Perturbative running of the topological angles Alessandro Valentiab Luca Vecchib aDipartamento di Fisica e Astronomia G. Galilei Universit a di Padova Italy

2025-05-02 1 0 497.62KB 25 页 10玖币

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Perturbative running of the topological angles

Alessandro Valenti a,b, Luca Vecchi b

aDipartamento di Fisica e Astronomia “G. Galilei”, Universit`a di Padova, Italy

bIstituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy

Abstract

We argue that in general renormalizable ﬁeld theories the topological angles may de-

velop an additive beta function starting no earlier than 2-loop order. The leading ex-

pression is uniquely determined by a single model-independent coeﬃcient. The associated

divergent diagrams are identiﬁed and a few methods for extracting the beta function in

dimensional regularization are discussed. We show that the peculiar nature of the topolog-

ical angles implies non-trivial constraints on the anomalous dimension of the CP-violating

operators and discuss how a non-vanishing beta function aﬀects the Weyl consistency

conditions. Some phenomenological considerations are presented.

alessandro.valenti@pd.infn.it

luca.vecchi@pd.infn.it

arXiv:2210.09328v2 [hep-ph] 20 Jan 2023

Contents

1 Motivations 2

2θin perturbation theory 4

2.1 βθfrom (extra)-ordinary diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Extracting βθfromoperatormixing ........................ 7

3βθin renormalizable QFTs 9

3.1 CPandﬂavorsymmetry............................... 10

3.2 The3-loopdiagrams ................................. 11

4 Implications 13

4.1 Consistencyconditions................................ 13

4.2 (In)Stability of UV solutions of the strong CP problem . . . . . . . . . . . . . . 14

5 Outlook 18

A CP-odd ﬂavor-invariants 19

References 23

1 Motivations

Relativistic four-dimensional quantum ﬁeld theories feature several renormalizable operators:

Yukawa, gauge, scalar self-interactions and the topological terms. The latter stand out of this

list because they are total derivatives. At the classical level they have no impact since they

do not alter the equations of motion; at the quantum mechanical level they do not introduce

Feynman vertices. Nevertheless, the topological angles θcan aﬀect physical observables if

non-perturbative eﬀects are taken into account [1,2,3,4] or if topological defects are present

[5]. In a semi-classical analysis one studies the quantum ﬂuctuations around a certain back-

ground ﬁeld conﬁguration, and θshows up in the path integral via a factor eiθν whenever the

background gauge ﬁelds are associated to a non-trivial topological charge ν.

Because θcan appear in observables, it is only natural to wonder about its renormaliza-

tion group evolution. Since θdoes not parametrize vertices for the quantum ﬂuctuations, it

is clear that the perturbative correlators and beta functions cannot depend on it. At most,

perturbatively θrepresents a counterterm necessary to reabsorb (ﬁnite and divergent) cor-

rections to CP-violating diagrams with external background gauge ﬁelds. In other words, in

perturbation theory θcan only get additively renormalized because of contributions induced

by the other couplings.

That θrenormalizes is out of the question. We can distinguish between ﬁnite and inﬁnite

renormalization eﬀects. Finite (threshold) corrections to θare very common. In generic

theories they occur at tree-level, when crossing a CP-violating fermion mass threshold, and

even more often at loop level. In the case of the QCD θangle, for example, ref. [9] found

that the leading order ﬁnite correction is a 3-loop eﬀect when matching to the eﬀective ﬁeld

theory below the W±mass. Inﬁnite corrections are more rare. A necessary condition for

these contributions to actually occur in a mass independent renormalization scheme is that

the quantum ﬁeld theory under consideration possesses polynomial CP-odd ﬂavor-invariant

combinations of the couplings (other than θ) that can contribute to

βθ=µdθ

dµ.(1.1)

In the Standard Model the ﬁrst such combination appears at a very high power in the Yukawa

couplings [6,7] indicating that, if present, the UV divergence that renormalizes θshould occur

at a prohibitive perturbative order. In view of this one may ask if such divergence exists at

all, and whether this high perturbative order is a general feature of renormalizable quantum

ﬁeld theories. To better assess these questions it would be useful to ﬁnd “toy” ﬁeld theories in

which θdevelops a beta function at a suﬃciently low order as to allow an explicit computation,

which may provide an indirect conﬁrmation of our expectations in the Standard Model. It

is not diﬃcult to ﬁnd non-renormalizable ﬁeld theories that induce divergent corrections to

θ. For example, adding cg2|H|2Ge

G/Λ2to the Standard Model Lagrangian gives the minimal

subtraction result βθ=−4cm2

H/Λ2, see [8]. However, the latter eﬀect is power-law suppressed

and is therefore practically irrelevant in the presence of a large gap between the IR scales and

the UV cutoﬀ. In addition, we will see there are a few non-trivial challenges that a calculation

of divergent contributions to θfaces within renormalizable theories, including the fact that the

regularization scheme adopted must be able to consistently deal with the so-called γ5problem.

In the non-renormalizable theory we mentioned above those challenges are not encountered

because the operator Ge

Gwas already present on the outset. The questions we are interested

in therefore better be addressed within renormalizable ﬁeld theories. Yet, surprisingly enough,

to the best of our knowledge nobody has ever found divergent corrections to θin that context.

While no concrete renormalizable example where θgets inﬁnitely renormalized is on the

market, there exists well-known instances in which one can rigorously prove that θis not

inﬁnitely renormalized at any order. An example is pure Yang-Mills, where this property

follows trivially because that theory does not satisfy the necessary condition stated above:

there are no ﬂavor-invariant CP-odd phases other than θ, and hence there is nothing that

can contribute perturbatively to βθ. The same holds for QCD with massive fermions, since

once the phases in the quark mass matrix are removed via anomalous chiral rotations of the

fermions, CP-violation is entirely encoded in θ. Another popular instance is provided by

supersymmetric gauge theories, where the exact one-loop running of the holomorphic gauge

coupling reveals that the theta angle does not run. The reason here is again the same: there

is no available CP-odd (holomorphic) combination of the other marginal couplings.

The main goals of this paper are to present concrete examples of four-dimensional renor-

malizable ﬁeld theories that can induce inﬁnite corrections to θ, as well as to identify the lead-

ing order structure of the beta function βθin any mass-independent renormalization scheme

(see Section 3); to discuss the subtleties encountered in a perturbative treatment of θas well

as to show how to concretely approach the calculation of βθin dimensional regularization (see

Sections 2); and ﬁnally to analyze the relevance of βθ(Section 4). 1

1It is worth dissipating a possible source of confusion right away. It is well-known that physical observables

must depend on a ﬂavor-invariant combination ¯

θof the topological angle and the other couplings of the theory

(e.g. the quark masses in QCD or the Yukawa couplings in the Standard Model). In a generic ﬁeld basis the

beta function of ¯

θreceives contributions from corrections to both θ, which represent the main subject of this

paper, and the other couplings as well (these latter corrections are those estimated in [6], strictly speaking).

Throughout the paper we will be mostly concerned with the basis-dependent parameter θ, the coeﬃcient of

the topological term. The physics of the QCD parameter ¯

θwill be discussed in Section 4.2.

Studying the RG evolution of θin a general renormalizable theory is not only of academic

interest. There are technical as well as potentially phenomenological reasons for doing it. At

a genuinely phenomenological level, studying the beta function of θin general theories might

potentially shed light on the absence of CP-violation in QCD, or more precisely on the neces-

sary properties that its UV completion must satisfy to account for experimental observations,

which in fact was the original motivation of our precursors [10,6]. Yet another reason is that

we do not know what theory will eventually be found to complete the SM at shorter distances.

It is perhaps not completely unconceivable that in such a UV-completion topological angles

occur in some particular observable, say because of the presence of magnetic monopoles or

because small-instanton eﬀects cannot be ignored. In these situations a renormalization group

evolution of θmay become phenomenologically relevant. At a more technical level, βθis an

obvious target for explicit calculations. The beta functions of the “ordinary” couplings of

general renormalizable ﬁeld theories have been explicitly calculated up to 2-loop order (see

[11,12,13] and more recent updates). Yet, strictly speaking, this program cannot be viewed

as fully complete until βθis also computed. On a completely diﬀerent note, the peculiar na-

ture of θencodes important information about the renormalization properties of the theory.

For example we will see that the independence of perturbative correlators on θtranslates into

a constraint on the anomalous dimension of the CP-odd operators of the theory. These and

other observations may turn out to be useful in concrete calculations.

2θin perturbation theory

In a semi-classical approach to quantum ﬁeld theory, the bare ﬁelds φ0are split into a classical

ﬁnite-action background φ0coverlapping with the vacuum, plus a quantum ﬂuctuation δφ0

that vanishes suﬃciently fast at the boundary. The path integral is deﬁned to include a

sum over all inequivalent background conﬁgurations (e.g. collective coordinates) along with

the functional integration over the ﬂuctuations around each background. Singling out the

topological term from the total gauge-ﬁxed action, 2which we schematically write as Stot =

S+ (g2

0θ0)/(32π2)RG0e

G0, this recipe produces the following generating functional

Z[J0] = X

φ0c

eig2

0θ0

32π2RG0ce

G0c+iRJ0φ0cZDδφ0eiS[φ0c+δφ0]+iRJ0δφ0(2.1)

φ0c

eig2

0θ0

32π2RG0ce

G0cb

Z[J0, φ0c].

The topological term is special because it is a total derivative. By construction the quantum

ﬂuctuations vanish at the boundary, so that any ﬂuctuation-dependent contribution to such a

term vanishes and RG0e

G0=RG0ce

G0creduces to an integral of the sole external background

ﬁelds, which we can take outside the functional integral in (2.1): the angles θdo not appear

in any interaction of the quantum ﬂuctuations but can act as counterterms in computations

with external background ﬁelds.

So far our discussion has been rather general. Yet, an actual evaluation of (2.1) is nec-

essarily regularization-scheme dependent. In the following we will specialize on dimensional

regularization (Dim-Reg), in which space-time is continued to ddimensions with coordinates

xµ=x¯µ, xˆµ, where ¯µ, ¯ν, · · · = 0,1,2,3 and ˆµ, ˆν, · · · denote the (d−4)-dimensional indices.

2Throughout the paper we assume the gauge-ﬁxing preserves the background gauge invariance.

In Dim-Reg the very deﬁnition of topological term forces us to deﬁne the Levi-Civita

tensor and deal with the famous γ5problem. So far the only known consistent prescription

is the ’t Hooft-Veltman-Breitenlohner-Maison scheme [14,15], where the Levi-Civita tensor

is a formal object ¯µ¯ν¯α¯

βthat carries only 4-dimensional indices. In other words, the (d−4)-

dimensional indices ˆµ, ˆν, · · · of an arbitrary vector do not contribute when contracted with

this tensor. An important implication is that

G0e

G0≡1

2G¯µ¯ν

0G¯α¯

0¯µ¯ν¯α¯

β=∂¯µK¯µ

0(2.2)

is 4-dimensional divergence of a (xµ-dependent) vector. Hence the regularized quantity

Rddx G0ce

G0ccontains a non-trivial residual (d−4)-dimensional integral and is not a topo-

logical term in ddimensions.

The d-dimensional continuation of (2.1) formally represents a set of regularized Green’s

functions. Such a path integral violates two of the familiar properties of the topological

angle, namely its periodicity in 2πand its role as compensator (spurion) of the abelian axial

symmetry. 3The technical reason for the ﬁrst loss boils down to the fact that, as a consequence

of (2.2), the bare angle θ0is not the coeﬃcient of a topological operator in the regularized

theory. The second loss occurs because anomalies are d-dependent; as a result, in d-dimensions

a shift of the coeﬃcient of Ge

Gdoes not fully compensate an axial rotation. An intuitive way

of arriving to the same conclusions is provided by dimensional analysis: the engineering

dimension of the bare coupling in Dim-Reg is [θ0] = d−4, and it is therefore impossible

for θ0to be periodic in 2πor even to shift via the dimensionless parameter of the axial

transformation while retaining its µ-independence in d-dimensions.

To recover the topological nature of the theta angle, as well as its role as a compensator

for abelian axial transformations, one has to derive the renormalized 4-dimensional version

of the path integral. In general this procedure requires a renormalization of theta as well.

Renormalization renders θa genuinely (4-dimensional) topological term and the background-

dependence in the 4-dimensional limit of the path integral (2.1) reduces to a dependence on

the topological index ν. The renormalized coupling θis periodic in 2πand transforms via a

shift under abelian axial rotations. This ensures in particular that physical amplitudes are

invariant under unitary ﬁeld re-deﬁnitions.

2.1 βθfrom (extra)-ordinary diagrams

For completeness we recall the standard prescription to extract the beta function within the

Minimal Subtraction scheme. The relation between the bare couplings θ0,ξ0i(the latter

symbol denotes all couplings except θ0) and the renormalized couplings θand ξiread (in

d= 4 −dimensions)

θ0=µ−[θ+Zθ],(2.3)

and ξ0i=µρi[ξi+Zξi]. By deﬁnition Zθ=P∞

n=1 −nZθ,n(ξi) contains no ﬁnite term, and simi-

larly for Zξi. The 4-dimensional beta functions βξ≡limd→4µdξ/dµ read βθ=ρiξi∂Zθ,1/∂ξi+

Zθ,1and similarly βξi=ρjξj∂Zi,1/∂ξj−ρiZi,1. In the above ∂Zθ/∂θ =∂Zξi/∂θ = 0 because

θdoes not appear in Feynman diagrams. As customary for ordinary couplings, also the beta

3In our discussion we implicitly assume the theory has integer topological index ν. The extension of our

arguments to theories in which νis rational is straightforward.

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PerturbativerunningofthetopologicalanglesAlessandroValentia;b,LucaVecchibaDipartamentodiFisicaeAstronomia\G.Galilei",UniversitadiPadova,ItalybIstitutoNazionalediFisicaNucleare,SezionediPadova,I-35131Padova,ItalyAbstractWearguethatingeneralrenormalizableeldtheoriesthetopologicalanglesmayde-velopana...

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Perturbative running of the topological angles Alessandro Valentiab Luca Vecchib aDipartamento di Fisica e Astronomia G. Galilei Universit a di Padova Italy

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