Nonlinear sigma models on constant curvature target manifolds: A functional renormalization group approach

Abstract

We study nonlinear sigma models on target manifolds with constant (positive or negative) curvature using the functional renormalization group and the background field method. We pay particular attention to the splitting Ward identities associated to the invariance under reparametrization of the background field. Implementing these Ward identities imposes to use the curvature as a formal expansion parameter, which allows us to close the flow equation of the (scale-dependent) effective action consistently to first order in the curvature. We shed new light on previous work using the background field method.

Full text

Nonlinear sigma models on constant curvature target manifolds:
A functional renormalization group approach
Alexander N. Efremov and Adam Rançon
Univ. Lille, CNRS, UMR 8523–PhLAM–Laboratoire de Physique des Lasers,
Atomes et Mol´ecules, F-59000 Lille, France
(Received 27 September 2021; accepted 8 October 2021; published 9 November 2021)
We study nonlinear sigma models on target manifolds with constant (positive or negative) curvature
using the functional renormalization group and the background field method. We pay particular attention to
the splitting Ward identities associated to the invariance under reparametrization of the background field.
Implementing these Ward identities imposes to use the curvature as a formal expansion parameter, which
allows us to close the flow equation of the (scale-dependent) effective action consistently to first order in
the curvature. We shed new light on previous work using the background field method.
DOI: 10.1103/PhysRevD.104.105003
I. INTRODUCTION
The nonlinear sigma models (NLSM) are a very rich
class of dynamical systems which spans many fields of
physics such as for example high energy physics, string
theory, statistical physics. For instance, the Oð4ÞNLSM
first appeared in the work of M. Gell-Mann and M. L´evy as
an effective model of pion-nucleon interaction [1]. More
recently L.D. Faddeev has shown that the Oð3ÞNLSM with
a topological term might appear in the confined phase of the
SUð2ÞYang–Mills theory if one performs the spin-charge
decomposition [2]. However there is no any proof of
existence of the quantum model at the present time.
NLSM on a two dimensional manifold, i.e., the string
world sheet, appear in string theory [3]. In general relativity
one can consider the metric tensor as a Goldstone boson
identified with the coset GLð4;RÞ=SOð1;3Þ[4].Itis
therefore a NLSM which is similar to the Skyrme model.
Furthermore one is often interested in the asymptotic safety
of this sigma-model in more than two dimensions. In the
language of Wilson’s renormalization group a theory is
asymptotically safe if the critical surface has a finite co-
dimension, i.e., Weinberg’s ultraviolet critical surface is
finite dimensional [5].
In statistical physics, NLSM are used to describe spin
systems, especially close to two dimensions [6,7]. In this
context, it is widely believed that the OðNÞNLSM belongs
to the same universality class than the OðNÞlinear sigma
model (a ϕ4theory), which has a non-trivial infrared fixed
point only in spatial dimensions 2<d<4(we only refer
to the case N>2for simplicity). This fixed point describes
the critical state between an ordered phase (described to a
weak coupling fixed point in the NLSM) and a disordered
phase (with massive modes, corresponding to a strongly
coupled theory). Importantly, this nontrivial Wilson-Fisher
fixed point merges with the Gaussian fixed point (in the
context of the linear model) at the upper critical dimension
dc¼4, meaning that critical behavior is captured by a free
field theory.
The NLSM on noncompact target-manifolds are also
relevant to the physics of Anderson localization [8,9], see
[10] for a review. In this context, a toy model corresponds
to the target space SOð1;N−1Þ=OðN−1Þ, in the “replica
limit”N→1[11] (see also [12–15] for applications of this
model in high-energy physics). The physics is expected to
be rather different from that of its compact counterpart. In
particular, it is believed that the upper critical dimension is
infinite, with nontrivial critical exponents in all dimensions
d>2, see [16] for a recent analysis of this issue.
Wilson’s Renormalization Group (RG) is the method of
choice to address phase transitions, developed originally in
statistical mechanics [17,18] and later extended to the field
theory [19]. An application of these ideas to the nonlinear
σ-model follows one of two directions. In the first approach
one considers the linear σ-model with an axillary nonlinear
constraint [20]. The second method is based on the
covariant Taylor expansion around a background field
[21] and is more natural for the σ-models which are not
multiplicatively renormalizable in general.
The functional RG (FRG) is a modern implementation of
the RG which allows for nonperturbative approximations,
see [22] for a recent review. The background field method
has been adapted to the FRG for applications in quantum
gravity and non-Abelian gauge theories. Note that the
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PHYSICAL REVIEW D 104, 105003 (2021)
2470-0010=2021=104(10)=105003(12) 105003-1 Published by the American Physical Society

invariance under reparametrization of the background field
give rise to the so-called splitting Ward identities [23,24].
Somewhat surprisingly, the FRG with the background field
method has only been used quite recently to study the OðNÞ
NLSM [25–27].In[25], the flow equation at lowest order
in the derivative expansion were obtained, and a nontrivial
fixed point is found for all d>2.In[26], the expansion is
pushed to the next order, and the fixed point seems to
disappear if all the allowed coupling constants are kept in
d¼3, where the existence of a nontrivial fixed point is
beyond doubt. In all these previous studies the authors have
not taken into account the splitting Ward identities to
organize their approximations.
Here, we revisit this problem, generalizing the analysis
to arbitrary constant curvature target-manifold. We use the
lowest order in the derivative expansion, as in [25],but
implement the splitting Ward identities explicitly, which
leads to different flow equations and emphasizes the
importance of the Ward identities in the background field
method for the nonlinear σ-model. Unfortunately, the
splitting Ward identity can only be written in terms of a
formal expansion in the curvature of the target-manifold,
and our flow equations are therefore only valid to lowest
order in derivatives and curvature.
The manuscript is organized as follows. In Sec. II we
define the model and review the background field method,
and describe the FRG in Sec. III. In Sec. IV we give the
splitting symmetry transformation for the symmetric mani-
folds up to second order in the Riemann tensor. An explicit
form of the transformation is needed to impose the splitting
Ward identities [23,24] for the FRG. As a consequence the
curvature becomes the main expansion parameter in our
work. The flow equations at lowest order in the curvature
are derived in Sec. V. In particular we find that the coupling
constants have different evolution equations, in contrast to
what happens in linear models. Using the Ward identities,
we are able to close the flow equations, the corresponding
beta functions are given in Sec. VI. We discuss our results,
and compare them to previous studies using the back-
ground field method, in Sec. VII.
II. MODEL AND BACKGROUND-FIELD
EXPANSION
For Ma simply connected D-dimensional (D¼N−1)
manifold of a constant curvature Kendowed with a metric
h, the Levy–Civita connection Dcompatible with the
metric Dh ¼0;ϕ−1∶M→Rda chart on M, the action
of the NLsM on the target space Mis defined as
SðϕÞ¼1
2tZx
hαβðϕÞ∂iϕα∂iϕβ;ð1Þ
where ∂iϕα¼∂
∂xiϕαðxÞ,α¼1…D,i¼1…d, and
Rx≡Rddx. Here t>0is a nonperturbative bare coupling
constant which is proportional to the temperature in the
Heisenberg model [7]. The theory is regularized by an
ultraviolet cutoff Λ. For a positive curvature manifold his
an elliptic metric. For a negative curvature manifold his a
hyperbolic metric. In the both cases his positive definite. In
the following, we will consider the case of a D-dimensional
sphere M¼SDfor K>0and the hyperbolic space HD
for K<0.
We consider the functional RG in the context of the
covariant background-field method, i.e., by writing
the field ϕðxÞin terms of a (fixed) background φðxÞand
the corresponding (fluctuating) normal field ξðxÞ∈
TφðxÞM[21]. Assume that there is a smooth map ϕsðxÞ
such that ϕ0ðxÞ¼φðxÞand ϕ1ðxÞ¼ϕðxÞwith _
ϕ0¼ξ.
We choose the curve ϕsin Mto coincide with the geodesic
between the initial and final points ϕ0,ϕ1, i.e.,
̈
ϕα
sþΓα
σγ _
ϕσ
s_
ϕγ
s¼0;Γαx
σzγz0¼Γα
σγðϕxÞδxz δxz0;ð2Þ
where Γα
σγðϕÞis the Christoffel symbol and αstands for the
multi-index ðα;x
1;…;x
dÞ. Then for a smooth functional f
we have the Taylor expansion
fðϕÞ¼fðϕ0Þþ_
ϕα
s
δfðϕÞ
δϕα


s¼0
þ1
2̈
ϕα
s
δfðϕÞ
δϕαþ_
ϕα
s_
ϕβ
s
δ2fðϕÞ
δϕαδϕβ


s¼0þ…;ð3Þ
which can be written as
fðϕÞ¼eξαDαfðφÞ;D
αfβðφÞ¼δfβðφÞ
δφαþΓβ
σαfσðφÞ:ð4Þ
The functional f½φ;ξ¼f½ϕðφ;ξÞ depends only on ϕand
not on the way the splitting between φand ξis done. In
other words for ˜
φa new expansion point and ˜
ξthe
corresponding new normal field such that ϕðφ;ξÞ¼
ϕð˜
φ;
˜
ξÞwe still have f½φ;ξ¼f½˜
φ;
˜
ξ. Such a functional
is called a “single field”functional [26]. This invariance
will impose strong constraints on the FRG functionals, as
discussed in Sec. IV. To calculate the expansion coefficients
one uses the standard relations
ξαDα∂iφλ¼∂iξλþΓλ
αγξα∂iφγ¼Diξλ;ð5Þ
DαDβ∂iφλ¼Rλ
βαγ∂iφγ;ð6Þ
where Rλ
βασ is the Riemann tensor,
Rλ
βασ ðφÞ¼δΓλ
βσ
δφα−
δΓλ
βα
δφσþΓλ
γαΓγ
βσ −Γλ
γσΓγ
βα:ð7Þ
ALEXANDER N. EFREMOV and ADAM RANÇON PHYS. REV. D 104, 105003 (2021)
105003-2

Note that Rλ
βασðφÞis ultra-local, i.e., it is proportional to the
product of three delta functions of the space coordinates.
Furthermore for a constant curvature manifold we have
Rαλβγ ¼KΠαλβγ ;Παλβγ ¼hαβ hλγ −hαγ hλβ :ð8Þ
Here and below hαβ ¼hαβðφÞdenotes the metric tensor at
the expansion point. Clearly the curvature tensor is cova-
riantly constant DγRλ
βασ ¼0.
III. FUNCTIONAL RG
The strategy of the FRG is to build a family of models,
indexed by a momentum scale k, which interpolates
between the semiclassical limit for k¼Λand the model
of interest for k→0. For this purpose, one introduces a
regulator term ΔSkin the action, which leaves the modes
with momentum larger than kuntouched while freezing
the low-momentum modes, implementing effectively
Wilson’s RG.
We first introduce the generating functional of n-point
connected Schwinger functions Wk½φ;j, which depends
on the background φand source j∈TφMlinearly coupled
to the normal field ξ[28–30],
eWk½φ;j¼ZDφðξÞe−S½φ;ξ−ΔSk½φ;ξþj:ξ:ð9Þ
For details see Appendix C. The measure
DφðξÞ¼ðDet −∂2
ΛÞD
2ffiffiffiffiffiffiffiffiffiffiffiffiffi
DethΛ
pe−UΛ½φ;ξY
α
dξα
ffiffiffiffiffiffi
2π
pð10Þ
corresponds to the invariant measure after the
change of variables from ϕto ξat fixed background.
It has been convenient to introduce UΛ½φ;ξ¼
−log ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Deth−1
ΛhΛðϕÞ
pjδϕ
δξ jÞ, corresponding to an ultralocal
term in the action which can be expanded in ξ.Itis
necessary to include this term to preserve the symmetries of
the background expansion explicitly, see Sec. IV. This term
contains the Dirac delta at zero δ0and thus it is meaningful
only in the presence of the ultraviolet regularization.
Introducing new constants ρi;Λ¼δ0, the expansion of
UΛin the normal fields ξreads [31]
UΛ½φ;ξ¼ρ2;ΛUð2Þ½φ;ξþρ4;ΛUð4Þ½φ;ξþoðξ4Þ;ð11Þ
where
Uð2Þ½φ;ξ¼1
6Rαβ Zx
ξαξβ¼KðD−1Þ
6Zx
ξαξα;
Uð4Þ½φ;ξ¼ 1
180 Rσ
αβγ Rγ
μνσ Zx
ξμξνξαξβ
¼K2ðD−1Þ
180 ZxðξαξαÞ2:ð12Þ
In perturbation theory, one usually uses dimensional
regularization, for which the ρi;Λvanish. In contrast, in
the FRG, one works (sometimes implicitly) with a momen-
tum cutoff Λ, implying a nonzero ρi;Λ, as was done in
particular in the FRG study of the NLsMs for example in
[25,26] (see however [32] for an attempt to reproduce the
β-function in the MS scheme with FRG).
For later convenience, we give the expansion of the
action to quadratic order in ξ[23,33,34]
S½ϕ¼1
2tZx
hαβðφÞ∂iφα∂iφβ−
1
tZx
ξαDi∂iφα
þ1
2tZx
ξαð−hαβD2þEαβ Þξβþoðξ2Þ;ð13Þ
with Eαβ ¼−KΠαλβγ ∂iφλ∂iφγ. More terms are given in
Appendix A.
Contrary to the action, the regulator term ΔSk½φ;ξ¼
1
2ξαRαβ;k½φξβis a “two-field”functional as it depends
independently on the fields φand ξ, and cannot be written
as a functional of ϕonly.
Introducing the classical fields ¯
ξα¼hξαi¼δWk
δjα, the
scale-dependent Wetterich’s effective action is defined as
a modified Legendre transform of Wk,
Γk½φ;¯
ξ¼−Wk½φ;jþj:¯
ξ−ΔSk½φ;¯
ξ:ð14Þ
The assumption that RΛ¼∞, see e.g., Appendix C,gives
the initial condition in the form
lim
k→ΛΓk;φ½φ;¯
ξ−
1
2Tr logð−∂2
ΛÞ−1ð−D2
ΛþRkÞ
¼SΛ½φ;¯
ξþUΛ½φ;¯
ξ:ð15Þ
We use in practice a regulator Rkwhich is finite at the
boundary, i.e., RΛ∝Λ2. Since we are only interested in
the behavior of the RG flow near fixed points, we will keep
the original boundary conditions unchanged and instead
consider the effective action at k¼Λas a perturbation of
the semiclassical model. It is believed that the trajectory of
the perturbed system on the phase diagram will remain
within a small distance from the trajectory of the model.
Since R0¼0, the functional Γk¼0½φ;¯
ξcoincides with the
Vilkovisky–Dewitt effective action [28].
The scale-dependent effective action obeys the exact RG
equation [35]
∂kΓk½φ;¯
ξ¼1
2Trð∂kRkðΓð2Þ
kþRkÞ−1Þ;ð16Þ
where the trace is over space and the internal degrees of
freedom. Here and below we use the following notation
NONLINEAR SIGMA MODELS ON CONSTANT CURVATURE …PHYS. REV. D 104, 105003 (2021)
105003-3

ΓðnÞ
α1…αn;k½φ;¯
ξ¼ δnΓk
δ¯
ξα1…δ¯
ξαn:ð17Þ
The exact flow equation is difficult to solve. Since we are
interested in Γk½φ;0and in the long-distance physics, it is
natural to restrict the effective action to a subspace of
functionals with a fixed number of derivatives. Howeverthe
normal fields ¯
ξare dimensionless and this truncation is not
enough to obtain a finite dimensional dynamical system. In
the background field method one usually retains only the
evolution equation for the background action Γk½φ;0
omitting the equations corresponding to n-point vertex
functions. To close the obtained dynamical system, we will
rely on the splitting Ward identities associated with the
splitting into the background and fluctuation fields. Since
these Ward identities are a formal series in the curvature K,
we will use Kas the main expansion parameter.
To leading order in Kand in derivatives, we use the
following ansatz
Γk½φ;¯
ξ¼ 1
2t0;k Zx
hαβðφÞ∂iφα∂iφβ−
1
t1;k Zx
¯
ξαDi∂iφα
þ1
2Zx
¯
ξα−
1
t2;k
hαβD2þυkEαβ þwkEγ
γhαβ¯
ξβ
þVk½φ;¯
ξþUk½φ;¯
ξ;ð18Þ
where to lowest order in K, we can use
Vk½φ;¯
ξ¼ 1
t3;k
Vð3Þ½φ;¯
ξþ 1
t4;k
Vð4Þ½φ;¯
ξþoðξ4Þ;
Uk½φ;¯
ξ¼ρ2;kUð2Þ½φ;¯
ξþρ4;kUð4Þ½φ;¯
ξþoðξ4Þ:ð19Þ
For VðnÞ½φ;¯
ξsee Appendix A, the functional UðnÞ½φ;¯
ξis
given in Eq. (12). All other terms generated by the
renormalization flow contribute to the second order in
K. This is why we do not include them in the truncation.
Comparing to the covariant Taylor expansion of the
action, Eq. (13), we find the initial conditions
ti;Λ¼t;
ρi;Λ¼δ0;
υΛ¼t−1;
wΛ¼0:ð20Þ
Note that while all ti;k are equal at the beginning of the flow,
this is not so for all k<Λ. However, they are not
independent, but related by the splitting Ward identities.
Finally, although wΛ¼0, the corresponding operator is
allowed by the symmetries, and will be generated during
the flow, and is of order Kin our ansatz.
IV. WARD IDENTITIES
A. Splitting symmetry on M
In flat models the split of the field ϕinto a classical
background φand the corresponding quantum fluctuation ξ
is linear, i.e., ϕ¼φþξ. This yields a very simple splitting
symmetry transformation: φ↦
˜
φ¼φþc,ξ↦
˜
ξ¼ξ−c
where cis a shift. In our case the split is nonlinear. To
proceed with the background field method we need the
transformation rule of the tangent vector ξunder an
infinitesimal small shift cof the expansion point,
φ_
λ↦
˜
φ_
λ¼ecαDαφ_
λ¼φ_
λþc_
λþoðcÞ:ð21Þ
Here and below the covariant derivative acting on the
dotted index is equivalent to the usual partial derivative.
Consider the covariant Taylor expansion of the coordinate
function
ϕ_
λ¼φ_
λþξ_
λ−X
n≥2
1
n!ξα1…ξαnM_
λ
α1…αn;ð22Þ
M_
λ
α1…αn¼−
1
n!X
π∈Sn
Dπ1…Dπ2φ_
λ;ð23Þ
where Snis the symmetry group on the indices α1…αn.Itis
convenient to define the covariant variation of the tangent
vector
δξα¼Dcξα¼cγDγξα:ð24Þ
Performing the shift of the expansion point in the Taylor
expansion (22) we obtain
0¼ε_
λ−X
n≥1
1
n!ελξα1…ξαnM_
λ
λα1…αn
−X
n≥2
1
n!cωξα1…ξαnðDωM_
λ
α1…αn−M_
λ
ωα1…αnÞ:ð25Þ
where ε_
λ¼c_
λþδξ_
λ. We are looking for the variation δξ in
the form
−δξ_
λ¼c_
λþX
∞
m¼2
L_
λ
ωβ1…βm
m!cωξβ1…ξβm:ð26Þ
Substitution in Eq. (25) δξ with the series yields a recurrent
relation
L_
λ
ωβ1…βn¼X
n−2
m¼1
n!
ðn−mÞ!m!M_
λ
β1…βmσLσ
ωβmþ1…βn
þM_
λ
ωβ1…βn−DωM_
λ
β1…βn:ð27Þ
ALEXANDER N. EFREMOV and ADAM RANÇON PHYS. REV. D 104, 105003 (2021)
105003-4

We have performed calculation for an arbitrary symmetric
manifold, DσRαλβγ ¼0. Denote by ¼
π∈Snthe equality under
the permutations of the symmetry group Sn. First we turn
our attention to the terms on the right-hand side which are
independent of the unknown coefficients L_
λ
ωβ1…βn,
M_
λ
ωπ1π2−DωM_
λ
π1π2¼
π∈S22
3R_
λ
π2π1ω;ð28Þ
M_
λ
ωπ1π2π3−DωM_
λ
π1π2π3¼
π∈S3−2Rσ
π1π2ωM_
λ
σπ3;ð29Þ
M_
λ
ωπ1π2π3π4−DωM_
λ
π1π2π3π4¼
π∈S44Rσ
π1π2ωM_
λ
σπ3π4
−
8
15 Rσ
π1π2ωR_
λ
π3π4σ:ð30Þ
Then using the recurrent relation we sequentially find the
first three coefficients
−δξα¼cαþ1
3Rα
μνωξμξνcω
−
1
45 Rα
μνγRγ
ρσωξμξνξρξσcωþoðK2Þ:ð31Þ
This geometrical transformation rule implies that given an
expansion point φand the action functional SðϕÞof the
nonlinear σ-model on a symmetric manifold the following
identity holds for the covariant Taylor expansion of
S½φ;ξ¼Sðϕðφ;ξÞÞ
cωDφω−hα
ωþ1
3Rα
μνωξμξν
−
1
45Rα
μνγRγ
ρσωξμξνξρξσþoðK2Þδ
δξαS½φ;ξ¼0:ð32Þ
The directional derivative cωDφωS½φ;ξcorresponds to the
parallel transport of ξalong cand has to be calculated with
the condition Dcξ¼0. Since the transformation (31) is
independent of the action there is a less laborious way to
obtain it by considering the splitting symmetry of the
expansion in question (see Appendix B).
B. Splitting Ward identities
With these results, we can now derive the corres-
ponding splitting Ward identities for our model, see also
[23,24,36].Forφ∈Mand c; j ∈TφMwe denote by
˜
φ¼φþcand by ˜
j∈T˜φMthe parallel transport of jfrom
φto ˜
φ,
eWk½˜
φ;
˜
j¼ZD˜
φð
˜
ξÞe−Ak½˜
φ;
˜
ξþ˜
j:
˜
ξ;
Ak½˜
φ;
˜
ξ¼S½˜
φ;
˜
ξ−ΔSk½˜
φ;
˜
ξ:ð33Þ
Since Dcðdet hðφÞÞ ¼ 0the measure is invariant under the
parallel transport,
D˜
φð˜
ξÞ¼DφðξÞ:ð34Þ
For an infinitesimally small cthis implies
eWk½˜φ;˜
j¼ZDφðξÞe−Ak½φ;ξ−δAk½φ;ξþjξ;
δA½φ;ξ¼DcAk½φ;ξ:ð35Þ
Then we change the variables ξ¼ξ0þδξ. From Eq. (31)
we obtain the Jacobian and the variation of Uφunder this
change
Y
α
dξα¼Y
α
d
˜
ξαe−δ0ðD−1ÞRxξ:cðK
3þK2
45 ξ2Þ;
UΛ½φ;ξ¼UΛ½φ;
˜
ξ−ρ2;Λ
KðD−1Þ
3Zx
ξ:c
−ρ4;Λ
K2ðD−1Þ
45 Zx
ξ2ξ:c þoðK2Þ:ð36Þ
Recall that ρΛ
2¼ρΛ
4¼δ0. Consequently the measure is
invariant also under the variation δξ
DφðξÞ¼Dφðξ0Þ:ð37Þ
If we did not include the functional UΛinto the definition
of the measure Dφwe would obtain an anomaly in the Ward
identities. Then the splitting identity (32) yields
eWk½˜
φ;
˜
j¼ZDφðξÞe−Ak½φ;ξþj:ξ−1
2hαβξαDcRkξβþðj−ξRkÞ:δξ:ð38Þ
To proceed further we introduce an auxiliary source γ,
S½φ;ξ;γ¼S½φ;ξþγ:δξ:ð39Þ
Thus for the directional derivative we obtain
DcWk½φ;j¼TrδWk½φ;j
δjRk−jWγ;k½φ;jþRk
δWγ;k½φ;j
δj
−
1
2TrδWk½φ;j
δjDcRk
δWk½φ;j
δj−DcRk
δ2Wk½φ;j
δjδj;ð40Þ
NONLINEAR SIGMA MODELS ON CONSTANT CURVATURE …PHYS. REV. D 104, 105003 (2021)
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Wγ;k½φ;j¼δWk½φ;j;γ
δγ jγ¼0:ð41Þ
Under the parallel transport Dcj¼0and Dc¯
ξ¼0.
Consequently for the directional derivative of the (true)
Legendre transform of Wk½φ;j,Fk½φ;¯
ξ¼Γk½φ;¯
ξþ
ΔSk½φ;¯
ξ,wehave
DcFk½φ;¯
ξ¼−DcWk½φ;j:ð42Þ
Eventually we get the Ward identity
DcþFγα;k½φ;¯
ξδ
δ¯
ξαFk½φ;¯
ξ−
1
2hαβ ¯
ξαRk¯
ξβ¼Nφk;
ð43Þ
Nφk¼TrδFγ;k½φ;¯
ξ
δ¯
ξRkþ1
2DcRkðFð2Þ
k½φ;¯
ξÞ−1Þ:
ð44Þ
For the Wetterich effective action the splitting Ward
identity has the form [24]
DcΓk½φ;¯
ξþΓγ;k½φ;¯
ξδΓk½φ;¯
ξ
δ¯
ξα¼Nφk;
Nφk¼TrδΓγ;k½φ;¯
ξ
δ¯
ξRkþ1
2DcRkðΓð2Þ
k½φ;¯
ξþRkÞ−1:
ð45Þ
C. Constraints from the splitting Ward identities to
linear order in K
To linear order in K, we choose the ansatz for the
insertion as follows
Γγα;k½φ;¯
ξ¼−ζ0;kcα−ζ2;k
3Rα
μνω ¯
ξμ¯
ξνcωþoðKÞ:ð46Þ
This form generalizes Eq. (31) by introducing two coupling
constants ζ0;k and ζ2;k. The ansatz is consistent with the
flow equation for Γγα;k to leading order in K.
The combination of Eq. (18), Eq. (46) and Eq. (45) gives
ζ0;k
tn;k
tnþ1;k ¼1þOðKÞ;ζ0;kζ2;k ¼1þOðKÞ;
t2;kυk¼1þOðKÞ;t
2;kwk¼OðKÞ:ð47Þ
Analysing the Ward identity Eq. (45), one finds that ρ2;k
is not an independent variable, but obeys
−ρ2;kζ0;k
δUð2Þ
δ¯
ξαcα¼TrRk
δΓγ;k
δ¯
ξðΓð2Þ
kþRkÞ−1þoðKÞ:
ð48Þ
On the right-hand side one only keeps the local term ¯
ξαcαto
leading order in K.
V. FRG FLOW EQUATIONS
A. Method and notations
In this section we compute the flow equations of the
various coupling constants to linear order in Kusing our
ansatz Eq. (18). For conciseness we use the following
notations
¯
ΓðnÞ
α1…αn;k ¼δnΓk½φ;¯
ξ
δ¯
ξα1…δ¯
ξαn


¯
ξ¼0ð49Þ
¯
Gk¼ð¯
Γð2Þ
kþRkÞ−1;ð50Þ
where
¯
Γð2Þ
k¼h1
t2;k ð−D2þm2
kÞþΣ;ð51Þ
with Σαβ ¼υkEαβ þwkEγ
γhαβ and m2
k¼KðD−1Þ
3ρ2;kt2;k .We
will see below that t2;k is of order K−1at the fixed point.
Consequently t2;kΣis of order K, while m2
kwill be of order
one. We choose the regulator function of the form
Rαβ;k½φ¼ 1
t2;k
hαβRkð−D2Þ;ð52Þ
with RkðωÞ¼ðk2−ωÞθðk2−ωÞ.
Then, for a sufficiently small K, we assume the existence
of the Neumann series
¯
Gk¼t2;kGh−1X
∞
n¼0ð−t2;kΣGh−1Þnð53Þ
where
G−1¼−D2þRkþm2
k:ð54Þ
Note that we do not expand ¯
Gin powers of m2
k.
To compute the trace, we use the heat kernel method
[37,38], that we outline briefly. The spectral decomposition
of a integral kernel freads
fαβðx; yÞ¼ X
ω∈σð−ΔÞ
ˆ
fðωÞψαωðxÞðψβω ðyÞÞ
¼Z∞
0
dsðL−1ˆ
fÞðsÞKαβ
xy ðsÞ;ð55Þ
ALEXANDER N. EFREMOV and ADAM RANÇON PHYS. REV. D 104, 105003 (2021)
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where −D2ψω¼ωψω. The heat kernel Ksatisfies the heat
equation ð∂s−D2ÞKxyðsÞ¼0.Fork2>k∂φk2
∞the
inverse Laplace transform ðL−1ˆ
fÞðsÞis small for all large
values of time, i.e., such that sk∂φk2
∞>1. Consequently
one can substitute the heat kernel with an asymptotic
expansion at small time
KxyðsÞ¼ð4πsÞ−d
2e−ðx−yÞ2
4sX
∞
m¼1
ð−1Þm
m!bmðx; yÞsm:ð56Þ
At the coincidence limit y→xthe leading heat kernel
coefficients are [39]
b0¼1;D
xib0¼0;ð57Þ
b1¼0;D
xib1¼DkΩki
6;
Ωαβki ¼−KΠαβλγ ∂kφλ∂iφγ:ð58Þ
For a spectral density ˆ
fðωÞ, we introduce
Qd
2−m½ˆ
f¼Z∞
0
dsðL−1ˆ
fÞðsÞsm
ð4πsÞd
2
;ð59Þ
that for d>2mis easier to calculate in the spectral
representation
Qd
2−m½
ˆ
f¼ 1
ð4πÞd
2Γðd
2−mÞZ∞
0
dω
ˆ
fðωÞωd
2−m−1:ð60Þ
In particular, for
H¼G∂kRk−
∂kt2;k
t2;k
RkG;ð61Þ
one finds
Qd
2−m½
ˆ
H¼ 2kdþ1−2m
ð4πÞd
2Γðd
2þ1−mÞðk2þm2
kÞ2
×1−k∂kt2;k
t2;kðd−2mþ2Þ:ð62Þ
B. Flow of the effective action
The flow equation of ¯
Γkis
∂k¯
Γk¼1
2Trð∂kRk¯
GkÞ:ð63Þ
To leading order in K, we obtain
∂kt0;k
t2
0;k ¼−t2;kðυkþDwkÞKðD−1ÞQd
2½
ˆ
H:ð64Þ
The flow of the one-point function ¯
Γð1Þ
kreads
∂k¯
Γð1Þ
α;k ¼−
1
2Trð∂kRk¯
Gk¯
Γð3Þ
α;k ¯
GkÞ:ð65Þ
Using the ansatz in Eqs. (18) and (19) we have
¯
Γð3Þ
μβα ¼X
π∈S3Zx
−2K
3t3;k
ΠπαλπβγðφxÞ∂iφλDi
γ
πμ;ð66Þ
¯
Γð4Þ
νμβα ¼X
π∈S4Zx
−K
3!t4;k ðΠπαλπβγðφxÞDi
γ
πμDi
λ
πν
þhπαπβðφxÞEπμπνðφxÞÞ
þK2ðD−1Þρ4;k
180 hπαπβðφxÞhπμπνðφxÞ:ð67Þ
Here hμzν¯zðφxÞ¼hμνðφxÞδxzδx¯zand Di
γx
μz¼∂xiδxzhγ
μþ
Γγ
μσ∂iφσ
xδxz, i.e., the covariant derivative with respect to
the upper index.
To leading order in Kthe equation has the form
∂k¯
Γð1Þ
α;k ¼−t2;k
2TrðH¯
Γð3Þ
α;kÞþoðKÞ:ð68Þ
This gives
∂kt1;k
t2
1;k ¼−2t2;k
t3;k
KðD−1Þ
3Qd
2½
ˆ
H:ð69Þ
Finally, the flow of the two-point function reads
∂k¯
Γð2Þ
αβ;k ¼1
2X
π∈S2
Trð∂kRk¯
Gk¯
Γð3Þ
πβ;k ¯
Gk¯
Γð3Þ
πα;k ¯
GkÞ
−
1
2Trð∂kRk¯
Gk¯
Γð4Þ
αβ;k
¯
GkÞ;ð70Þ
where S2is the symmetry group on two indices α,β.To
leading order in K, the equation is
∂k¯
Γð2Þ
ν¯
ν;k ¼−t2;k
2TrðH¯
Γð4Þ
ν¯
ν;kÞþoðKÞ:ð71Þ
It is convenient to write this flow using an auxiliary
generating functional
Fðξ;¯
ξÞ¼t2;k
2TrðH¯
Γð4Þ
νμ;kÞξμ¯
ξν;ð72Þ
which reads after expansion to leading order in Kand to
second order in derivatives
F¼Zx
ξαð−ðD−1Þr0D2þl1Þ¯
ξα
þðr0hαβEγ
γþðDþ4Þr0EαβÞξα¯
ξβþoðKÞ:ð73Þ
NONLINEAR SIGMA MODELS ON CONSTANT CURVATURE …PHYS. REV. D 104, 105003 (2021)
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The auxiliary constants are as follows
ri¼−Kt2;k
3t4;k
Qd
2þi½
ˆ
H;
l1¼d
2ðD−1Þr1þK2ðDþ2ÞðD−1Þρ4;kt2;k
45 Qd
2½
ˆ
H:ð74Þ
Equation (71) yields the evolution equation for the constant
t2;k
∂kt2;k
t2
2;k ¼−t2;k
t4;k
KðD−1Þ
3Qd
2½
ˆ
H:ð75Þ
From Eq. (73), one can also obtain the flow of υk,wkand
ρ2;k, although these will not be needed as they are fixed by
the splitting Ward identity.
For flat models one makes the usual substitution
ti;k ¼tkwhere tkis a unique renormalized coupling
constant. This makes possible to retain only the evolution
equations for 1PI vertex functions ¯
ΓðnÞwith n<2.For
nonlinear σ-models the substitution ti;k ¼tkwould give
incorrect flow equations [e.g., comparing Eqs. (69)
and (75)].
C. Flow of Γγ;k
The flow equation of Γγ;k reads
∂kΓγ;k ¼−
1
2Trð∂kRk¯
Gk¯
Γð2Þ
γ;k ¯
GkÞ;ð76Þ
that at leading order in Ktakes the form
∂kΓγα;k ¼−t2;k
2TrðH¯
Γð2Þ
γαÞ
¼−KðD−1Þ
3t2;kζ2;k Qd
2½
ˆ
HcαþOðKÞ:ð77Þ
Consequently we have
∂kζ0;k ¼KðD−1Þ
3ζ2;kt2;k Qd
2½
ˆ
H:ð78Þ
Finally, for our choice of regulator function Eq. (52), the
Ward identity Eq. (48) written in terms of m2
kreads
m2
k¼ζ2;kt2;k
ζ0;k
KðD−1Þ
3Qd
2½ˆ
R
ˆ
G;ð79Þ
with
Qd
2½ˆ
R
ˆ
G¼ kdþ2
ð4πÞd
2Γðd
2þ2Þðk2þm2
kÞ:ð80Þ
VI. β-FUNCTIONS AND FIXED POINT ANALYSIS
Using Ward identities Eq. (47) the flow of t0;k Eq. (64)
can be written to leading order in Kas
∂kt0;k
t2
0;k ¼−KðD−1ÞQd
2½
ˆ
H;ð81Þ
while that of t1;k,t2;k and ζ0;k takes the simple form
∂kt1;k
t1;k ¼2
3
∂kt0;k
t0;k
;∂kt2;k
t2;k ¼1
3
∂kt0;k
t0;k
;∂kζ0;k
ζ0;k ¼−
1
3
∂kt0;k
t0;k
:
ð82Þ
Furthermore, recalling that m2
k¼KðD−1Þ
3ρ2;kt2;k and using
Eqs. (47) and (79), one finds m2
kas a function of t0;k,
m2
k¼2sd
3ðdþ2Þ
kdþ2t0;k
k2þm2
k
;ð83Þ
with sd¼KðD−1Þ
ð4πÞd
2Γðd
2þ1Þ. Since Qd
2½
ˆ
Hdepends on ∂kt2;k
t2;k and m2
k,
this allows to write the flow equation of t0;k in terms of t0;k
only,
k∂kt0;k ¼−
2sdkdþ2t2
0;k
ðk2þm2
kÞ21−k∂kt0;k
3ðdþ2Þt0;k:ð84Þ
To analyze the flow equations, it is convenient to
introduce dimensionless variables ˜
t0;k ¼kd−2t0;k and
˜
m2
k¼k−2m2
k. For the latter, by keeping only the positive
root when solving Eq. (83), we obtain
˜
m2
k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ8sd
˜
t0;k
3ðdþ2Þ
q−1
2:ð85Þ
Defining the β-function, β0¼k∂k
˜
t0;k, our final result is
β0¼ðd−2Þ˜
t0;k −
4sd
˜
t2
0;k
1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ8sd
˜
t0;k
3ðdþ2Þ
q:ð86Þ
A fixed point is a scale independent solution, i.e., β0¼0.
There are two fixed points associated to this β-function, the
trivial fixed point ˜
t
0;k ¼0, which is attractive in the infrared
and corresponds to the low-temperature phase, and a
nontrivial fixed point
˜
t
0¼2ðdþ1Þðd−2Þ
3ðdþ2Þsd
;ð87Þ
for d>2and if sdis positive. For K>0, the model is the
usual OðDÞNLsM, while for K<0, the fixed point is
physical in the formal limit D<1, and in particular for
ALEXANDER N. EFREMOV and ADAM RANÇON PHYS. REV. D 104, 105003 (2021)
105003-8

D¼0. Expanding the β-function at ˜
t
0we obtain the
linearized equation
k∂k
˜
t0;k ¼−ν−1ð˜
t0;k −
˜
t
0Þþoð˜
t0;k −
˜
t
0Þ;
ν−1¼ðd−2Þ1−d−2
5dþ2;ð88Þ
with ν−1the critical exponent governing the divergence of
the correlation length close to criticality. In particular for
d¼3we have ν−1¼16=17, for all Dand Ksuch that
s3>0. Clearly the fixed point is repulsive. For d¼2þϵ,
ϵ→0, we recover the standard one-loop result ν−1¼ϵ
[7,9]. At the fixed point the Ward identities (85),(82) give
˜
m2
k¼d−2
3ðdþ2Þ;ρ2¼3ΛdðkΛ−1Þ2þd−2
3
2ðdþ1Þð4πÞd
2Γðd
2þ1Þ;ð89Þ
ζ−1
2¼ζ0¼ðkΛ−1Þd−2
3;t
2;k ¼˜
t
0Λ2−dðkΛ−1Þ2−d
3:ð90Þ
VII. DISCUSSION AND CONCLUSION
We have computed the FRG flow equation of NLSM
with constant curvature using the background field method,
to lowest order in the derivative expansion. In order to
implement consistently the splitting Ward identities
induced by the background field reparametrization invari-
ance, we have also performed a formal expansion in the
curvature, keeping terms to lowest order in K. The beta
functions we have obtained are different from those of the
previous studies using the same method [25,26], corre-
sponding to different critical exponents (if one stays at the
same order of the derivative expansion). Let us comment on
the main difference between these works and ours.
In [25], we note that the “mass”term induced by the
measure is neglected, and that all the ti;k are assumed to be
identical, i.e., ti;k ¼t0;k, in our notations. This is obviously
not consistent with the splitting Ward identities derived
above. In [26],a“wave function renormalization”is
introduced for the fluctuating field, corresponding here
to t0;k=t2;k , as well as a mass term. No connection with the
splitting Ward identities is made, and the mass has an
independent flow, whereas we have shown that it is fixed by
the Ward identities. Therefore, their flow equations at the
lowest order in the derivative expansion are different from
ours. It has been noted in [26] that if one includes all
coupling constants at the next order of the derivative
expansion, the nontrivial fixed point disappears. One could
hope that using an ansatz that obeys the Ward identities to
second order in Kwill cure this problem.
One aspect which is identical in our work and [25] is that a
nontrivial fixed point is found in all dimensions d>2,
which, if confirmed, implies that there is no upper critical
dimension. While this is expected for noncompact NLSM, as
discussed in the Introduction, the fact that we find the same
result for the OðNÞNLSM questions the validity of the
approach. Indeed, on the lattice, the OðNÞNLSM corre-
sponds to a OðNÞspin model, for which there is no doubt
that the upper critical dimension is dc¼4.If,andhow,the
present method is able to recover this result is still an open
question. (We note in passing that a lattice FRG approach of
the OðNÞNLSM, not using the background field method but
taking the nonlinear constraint into account exactly, does not
suffer from this problem. Indeed, the flow equations are
formally the same than that of the corresponding linear
sigma model, and only the initial condition is different,
which does not affect the fixed point properties [40].) It has
been argued that the 2þϵexpansion of the OðNÞNLSM
does not describe the Wilson-Fisher fixed point at ϵ¼1,as
it cannot capture the topological excitations that drive the
transition, e.g., the hedgehogs excitations for N¼3[41].It
could well be that the background field method, even
supplemented with a functional RG approach, is incapable
to capture the correct physics far from d¼2. We hope that
the expansion to the next order in derivatives and curvature
will help to answer these questions.
ACKNOWLEDGMENTS
We thank R. Percacci and A. Codello for correspondence
about their work and discussions. AR thanks D. Mouhanna
for insightful discussions on the NLSM, as well I. Balog for
very useful discussions on the SOð1;NÞmodel. AE is
greatful to B. Arras for giving the opportunity to do this
work. For all tensor calculus we have used Cadabra [42].
This is an extremely lightweight, latex friendly and com-
pletely free software tool. Needless to say how easy tensor
algebra has nowadays become. This work was supported by
Agence Nationale de la Recherche through Research Grants
No. QRITiC I-SITE ULNE/ ANR-16-IDEX-0004 ULNE,
the Labex Centre Europ´een pour les Math´ematiques, la
Physique et leurs Interactions (CEMPI) Grant No. ANR-11-
LABX-0007-01, the Programme Investissements d’Avenir
ANR-11-IDEX-0002-02, reference No. ANR-10-LABX-
0037-NEXT and the Ministry of Higher Education and
Research, Hauts-de-France Council and European Regional
Development Fund (ERDF) through the Contrat de Projets
État-Region (Contrat Plan Etat-R´egion (CPER) Photonics
for Society, P4S).
APPENDIX A: COVARIANT EXPANSION OF
THE ACTION
There is a variety of sources where the reader can find the
covariant expansion of the NLSM, see e.g., [23,33,34],
ZRdð∂ϕÞ2
2¼ZRdð∂φÞ2
2−ξγDi∂iφγþξαðC−1
αβ þEαβÞξβ
2
þX
5
n¼3
VðnÞ½φ;ξþoðξ5Þ;ðA1Þ
NONLINEAR SIGMA MODELS ON CONSTANT CURVATURE …PHYS. REV. D 104, 105003 (2021)
105003-9

C−1
αβ ¼hαβð−D2Þ;
Eαβ ¼−Rλαγβ ∂iφλ∂iφγ¼−KΠαλβγ ∂iφλ∂iφγ;ðA2Þ
Vð3Þ½φ;ξ¼−
2
3Rσαγβξαξβ∂iφσDiξγ
¼−2K
3ZRd
ξαξβΠαλβγ ∂iφλDiξγ;ðA3Þ
Vð4Þ½φ;ξ¼−
1
3! RγασβξαξβðDiξγDiξσ
−Rγ
α0σ0β0ξα0ξβ0∂iφσ∂iφσ0Þ
¼−K
3! ZRd
ξαξβðΠαλβγ DiξγDiξλþξ2EαβÞ;ðA4Þ
Vð5Þ½φ;ξ¼ 2
15 ZRd
RμαβλRλ
ρσνξαξβξρξσDiξν∂iφμ
¼2K2
15 ZRd
ξ2Παλβγ ξαξβDiξλ∂iφγ;ðA5Þ
where we have factored out the factor t−1in the action.
APPENDIX B: SPLITTING SYMMETRY
OF THE ACTION
We would like to give a simple method to obtain the
symmetry transformation in Eq. (31). Indeed the covariant
Taylor expansion of the action Eq. (A1) is independent of
the point φ. To proceed we have to retain in the expansion
all terms quadratic in the Riemann tensor. The directional
derivative vanishes at the first and third orders in ξiff
L_
α
ωβ ¼L_
α
ωβ1β2β3¼0. The definition of L_
α
ωβ1…βmis given in
(26). Then for man even integer we put
L_
αωβ1β2…βm−1βm¼X
π∈Smðam1h_
αωhπβ1πβ2
…hπβm−1πβm
þam2h_
απβ1hωπβ2
…hβm−1πβmÞ:ðB1Þ
The derivative vanishes at the second and fourth orders
in ξiff
a21 ¼−K
3;a
22 ¼K
3;a
41 ¼−K2
45 ;a
42 ¼K2
45 :
ðB2Þ
Once again this yields the symmetry transformation given
in (31).
APPENDIX C: WILSON–POLCHINSKI
EQUATION
Most equations of this appendix are complementary to
those of the main text. However we believe they are likely
useful for the reader. Let CkΛbe a regularized propagator
such that
CΛΛ ¼0;lim
k→0
Λ→∞
C−1
kΛ¼−D2:ðC1Þ
For ∀j∈DðRd;TφMÞone can write the partition func-
tional in the form
ZkΛ½φ;j¼e−1
2Rð∂φÞ2−Γ1;kΛ½φþ ˜
WkΛ½φ;jþDi∂iφ:ðC2Þ
Here Γ1;kΛis a normalization coefficient,
Γ1;kΛ½φ¼1
2Tr logðð−∂2
ΛÞ−1h−1C−1
kΛÞ:ðC3Þ
The generating functional of connected Schwinger func-
tions ˜
WkΛ½φ;jis
e
˜
WkΛ½φ;j¼ZdμkΛðξÞe−LΛ½φ;ξþξαjα;ðC4Þ
where dμkΛis a Gaussian measure on a finite dimensional
Borel cylinder set [43],
dμkΛðξÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
det C−1
kΛ
qY
α
dξα
ffiffiffiffiffiffi
2π
pe−1
2ξαC−1
kΛαβξβ:ðC5Þ
At 1-loop the bare reduced action is (see Appendix A)
LΛ½φ;ξ¼1
t0;Λ
−1hαβ∂φα∂φβþ1−
1
t1;ΛξαDi∂iφαþ1
21
t2;Λ
−1ξαC−1
0Λαβξβ
þυΛ
2Eαβξαξβþ1
t3;Λ
Vð3Þ½φ;ξþ 1
t4;Λ
Vð4Þ½φ;ξþρ2;ΛUð2Þ½φ;ξþρ4;ΛUð4Þ½φ;ξþoðξ4Þ:ðC6Þ
The usual way to give a meaningful interpretation of Γ1;kΛ
is to consider a stationary point of the free energy,
δW0Λ½φ;j
δj¼0:ðC7Þ
Using convexity of the effective action one shows that at
this point the normalization coefficient (C3) is the effective
action at 1-loop [44].
It is convenient to define the reduced effective action
˜
ΓkΛ½φ;¯
ξ
ALEXANDER N. EFREMOV and ADAM RANÇON PHYS. REV. D 104, 105003 (2021)
105003-10

Lð˜
WkΛ½φ;·Þð¯
ξÞ¼1
2
¯
ξαC−1
kΛαβ
¯
ξβþ˜
ΓkΛ½φ;¯
ξ;ðC8Þ
where Lð·Þis the Legendre transform. Then the Wilson–
Polchinski equation [19,45,46] is
∂k
˜
ΓkΛ½φ;¯
ξ¼1
2Trð∂kCkΛ
˜
Γð2Þ
kΛ½φ;¯
ξð1þCkΛ
˜
Γð2Þ
kΛ½φ;¯
ξÞ−1Þ;
ðC9Þ
˜
ΓΛΛ½φ;¯
ξ¼LΛ½φ;¯
ξ:ðC10Þ
Substituting C−1
kΛ¼C−1
0ΛþRinto the Wetterich effective
action (14) we obtain
ΓkΛ½φ;¯
ξ¼1
2hαβ∂iφα∂iφβ−¯
ξαDi∂iφαþ1
2
¯
ξαC−1
0Λαβ
¯
ξβ
þΓ1;kΛ½φ;¯
ξþ˜
ΓkΛ½φ;¯
ξ:ðC11Þ
It follows that this action satisfies the following boundary
condition
lim
k→ΛΓkΛ½φ;¯
ξ−
1
2Tr logð−∂2
ΛÞ−1h−1C−1
kΛ
¼SΛ½φ;¯
ξþUΛ½φ;¯
ξ:ðC12Þ
On the right-hand side we used ti;Λ¼t, see Eq. (20).
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