{\cal
O}(\epsilon^{0})\;,\end{split}$ (4.68)

where the objects $T^{\alpha\beta}_{AB}$ have been defined in (4.56). However,
the two diagrams combine in a non-trivial way to form the Lorentz and gauge-
invariant term

$\begin{split}\langle S_{WW}\rangle-\frac{1}{2}\,\langle
S_{W}^{2}\rangle_{1{\rm PI}}&=\frac{1}{16\pi\epsilon}\int
d^{2}x\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\,W_{\mu\,AB}\,W_{\nu\,CD}\,\big{(}\eta^{AC}\,\eta^{BD}-h^{AC}\,h^{BD}\big{)}+{\cal
O}(\epsilon^{0})\\\ &=\frac{1}{8\pi\epsilon}\int
d^{2}x\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\,P_{\mu\,AB}\,P_{\nu}{}^{AB}+{\cal
O}(\epsilon^{0})\\\ &=\frac{1}{32\pi\epsilon}\int
d^{2}x\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\,\partial_{\mu}{\cal
H}_{MN}\,\partial_{\nu}{\cal H}^{MN}+{\cal O}(\epsilon^{0})\;.\end{split}$
(4.69)

The final contributions to $\Gamma_{1l\,\partial\varphi\partial\varphi}$,
namely $\langle S_{{\cal F}{\cal F}}\rangle$ and $\langle S^{2}_{\cal
F}\rangle_{1{\rm PI}}$, behave in a similar manner: the separate diagrams are
not Lorentz invariant, being given by

$\begin{split}\langle S_{{\cal F}{\cal F}}\rangle&=\frac{1}{2\lambda}\int
d^{2}x\,\langle\xi^{a}\xi^{b}\rangle\,\partial_{1}\varphi^{\mu}\,{\cal F}_{\mu
a}^{M}\,{\cal F}_{\nu b}^{N}\,\big{(}\partial_{1}\varphi^{\nu}\,{\cal
H}_{MN}-i\,\partial_{2}\varphi^{\nu}\,\eta_{MN}\big{)}\\\
&=-\frac{1}{4\pi\epsilon}\int d^{2}x\,{\cal F}_{\mu\lambda}^{M}\,{\cal
F}_{\nu}{}^{\lambda\,N}\,\big{(}\partial_{1}\varphi^{\mu}\partial_{1}\varphi^{\nu}\,{\cal
H}_{MN}-i\,\partial_{1}\varphi^{\mu}\partial_{2}\varphi^{\nu}\,\eta_{MN}\big{)}+{\cal
O}(\epsilon^{0})\end{split}$ (4.70)

and

$\begin{split}-\frac{1}{2}\,\langle S^{2}_{\cal F}\rangle_{1{\rm
PI}}&=-\frac{1}{2\lambda^{2}}\int d^{2}xd^{2}y\,\big{[}{\cal F}_{\mu
a}^{N}\,(\partial_{1}\varphi^{\mu}\,{\cal
H}_{MN}-i\,\partial_{2}\varphi^{\mu}\,\eta_{MN})\,E^{M}{}_{A}\big{]}(x)\\\
&\times\,\big{[}{\cal F}_{\nu b}^{Q}\,(\partial_{1}\varphi^{\nu}\,{\cal
H}_{PQ}-i\,\partial_{2}\varphi^{\nu}\,\eta_{PQ})\,E^{P}{}_{B}\big{]}(y)\,\big{\langle}\xi^{a}(x)\,\partial_{1}\zeta^{A}(x)\,\xi^{b}(y)\,\partial_{1}\zeta^{B}(y)\big{\rangle}\\\
&=\frac{1}{8\pi\epsilon}\int d^{2}x\,{\cal F}_{\mu\lambda}^{M}\,{\cal
F}_{\nu}{}^{\lambda\,N}\,\Big{[}\big{(}\partial_{1}\varphi^{\mu}\partial_{1}\varphi^{\nu}-\partial_{2}\varphi^{\mu}\partial_{2}\varphi^{\nu}\big{)}\,{\cal
H}_{MN}-2i\,\partial_{1}\varphi^{\mu}\partial_{2}\varphi^{\nu}\,\eta_{MN}\Big{]}+{\cal
O}(\epsilon^{0})\;,\end{split}$ (4.71)

but combine into the Lorentz invariant term

$\langle S_{{\cal F}{\cal F}}\rangle-\frac{1}{2}\,\langle S^{2}_{\cal
F}\rangle_{1{\rm PI}}=-\frac{1}{8\pi\epsilon}\int
d^{2}x\,\partial^{\alpha}\varphi^{\mu}\partial_{\alpha}\varphi^{\nu}\,{\cal
H}_{MN}\,{\cal F}_{\mu\lambda}^{M}\,{\cal F}_{\nu}{}^{\lambda\,N}+{\cal
O}(\epsilon^{0})\;.$ (4.72)