Pauli-Term-Induced Fixed Points in d-dimensional QED Holger Gies1 2 3Kevin K. K. Tam4 1and Jobst Ziebell1 1Theoretisch-Physikalisches Institut Abbe Center of Photonics

2025-05-02 0 0 1.13MB 11 页 10玖币
侵权投诉
Pauli-Term-Induced Fixed Points in d-dimensional QED
Holger Gies,1, 2, 3 Kevin K. K. Tam,4, 1 and Jobst Ziebell1
1Theoretisch-Physikalisches Institut, Abbe Center of Photonics,
Friedrich-Schiller-Universit¨
at Jena, Max-Wien-Platz 1, 07743 Jena, Germany
2Helmholtz-Institut Jena, Fr¨
obelstieg 3, 07743 Jena, Germany
3GSI Helmholtzzentrum f¨
ur Schwerionenforschung, Planckstr. 1, 64291 Darmstadt, Germany
4Max Planck School of Photonics, Hans-Kn¨
oll-Straße 1, 07745 Jena, Germany
(Dated: October 24, 2022)
We explore the fixed-point structure of QED-like theories upon the inclusion of a Pauli spin-
field coupling. We concentrate on the fate of UV-stable fixed points recently discovered in d= 4
spacetime dimensions upon generalizations to lower as well as higher dimensions for an arbitrary
number of fermion flavors Nf. As an overall trend, we observe that going away from d= 4 dimensions
and increasing the flavor number tends to destabilize the non-Gaussian fixed points discovered in
four spacetime dimensions. A notable exception is a non-Gaussian fixed point at finite Pauli spin-
field coupling but vanishing gauge coupling, which also remains stable down to d= 3 dimensions
and for small flavor numbers. This includes also the range of degrees of freedom used in effective
theories of layered condensed-matter systems. As an application, we construct renormalization group
trajectories that emanate from the non-Gaussian fixed point and approach a long-range regime in
the conventional QED3universality class that is governed by the interacting (quasi) fixed point in
the gauge coupling.
I. INTRODUCTION
The Pauli term, denoting the coupling between the
electron spin and the electromagnetic field, plays an in-
teresting role in quantum electrodynamics (QED): it un-
dergoes finite renormalization [1] while receiving contri-
butions from all scales [2, 3]; in the effective action, it
parameterizes the famous anomalous magnetic moment
of the electron [4, 5] which has been measured and com-
puted to an extraordinary precision [6–8]; and from a
Wilsonian viewpoint, it corresponds to a perturbatively
nonrenormalizable dimension-5 operator and thus has the
least possible finite distance to the set of renormalizable
operators in QED theory space.
Specifically the last property makes the Pauli term a
candidate for a relevant interaction in a coupling regime
where nonperturbative interactions set in. In fact, a re-
cent study [9] provides evidence that the observed long-
range properties of (pure) QED can be extended to high-
energy scales along renormalization group (RG) trajec-
tories that exhibit a sizable Pauli-term contribution. Re-
markably, a systematic next-to-leading order expansion
of the effective action features nonperturbative ultravi-
olet (UV) stable fixed points that give rise to a UV-
complete version of QED within an asymptotic-safety
scenario [10, 11]. The possibility that a sizable Pauli
term could be sufficient for QED to evade the infamous
Landau-pole problem [12] had already been suggested in
[13] on the basis of an effective-field-theory study.
It is important to note that the Pauli-term-induced
UV-completion of QED also evades the conclusion of
QED triviality from previous analyses of QED in the
strong-coupling regime on the lattice [14] as well as us-
ing the functional RG [15]. The reason is that the
Pauli term goes beyond the chirally invariant subspace of
massless QED; moreover, one of the non-Gaussian fixed
points occurs at vanishing gauge coupling. QED trivial-
ity observed in [14, 15] relies on the observation of chi-
ral symmetry breaking induced by a strong gauge cou-
pling which prohibits the connection of a strongly cou-
pled high-energy regime to the observed phase with a
light electron. As a caveat, we should mention that the
potential of the Pauli term to generate a heavy electron
mass has not yet been explored.
The findings of [9] serve as a strong inspiration to study
the Pauli term and the fate of the corresponding non-
perturbative fixed points also in dimensions larger and
smaller than d= 4. Specifically d= 3 is a relevant case,
since QED3serves as an effective theory for the long-
range excitations of various layered condensed-matter
systems [16–24] including graphene and cuprate super-
conductors. In this context, also the dependence of the
renormalization structure of the theory on the number of
fermion flavors Nfis of substantial interest: the question
as to whether the long-range properties of QED3depend
on the flavor number and whether quantum phase tran-
sitions as a function of Nfexist has a long tradition in
the literature [25–61].
At the same time, it is interesting to explore the fea-
tures of the system towards higher dimensions: whereas
perturbative renormalization clearly favors four dimen-
sional spacetime as a setting where interacting field the-
ories of scalar and fermionic matter with gauge interac-
tions can exist over a wide range of scales, the possibility
of nonperturbative fixed points appears to loosen this
connection between the existence of interacting quantum
field theories and the observed dimensionality of space-
time. Still, various studies find that evidence for nonper-
turbative UV completions appears to become less robust
beyond d= 4 dimensions [62–72].
We start by introducing the model in general dimen-
sions in Sect. II. Section III briefly reviews the results
arXiv:2210.11927v1 [hep-th] 21 Oct 2022
2
derived in [9] as a reference point for our investigations of
lower (Sect. IV) and higher (Sect. V) dimensional space-
times. Our conclusions are given in Sect. VI.
II. QEDdWITH A PAULI TERM
We investigate QEDd, i.e. QED in dspacetime di-
mensions, with NfDirac flavors ψainteracting with an
electromagnetic field Aµ. In addition to the conventional
gauge interaction, we include a Pauli spin-field coupling
already on the level of the bare action. Throughout this
work, we use Euclidean conventions in which the bare
action satisfying Osterwalder-Schrader positivity reads
S=Zx
1
4Fµν Fµν +¯
ψai/
D[A]ψai¯m¯
ψaψa+i¯κ¯
ψaσµν Fµν ψa,
(1)
Here the covariant derivative is defined as Dµ[A] =
µi¯eAµ. All mass and coupling parameters are un-
derstood to denote bare quantities. While the kinetic
term is chirally invariant, the mass term ¯mand the
Pauli term ¯κbreak chiral symmetry explicitly.
In this work, we study the renormalization flow of the
couplings and the mass, also allowing for wave function
renormalizations Zψ,A that renormalize the fields of a
Wilsonian-type action of the form of Eq. (1) multiplica-
tively, ψpZψψ,AZAA. This allows us to de-
fine the corresponding renormalized couplings. For our
present goal of searching for fixed points, where the the-
ory becomes (quantum) scale invariant, it is useful to
introduce dimensionless renormalized quantities:
e=kd
22¯e
ZψZA
, m =¯m
Zψk, κ =kd
21¯κ
ZψZA
,(2)
where kdenotes an RG scale that is used to parame-
terize RG trajectories in the space of couplings. The
exponents of kreflect the canonical scaling of the corre-
sponding operators. For instance, the gauge coupling e
is power-counting marginal in d= 4, relevant in d < 4
and irrelevant in higher dimensions. The mass term is
a relevant operator in any dimension, whereas the Pauli
term is power-counting irrelevant in all dimensions d > 2.
Note, however, that the Pauli term, being a dimension-5
operator in d= 4, has the smallest possible distance to
marginality in an operator expansion of the action. In
addition, it is a leading term in a derivative expansion,
in which the only other dimension-5 term ¯
ψ/
D/
Dψ is
subleading.
Even though the canonical scaling fully governs RG (ir-
)relevance in the perturbative regime, where corrections
to scaling can only be logarithmic according to Wein-
berg’s theorem [73], nonperturbative phenomena can be
characterized by large anomalous dimensions and thus
strongly affect canonical scaling. In both d= 4 and
d= 3, the nonperturbative phenomenon of chiral sym-
metry breaking is a prime example for this: at large cou-
pling, the anomalous dimensions of the fermionic self-
interaction operators of the type Oψ4(¯
ψψ)2can result
in RG relevance and induce a chiral condensate. This can
occur in both d= 4 [74–76], where Oψ4is a dimension-6
operator, as well as in d= 3 [35, 36, 44, 77] where it is
a dimension-4 operator. These observations put an even
stronger emphasis on the question of a possible RG rel-
evance of the Pauli term in the nonperturbative regime,
as it is closest to marginality.
We approach the answer to this question by studying
the phase diagram of QEDd. More specifically, we use the
Wetterich equation [78–81], a functional RG flow equa-
tion, and determine the βfunctions of the couplings e,
m, and κ. Using t= ln kas a flow parameter, the β
functions can be written as
te=βe=ed
22 + ηψ+ηA
2+ ∆βe,(3)
tm=βm=m(1 ηψ)+∆βm,(4)
tκ=βκ=κd
21 + ηψ+ηA
2+ ∆βκ,(5)
where ηψ,A denote the anomalous dimensions obtained
from
ηψ=tln Zψ, ηA=tln ZA.(6)
In Eqs. (3)-(5), we highlighted the dimensional scaling
terms explicitly, which reflect the canonical scaling ex-
ponents already displayed in Eq. (2) together with the
anomalous dimensions ηψ,A. The last terms ∆βe,m,κ ab-
breviate the quantum (loop) contributions. Their explicit
forms have been computed in [9] and are summarized in
Appendix A to next-to-leading order in a systematic ex-
pansion scheme of the Wetterich equation. Structurally,
these fluctuation contributions depend on
β= ∆β(e, m, κ;ηψ, ηA|d, dγNf),(7)
where the anomalous dimensions satisfy algebraic equa-
tions also listed in App. A and can be expressed as func-
tions of the couplings as well. In addition to a parametric
dependence on the dimension d, the βfunctions also de-
pend on the product dγNfcounting the number of spinor
degrees of freedom: in addition to the flavor number Nf,
dγdenotes the dimensionality of the representation of
the Dirac algebra. The irreducible representations satisfy
dγ= 2bd/2cwhich is used below unless specified other-
wise.
Rather generally, the βfunctions (3)-(5) are not uni-
versal, but also depend on the details of the regulariza-
tion. Even perturbatively, only the one- and two-loop co-
efficient of the marginal coupling ein d= 4 are universal
in a mass-independent scheme. Since we include the run-
ning of the mass and pay attention to threshold effects,
we work with a standard mass-scale-dependent functional
RG scheme. Nevertheless, the existence of fixed points of
the RG, where all βfunctions vanish, is a universal state-
ment. Summarizing the couplings and the beta functions
in vector-like quantities, g= (e, κ, m), β= (βe, βκ, βm),
3
a fixed point gsatisfies β(g=g) = 0. In addition, the
critical exponents θIat a fixed point are also universal.
Linearizing the βfunctions near a fixed point, the flow
is governed by the stability matrix B, the eigenvalues of
which determine the critical exponents,
Bij =βgi
gjg=g
, θI=eig(B).(8)
Since we work in an approximation scheme, our results
for the universal quantities listed in the following also
exhibit only approximate universality and thus depend
on the scheme to some extent. For concrete computa-
tions, we use the partially linear regularization scheme
which is known to be optimized for fast convergence to
universal results in a class of approximations also used
here [82, 83].
Positive (negative) critical exponents θi>0 (θi<0)
are associated with relevant (irrelevant) directions. The
eigendirections associated with the positive exponents
span the UV critical surface of trajectories emanating
from the fixed point. The dimensionality of this surface,
and thus the number of positive exponents, is equal to
the number of physical parameters to be fixed in order to
render the long-range behavior of the theory fully com-
putable. (Eigenvalues θI= 0 denote marginal directions;
here, higher orders beyond the linearized flow determine
relevance or irrelevance. For instance, the QED4gauge
coupling eis marginally irrelevant.)
In addition to gauge symmetry as a local redundancy,
the action has a global U(Nf) flavor symmetry, whereas
an extended U(Nf)L×U(Nf)Rchiral symmetry is present
only in the absence of the mass and the Pauli term, m= 0
and κ= 0. In the general case, the action is also invariant
under a discrete axial rotation of the spinors by an angle
of π/2 and a simultaneous sign flip of κand m, as well as
under charge conjugation and a simultaneous sign flip of
eand κ. These latter Z2symmetries are also visible on
the level of the flow equations which remain invariant un-
der (e, κ, m)(e, κ, m) and (e, κ, m)(e, κ, m)
as well as combinations thereof. Correspondingly, fixed
points can exist in these sign-flip multiplicities, but, of
course, describe one and the same universality class.
In the following, results for fixed-point searches in
various dimensions and for various fermion degrees of
freedom (flavor number, spinor representation) are pre-
sented. As fixed points at finite couplings are an inher-
ently nonperturbative phenomenon, it is important to
discuss consistency criteria in the absence of a generically
small control parameter. Our systematic approximation
scheme represents a combined expansion in operator di-
mension and in derivatives. In this sense, the inclusion of
wave function renormalizations Zψ,A already represents
a next-to-leading order contribution. As a quantitative
control, we can compare to the leading-order result which
is obtained by ignoring wave-function-renormalization ef-
fects in loop terms. In practice, this corresponds to set-
ting ηψ,A = 0 inside the loop terms (technically, inside
the threshold functions, see App. A), but retaining them
in the scaling terms displayed in Eqs. (3)-(5).
We list results only for fixed points where the tran-
sition from leading- to next-to-leading-order results are
quantitatively controlled. This control is implemented by
demanding that the anomalous dimensions remain suffi-
ciently small, |ηψ,A|.O(1). From a technical view-
point, the flow equations (3)-(5) feature rational func-
tions of high order in the couplings on the right-hand
sides. Generically, they exhibit a large number of fixed-
points, most of which do not satisfy our quality criteria
and are thus considered as artifacts of our approxima-
tion. Only a small number fulfills the consistency condi-
tions in a remarkably stable manner. These are the ones
presented in the following sections.
III. PAULI-TERM FIXED POINTS IN d= 4
As a reference point, we start by reviewing the phase
diagram of QED4with a Pauli term for Nf= 1 as has
been found in the literature [9]. Generalizations to differ-
ent dimensions and different flavor numbers can be well
understood by analyzing the similarities and differences
to this reference case.
In addition to the Gaussian fixed point Acharacter-
ized by vanishing couplings, we find two interacting fixed
points that satisfy all our consistency criteria, cf. Tab.I.
Both these fixed points Band Coccur at finite Pauli κ
but vanishing gauge coupling. Fixed point Balso oc-
curs at finite mass parameter mand has full Z2×Z2-fold
multiplicity, whereas for fixed point Conly the charge
conjugation multiplicity is pertinent.
e κ m multiplicity nphys θmax ηψηA
A: 0 0 0 1 1.00 0.00 0.00
B: 0 5.09 0.328 Z2×Z22 3.10 1.38 0.53
C: 0 3.82 0 Z23 2.25 1.00 0.00
TABLE I. Fixed points of d= 4 dimensional QED.
At the Gaussian fixed point A, only the mass is a rel-
evant direction. At fixed point B, the direction towards
finite gauge coupling also represents a relevant direction.
Fixed point Cfeatures 3 relevant directions, implying
that all couplings in the action correspond to physical
parameters that define the long-range behavior of the
system. The flow towards the long-range IR is visual-
ized in the phase diagram of the (κ, m) plane at e= 0
in Fig. 1. Note that the rapid flow near the maxis to-
wards large values of mreflects the fact that mdenotes
a dimensionless mass parameter increasing as m1/k
for k0 if the physical mass approaches a finite value.
As discussed in [9], UV-complete trajectories that
agree with the observed long-range behavior of pure
QED4can be constructed with fixed point Cas a UV fixed
point. Even though UV-complete trajectories emanating
from B, of course, also exist, their long-range behavior
摘要:

Pauli-Term-InducedFixedPointsind-dimensionalQEDHolgerGies,1,2,3KevinK.K.Tam,4,1andJobstZiebell11Theoretisch-PhysikalischesInstitut,AbbeCenterofPhotonics,Friedrich-Schiller-UniversitatJena,Max-Wien-Platz1,07743Jena,Germany2Helmholtz-InstitutJena,Frobelstieg3,07743Jena,Germany3GSIHelmholtzzentrumfu...

展开>> 收起<<
Pauli-Term-Induced Fixed Points in d-dimensional QED Holger Gies1 2 3Kevin K. K. Tam4 1and Jobst Ziebell1 1Theoretisch-Physikalisches Institut Abbe Center of Photonics.pdf

共11页,预览3页

还剩页未读, 继续阅读

声明:本站为文档C2C交易模式,即用户上传的文档直接被用户下载,本站只是中间服务平台,本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私,请立即通知玖贝云文库,我们立即给予删除!
分类:图书资源 价格:10玖币 属性:11 页 大小:1.13MB 格式:PDF 时间:2025-05-02

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务
/ 11
客服
关注