The QCD Adler function and the muon g 2 anomaly from renormalons
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
ACFI-T21-12
The QCD Adler function and the muon g − 2 anomaly from renormalons
Alessio Maiezza1, ∗ and Juan Carlos Vasquez2, †
1
Rue.r Bošković Institute, Bijenička cesta 54, 10000, Zagreb, Croatia,
2
Amherst Center for Fundamental Interactions, Department of Physics,
University of Massachusetts, Amherst, MA 01003, USA.
We describe the Adler function in Quantum Chromodynamics in terms of a transseries representation
within a resurgent framework. The approach is based on a Borel-Ecalle resummation of the infrared
renormalons, which, combined with an effective running for the strong coupling, gives values in
excellent quantitative agreement with the so-called experimental Adler function data. We discuss
its impact on the muon’s anomalous magnetic moment. Without the need for new physics and using
our expression for the Adler function, we show that it is possible to saturate the current discrepancy
between the best Standard Model value for the muon’s anomalous magnetic moment and the one
obtained from data.
arXiv:2111.06792v2 [hep-ph] 22 Nov 2021
Introduction. The description of the Quantum leaving one arbitrary constant fixed from data. It repre-
Chromodynamics (QCD) at the hadronic scale is a sents an improvement to all the known renormalon-based
formidable challenge because of the failure of perturba- evaluations in QFT and QCD. The inability to calculate
tion theory, the only analytical and stand-alone tool to this arbitrary constant is due to the non-existence of a
deal with quantum field theory (QFT). The Adler func- semiclassical limit for renormalons. Because of technical
tion [1] is a fundamental quantity used to describe the details that we shall discuss, the transseries for the Adler
QCD non-perturbative effects at the hadronic scale. Its function has three free constants to be determined from
theoretical description is essential since it appears in any the data.
process involving the QCD corrections due to the vac-
uum polarization function. Lattice QCD provides a the- In this work, we show for the first time that the new
oretical representation of the Adler function [2]. How- approach of Ref. [17] can accurately describe the Adler
ever, an analytical understanding would be beneficial. function in the entire infrared (IR) regime, provided one
Based on the notion of renormalons [3–6] and Operator- properly deal with the Landau pole [32] singularity. To
Product-Expansion [7, 8], there are non-perturbative an- this aim, we adopt an effective running for the strong
alytical evaluations for the Adler function and QCD ob- coupling αs that considers the confinement and prevents
servables [9–12]. Other methods use integral representa- the coupling from diverging [33]. The non-perturbative
tions [13–15]. Although all these analytical approaches running we adopt is such that the strong coupling freezes
reproduce all the qualitative features of the “experimen- at low energy [34] – see the review [35] for typical non-
tal” Adler function [16] at the hadronic scale, the quan- perturbative running for αs . The final result is shown in
titative agreement is not excellent. Fig. 1. Our result features three parameters in contrast
The recent analytical approach to the Adler function to conventional renormalon approaches with an infinite
of Ref. [17] distinguishes from previous ones because it is number of arbitrary constants. To illustrate the predic-
based on renormalons and the resurgence theory. First tivity of the transseries representation, we also compare
proposed by Ecalle [18] in a purely mathematical context, it with a fit including the same number of free parame-
it has found fertile ground in QFT [19–27]. Renormal- ters (three) but in conventional renormalon-based evalu-
ized perturbation theory controls the finiteness of QFT ation. In this case, there is not a quantitative agreement
in the proper regime. Therefore, it is an appealing possi- between the theory and data for energies below ≈ 1.3
bility to analytically continue it to the non-perturbative GeV.
regime. For this reason, resurgence may represent a good
candidate for a foundational, analytical approach to a The application to the g − 2 discrepancy [36] pro-
non-perturbative QFT. vides a straightforward illustration of the usefulness of
In the specific framework of ordinary-differential- our transseries representation. We shall show two main
equation [28, 29], a resurgent approach to renormaliza- results. First, our expression can reproduce the best the-
tion group was proposed in Refs [30, 31]. In this new ap- oretical value for aµ given in Refs. [37, 38]. Second, as-
proach, the renormalization group equation (RGE) was suming that the QCD vacuum polarization function sat-
written as a non-linear ODE in the coupling constant. urates the g − 2 discrepancy [39] and by doing a small
The resulting theory was then applied to the QCD Adler modification of the parameters, we can explain the aµ
function [17], where the renormalons can be resummed, discrepancy reported in Ref. [36]. Whereby the g − 2
discrepancy can be explained by including previously in-
calculable non-analytic corrections αs to the QCD vac-
uum polarization function – in agreement with the most
∗ alessiomaiezza@gmail.com recent lattice computation [40], and the experimental re-
† jvasquezcarm@umass.edu sult reported in Ref. [36].2
1.4 parametrizing simple pole ambiguity due to the first non-
zero renormalon; another constant C stemming from the
1.2 Borel-Ecalle resummation of quadratic renormalons; a
1.0
constant K related to the n! behavior in the perturba-
mc tive series [9], which in the case of renormalons and unlike
0.8 ·
D(Q)
Exp. points instantons [44], cannot be determined using semiclassical
— Resurgence
methods [5]. The inability to determine those constants
0.6 PT
-- 1 renormalon from first principles is a well-known problem, and it has
0.4 — 2 renormalons been recently linked to foundational issues to construct
an unambiguous QFT starting from the free fields [45].
0.2
The original fermion bubble graph contribution to the
0.0
0.0 0.5 1.0 1.5 2.0 2.5 Adler function was calculated in Ref. [46]. In Ref. [17],
Q(GeV) we rewrote the fermion bubble graph contribution of
Ref. [46] such that the pole structure of the Borel trans-
FIG. 1: Adler function in the energy range (0, 2.5) GeV. form was apparent. After applying the Borel-Ecalle re-
The dashed green line is the perturbation theory summation of Refs. [30, 31], the transseries expression of
approximation of the Adler function. Solid Black line the Adler function is of the form
corresponds to the resurgent expressions (4) and (7).
Dashed and solid gray lines correspond to the
approximation of the Adler function including the first 4π 2
Dresurg. (Q) = D0 (Q) − c1 e β0 αs (Q2 )
and second renormalon power corrections. β0
ap
1 1
+ Ce β0 αs (Q2 )
D1 (Q2 ) , (4)
αs (Q2 )
a. The Adler function. The Adler function D(Q)
is defined as
dΠ (Q) where ap = 1 + O(β1 /β02 ). The function D0 (Q) contains
D (Q) = 4π 2 Q2 , (1) the perturbative expression up to O(αs4 ) shown in Eq. (3)
dQ2
and is given by:
where Π(Q) is determined via
Z
D0 (Q) = Dpert (Q) + DK (Q) , (5)
−i d4 x e−iqx h0 |T (jµ (x)jν (0))| 0i
= qµ qν − q 2 gµν Π (Q) ,
(2) P∞
where DK (Q) ∝ n=0 K β0n αsn+1 n!. Following the for-
being q the transferred momentum, Q2 = −q 2 and malism of Ref. [29], we then regularize the n! diver-
jµ = q̄γµ q two massless quark currents. In perturbation gence in DK by taking the Cauchy principal value for
theory, the Adler function is given by the Laplace integral such that
∞
αs X n n 2
3
Dpert (Q) = 1 + α [dn (−β0 ) + δn ] . (3) e αs β0 Γ 0, αs2β0 2e αs β0 Γ 0, αs3β0
π n=0 s DK (Q)
= + +
2K β0 3αs β02
We use the convention for the beta function β(αs ) = 2(p+1)
2(p+1)
µ2 dα 2 3 4 2 ∞ α β − 2(p + 1)e αs β0
Γ 0,
dµ2 = β0 αs + β1 αs + O(αs ) , 2π β0 = −11 + 3 nf ,
s s 0
X α β
s 0
+
where nf is the number of active flavors, αs = gs2 /4π 3β 2 α p(p + 1)(2p + 1)
p=1 0 s
and gs denotes the SU (3) gauge coupling. The Adler
function is known up to n = 3 [9, 41–43] or up to O(αs4 ), 2p+3
2 (2p + 3)e αs β0 Γ 0, 2p+3
αs β0 − αs β0
from which the coefficients dn and δn can be taken from . (6)
Ref. [9]. We estimate the large order behavior in Eq. (3) 3β02 αs (p + 1)(2p + 1)(2p + 3)
using the “Naive non-abelianization”, namely the δn ∼ 0
for n ≥ 4, such that we are able to estimate dn to all
orders from the fermion bubble graphs [9].
In the above expression, the terms up to O(αs4 ) must be
b. Resurgent Adler function. In Ref. [17], we re-
removed in DK (Q) to prevent the double counting of this
summed the IR renormalon contribution to the QCD
contribution in Dpert (Q).
Adler function using the Borel-Ecalle resummation of
Refs. [30, 31]. After resuming the renormalons using Finally, the function D1 (Q2 ) is found from D0 [30] us-
the new framework, the expression for the Adler func- ing resurgent relations [30, 31]. Choosing the renormal-
tion features three arbitrary constants: one constant c1 ization scale µ2 = Q2 e−5/3 and neglecting the two-loop3
1
corrections proportional to β1 , one finds Parameter Low energy fit aµ discrepancy
K 1.42226 1.45113
8πK h α 1β 1 2
C 0.62943 0.80716
D1 (Q) = 2e s 0 − e αs β0
+ 1 log 1 − e αs β0
3αs β02 c1 0.03261 0.06148
1 1 i Λ 731 MeV 713 MeV
− 2 e αs β0 + 1 tanh−1 e αs β0 . (7)
TABLE I: Numerical value of the constants in Eq. (4).
The next step is to implement the non-perturbative run- Central column represents the low energy (Q . 1.3
ning for the coupling αs (µ) to be used in Eq. (4). GeV) fit shown in Fig. 1. Third column shows the
c. Effective running and the QCD Adler func- values reproducing the experimental g − 2 discrepancy.
tion at low energies. In Ref. [47], the authors ex-
plored the possibility that the QCD running coupling
can be effectively extrapolated in a process-independent In Fig. 1, we show the Adler function in the energy
way to smaller momenta of the order of the hadronic range Q = (0, 2.5) GeV. We see no appreciable difference
scale. The non-perturbative physics should reveal itself at energies Q = (1.3, 2.5) GeV between the expressions
smoothly in inclusive observables. Consequently, it is coming from the first two power corrections (gray lines)
meaningful to extend the notion of the perturbative QCD and the resurgent result (solid black line). Conversely,
coupling to zero energy. This logic applies to the QCD in the low energy range Q = (0, 1.3) GeV, the solid and
Adler function. dashed gray lines fail to describe the Adler function, while
The transseries provided in Ref. [17] is capable of the solid black line successfully follows the behavior of
fitting the experimental Adler function up to energy the data in the whole range. To our knowledge, this is
≈ 0.7GeV. The failure below that energy is due to the the first time that resurgence formalism provides a clear
unphysical Landau pole of the perturbative running of phenomenological result for QCD, in particular an ex-
αs – and not the Borel-Ecalle resummation formalism. pression for the Adler function valid at all energies. In
To overcome this difficulty, in this work, we use an effec- Tab. I, we show the values for the parameters entering in
tive running coupling 2 valid up to zero energy in which the transseries for the Adler function in Eq. (4).
αs goes to a constant value. More specifically, we use d. The anomalous magnetic moment of the
Cornwall’s coupling [33], which is one of the simplest an- muon. The magnetic moment of the muon µ ~ directed
alytic non-perturbative models for the running of αs and along its spin ~s is given by
given by [34]
Qe
4π µ
~ =g ~s , (9)
αs (Q) = , (8) 2mµ c
11 ln (z + χg ) − 2nf ln (z + χq ) /3
where Qe is the electric charge, mµ is the muon mass, c
where z = Q2 /Λ2 , nf is the number of flavors, χg = is the speed of light, and Dirac’s theory predicts g = 2.
4m2g /Λ2 , χq = 4m2q /Λ2 , the light constituent quark mass Quantum effects correct the value g = 2 and the devia-
mq = 350 MeV, the gluon mass mg ' 500 MeV, and Λ tion is parameterized as aµ = (g − 2)/2. In this work, we
denotes the QCD hadronic (non-perturbative) scale. We only consider the so-called hadronic vacuum polarization
shall determine Λ by fitting the experimental data of the (h.v.p.)
contribution (h.v.p.) aµ – for comprehensive works
Adler function. The result is graphically illustrated in on this subject see Refs. [37, 38, 51–57].
Fig. 1. The leading order hadronic vacuum polarization con-
We find that the typical running for αs reproducing the tribution in terms of the QCD Adler function is of the
Adler function is such that at low energies, αs (0) ' 1.6, form [53, 58]
which is in the ballpark of known results in the literature.
See Ref. [35] and references therein for a detailed discus- α 2 Z 1
dx
sion about the low energy behavior of the QCD running a(h.v.p.)
µ = 2π 2
(1 − x)(2 − x)D (Q) , (10)
π 0 x
coupling in several non-perturbative approaches.
where α ' 1/137 q is the electromagnetic coupling con-
x2
stant, and Q = 1−x m2µ .
1 To check the correctness of Eq. (7), we proved that the resurgence By using Eqs. (4) and (10), we shall show two results.
relations of Ref. [29] can also be derived using Ecalle bridge equa- First, we correctly reproduce the leading contribution to
tion obtained from the RGE. We will discuss the latter point in aµ from the QCD vacuum polarization function. Sec-
a separate publication. ond, by a slight modification of the best fit values for the
2 It has been argued that the operation of power corrections to
constants K, c1 , and C shown in Tab. I, we can accom-
physical observables and making the coupling αs effective (ana-
lyzation) do not commute, and this would lead to an ambiguity modate the g − 2 muon anomaly [36] (consistent with the
in Eq. (4) [48–50]. However, since the Eq. (4) is intrinsically am- most recent lattice evaluation [40]), minimally modifying
biguous, one can reabsorb the aforementioned ambiguity in the the Adler function for energies below ∼ 0.7 GeV. This
definition of the fitted parameters (e.g. ”C”). possibility was recently raised in Ref. [39].4
1.2 -- aμ discrepancy Following Ref. [39], we assume the g − 2 discrepancy
— fit D at low energy
can be solely explained by modifying the SM vacuum
experimental D
1.0 polarization function contribution. It has been argued
in Ref. [39] that the data for the hadronic cross-section
0.8
σ(e+ e− → hadrons) may have some missed contribu-
0.6
D(Q)
0.6 tions for Q . 0.7 GeV. These missed contributions open
the possibility of explaining the g−2 discrepancy by devi-
0.4 0.45 ations of the e+ e− cross-section measurement for Q < 0.7
GeV. In the same Ref. [39], the authors provide two ad
0.2 hoc models to change the data and saturate the gap.
0.3
Hence, we require the Adler function to match the ex-
0.0 0.26 0.3 0.34 0.38
perimental data for energies Q ≥ 0.7 GeV. Notice that
0.0 0.2 0.4 0.6 0.8 1.0 1.2 the deviation of C with respect the one in Tab. I (cen-
Q(GeV)
tral column) is about 30% for c1 and C and around 1%
FIG. 2: The Adler function in the energy range (0, 1.3) for K. The plot for the Adler function, corresponding to
GeV. The purple region denotes the “experimental” the values for C, c1 and K in Tab. I (central column) is
Adler function from tau data [59]. The black line shown in Fig. 2 and represented with the dashed red line.
represent the Adler function as in Fig. 1. For a slightly The deviation concerning the average value aµ ' 6.9 ×
different value of the constants C, K, c1 , the dashed, red 10−8 of Refs. [37, 38] is due to non-perturbative (non-
line represents the Adler function saturating the muon analytic) contributions in the strong coupling constant
g − 2 discrepancy between experiments and predictions. αs , which were calculated using the resurgence frame-
The inset is a zoom on the region of interest. work of Refs [30, 31]. These non-analytic contributions
become dominant for αs ∼ 1. The non-perturbative
electro-weak corrections are sub-leading since the numer-
e. Reproducing the best theoretical value for ical values of the electromagnetic and weak couplings re-
the anomalous magnetic moment of the muon. In main small at the muon mass-energy scale.
evaluating the integral in Eq. (10), the Adler function Summary and outlook. We have shown that the
D(Q) needs to be evaluated in the energy range [0, ∞). resurgent, analytical expression for the Adler function in
Following Ref. [16], one has to split D in√two branches, Eq. (4) gives an excellent quantitative agreement with
using the perturbative estimate for Q > 1.6 GeV and data at the hadronic scale. We have used Cornwall’s
√
the data for 6 1.6 GeV. The Eq. (4) provides a good coupling to model the running of αs whose value freezes
estimate of data, thus the Adler function used in the at low energies with αs (0) ∼ O(1). The latter ensures the
evaluation of Eq. (10) is given by applicability of the resurgent approach to renormalons
and renormalization group equation of Ref. [31], which is
√
based on non-linear, ordinary differential equations [29].
Dresurg. (Q) Q 6 √1.6 GeV
D(Q) = (11) As a result, Eq. (4) features three arbitrary parameters,
Dpert. (Q) Q > 1.6 GeV .
in contrast to conventional renormalon-based evaluations
Using the values of the low energy fit in Tab. I, we get with an infinite number of arbitrary constants. We have
for the leading contribution of the hadronic vacuum po- determined those parameters from data, shown in Tab. I,
larization: and our representation of the Adler function is drawn in
Fig. 1.
a(h.v.p.)
µ = 6.85024 × 10−8 . (12) We have also addressed the implications and the in-
terplay with the muon’s magnetic moment. We can re-
It is remarkably close to the averaged value aµ ' 6.9 × produce both the leading order value for the vacuum po-
10−8 reported in Refs. [37, 38], based on a number of larization hadronic contribution predicted by dispersive
independent evaluations from e+ e− → hadrons data and approaches, as well as the most recent value consistent
τ -spectral function. The corresponding behavior of the with the measurement of the magnetic moment of the
Adler function in [0, 1.3] GeV is shown in Fig. 2 (solid muon reported in Ref. [36], which is compatible with the
black line) together with the uncertainties represented most recent lattice calculation [40]. As proof of concept,
by the light blue band. our result shows that the muon g − 2 discrepancy can
f. Saturating the g-2 experimental discrepancy be entirely explained without resorting to new physics.
of the muon anomalous magnetic moment of the Instead, it can be explained by considering non-analytic
muon. In this section we find the modification of the contributions in the strong coupling constant αs calcu-
numerical value of the constants K, c1 and C shown in lated using resurgence theory.
Tab. I (central column), to saturate the gap between the In particular, fitting the available data for Adler func-
(h.v.p.)
average value of aµ ' 6.9 × 10−8 [37, 38] and the tion using Eq. (4), we get aµ ' 6.85 × 10−8 , notably close
(h.v.p.)
value aµ ' 7.15 × 10−8 consistent with the Muon to the average value ' 6.9 × 10−9 . A small modification
g − 2 Collaboration [36] experimental result. of the (fitted) parameters shown in Tab. I can explain5
the SM discrepancy for aµ . As shown in Fig. 2, the only may be on the determination of the heavy quark pole
effect is of slightly spoiling the behavior of the Adler func- mass [66, 67] and on the static quark-antiquark poten-
tion at energies . 0.7 GeV, range in which data may be tial [68]. These processes open the possibility of testing
not complete due to missed contributions in the hadronic the universality of QCD running coupling and of the con-
cross section σ(e+ e− → hadrons). stants found in Tab. I, in the spirit of Ref. [47].
We expect our result to be applicable for other relevant
processes involving the two-point Green function at the Acknowledgements. JCV was supported in part un-
hadronic scale. An example may be the event shape ob- der the U.S. Department of Energy contract DE-
servables in e+ e− collisions [60–65]. Other applications SC0015376.
[1] S. L. Adler, Some simple vacuum-polarization [18] J. Écalle, Six lectures on transseries, analysable
phenomenology: e+ e− → hadrons; the muonic-atom functions and the constructive proof of dulac’s
x-ray discrepancy and gµ − 2, Phys. Rev. D 10 (Dec, conjecture, .
1974) 3714–3728. [19] P. C. Argyres and M. Unsal, The semi-classical
[2] A. Francis, B. Jäger, H. B. Meyer and H. Wittig, New expansion and resurgence in gauge theories: new
representation of the adler function for lattice qcd, perturbative, instanton, bion, and renormalon effects,
Physical Review D 88 (Sep, 2013) . JHEP 08 (2012) 063, [1206.1890].
[3] D. J. Gross and A. Neveu, Dynamical symmetry [20] G. V. Dunne and M. Unsal, Resurgence and
breaking in asymptotically free field theories, Phys. Rev. Trans-series in Quantum Field Theory: The CP(N-1)
D 10 (Nov, 1974) 3235–3253. Model, JHEP 11 (2012) 170, [1210.2423].
[4] B. Lautrup, On high order estimates in qed, Physics [21] D. Dorigoni, An Introduction to Resurgence,
Letters B 69 (1977) 109–111. Trans-Series and Alien Calculus, Annals Phys. 409
[5] G. ’t Hooft, Can We Make Sense Out of Quantum (2019) 167914, [1411.3585].
Chromodynamics?, Subnucl. Ser. 15 (1979) 943. [22] I. Aniceto, G. Basar and R. Schiappa, A Primer on
[6] G. Parisi, The Borel Transform and the Resurgent Transseries and Their Asymptotics, Phys.
Renormalization Group, Phys. Rept. 49 (1979) 215–219. Rept. 809 (2019) 1–135, [1802.10441].
[7] K. G. Wilson and W. Zimmermann, Operator product [23] P. J. Clavier, Borel-Ecalle resummation of a two-point
expansions and composite field operators in the general function, 1912.03237.
framework of quantum field theory, Comm. Math. Phys. [24] M. Borinsky and G. V. Dunne, Non-Perturbative
24 (1972) 87–106. Completion of Hopf-Algebraic Dyson-Schwinger
[8] M. Shifman, Yang-Mills at Strong vs. Weak Coupling: Equations, Nucl. Phys. B 957 (2020) 115096,
Renormalons, OPE And All That, 2107.12287. [2005.04265].
[9] M. Beneke, Renormalons, Phys. Rept. 317 (1999) [25] T. Fujimori, M. Honda, S. Kamata, T. Misumi,
1–142, [hep-ph/9807443]. N. Sakai and T. Yoda, Quantum phase transition and
[10] M. Shifman, New and Old about Renormalons: in Resurgence: Lessons from 3d N = 4 SQED,
Memoriam Kolya Uraltsev, Int. J. Mod. Phys. A 30 2103.13654.
(2015) 1543001, [1310.1966]. [26] O. Costin and G. V. Dunne, Resurgent extrapolation:
[11] G. Cvetič, Renormalon-motivated evaluation of QCD rebuilding a function from asymptotic data. Painlevé I,
observables, Phys. Rev. D 99 (2019) 014028, J. Phys. A 52 (2019) 445205, [1904.11593].
[1812.01580]. [27] O. Costin and G. V. Dunne, Physical Resurgent
[12] I. Caprini, Conformal mapping of the Borel plane: going Extrapolation, Phys. Lett. B 808 (2020) 135627,
beyond perturbative QCD, Phys. Rev. D 102 (2020) [2003.07451].
054017, [2006.16605]. [28] O. CostinInternational Mathematics Research Notices
[13] D. V. Shirkov and I. L. Solovtsov, Analytic model for 1995 (1995) 377.
the QCD running coupling with universal alpha-s (0) [29] O. Costin, Asymptotics and Borel Summability.
value, Phys. Rev. Lett. 79 (1997) 1209–1212, Monographs and Surveys in Pure and Applied
[hep-ph/9704333]. Mathematics. Chapman and Hall/CRC (2008), .
[14] A. V. Nesterenko, Adler function in the analytic [30] A. Maiezza and J. C. Vasquez, Non-local Lagrangians
approach to QCD, eConf C0706044 (2007) 25, from Renormalons and Analyzable Functions, Annals
[0710.5878]. Phys. 407 (2019) 78–91, [1902.05847].
[15] G. Cvetic and C. Valenzuela, Analytic QCD: A Short [31] J. Bersini, A. Maiezza and J. C. Vasquez, Resurgence of
review, Braz. J. Phys. 38 (2008) 371–380, [0804.0872]. the Renormalization Group Equation, Annals Phys. 415
[16] S. Peris, M. Perrottet and E. de Rafael, Matching long (2020) 168126, [1910.14507].
and short distances in large N(c) QCD, JHEP 05 [32] L. D. Landau, Niels Bohr and the Development of
(1998) 011, [hep-ph/9805442]. Physics, Pergamon Press. London (1955) .
[17] A. Maiezza and J. C. Vasquez, Resurgence of the QCD [33] J. M. Cornwall, Dynamical Mass Generation in
Adler function, Phys. Lett. B 817 (2021) 136338, Continuum QCD, Phys. Rev. D 26 (1982) 1453.
[2104.03095]. [34] J. Papavassiliou and J. M. Cornwall, Coupled fermion
gap and vertex equations for chiral-symmetry breakdown6
in qcd, Phys. Rev. D 44 (Aug, 1991) 1285–1297. [51] A. Czarnecki and W. J. Marciano, The Muon
[35] A. Deur, S. J. Brodsky and G. F. de Teramond, The anomalous magnetic moment: Standard model theory
QCD Running Coupling, Nucl. Phys. 90 (2016) 1, and beyond, in 5th International Symposium on
[1604.08082]. Radiative Corrections: Applications of Quantum Field
[36] Muon g − 2 Collaboration collaboration, B. Abi, Theory to Phenomenology, 9, 2000. hep-ph/0010194.
T. Albahri, S. Al-Kilani, D. Allspach, L. P. Alonzi, [52] A. Czarnecki and W. J. Marciano, The Muon anomalous
A. Anastasi et al., Measurement of the positive muon magnetic moment: A Harbinger for ’new physics’, Phys.
anomalous magnetic moment to 0.46 ppm, Phys. Rev. Rev. D 64 (2001) 013014, [hep-ph/0102122].
Lett. 126 (Apr, 2021) 141801. [53] M. Knecht, The Anomalous magnetic moment of the
[37] J. P. Miller, E. de Rafael and B. L. Roberts, Muon muon: A Theoretical introduction, Lect. Notes Phys.
(g-2): Experiment and theory, Rept. Prog. Phys. 70 629 (2004) 37–84, [hep-ph/0307239].
(2007) 795, [hep-ph/0703049]. [54] M. Davier and W. J. Marciano, The theoretical
[38] J. P. Miller, E. de Rafael, B. L. Roberts and prediction for the muon anomalous magnetic moment,
D. Stöckinger, Muon (g-2): Experiment and Theory, Ann. Rev. Nucl. Part. Sci. 54 (2004) 115–140.
Ann. Rev. Nucl. Part. Sci. 62 (2012) 237–264. [55] F. Jegerlehner and A. Nyffeler, The Muon g-2, Phys.
[39] A. Keshavarzi, W. J. Marciano, M. Passera and Rept. 477 (2009) 1–110, [0902.3360].
A. Sirlin, Muon g − 2 and ∆α connection, Phys. Rev. D [56] T. Aoyama et al., The anomalous magnetic moment of
102 (2020) 033002, [2006.12666]. the muon in the Standard Model, Phys. Rept. 887
[40] S. Borsanyi et al., Leading hadronic contribution to the (2020) 1–166, [2006.04822].
muon magnetic moment from lattice QCD, Nature 593 [57] G. Cvetič and R. Kögerler, Lattice-motivated QCD
(2021) 51–55, [2002.12347]. coupling and hadronic contribution to muon g − 2, J.
[41] S. G. Gorishnii, A. L. Kataev and S. A. Larin, The Phys. G 48 (2021) 055008, [2009.13742].
O(αs3 )-corrections to σtot (e+ e− → hadrons) and [58] B. e. Lautrup, A. Peterman and E. de Rafael, Recent
Γ(τ − → ντ + hadrons) in QCD, Phys. Lett. B 259 developments in the comparison between theory and
(1991) 144–150. experiments in quantum electrodynamics, Phys. Rept. 3
[42] L. R. Surguladze and M. A. Samuel, Total hadronic (1972) 193–259.
cross-section in e+ e- annihilation at the four loop level [59] M. Davier, A. Hocker and Z. Zhang, The Physics of
of perturbative QCD, Phys. Rev. Lett. 66 (1991) Hadronic Tau Decays, Rev. Mod. Phys. 78 (2006)
560–563. 1043–1109, [hep-ph/0507078].
[43] A. L. Kataev and V. V. Starshenko, Estimates of the [60] B. R. Webber, Estimation of power corrections to
higher order QCD corrections to R(s), R(tau) and deep hadronic event shapes, Phys. Lett. B 339 (1994)
inelastic scattering sum rules, Mod. Phys. Lett. A 10 148–150, [hep-ph/9408222].
(1995) 235–250, [hep-ph/9502348]. [61] A. V. Manohar and M. B. Wise, Power suppressed
[44] L. N. Lipatov, Multi-Regge Processes and the corrections to hadronic event shapes, Phys. Lett. B 344
Pomeranchuk Singularity in Nonabelian Gauge (1995) 407–412, [hep-ph/9406392].
Theories, in Proceedings, XVIII International [62] G. P. Korchemsky and G. F. Sterman, Nonperturbative
Conference on High-Energy Physics Volume 1: July corrections in resummed cross-sections, Nucl. Phys. B
15-21, 1976 Tbilisi, USSR, pp. A5.26–28, 1976. 437 (1995) 415–432, [hep-ph/9411211].
[45] A. Maiezza and J. C. Vasquez, On Haag’s Theorem and [63] Y. L. Dokshitzer and B. R. Webber, Calculation of
Renormalization Ambiguities, Found. Phys. 51 (2021) power corrections to hadronic event shapes, Phys. Lett.
80, [2011.08875]. B 352 (1995) 451–455, [hep-ph/9504219].
[46] M. Neubert, Scale setting in QCD and the momentum [64] R. Akhoury and V. I. Zakharov, On the universality of
flow in Feynman diagrams, Phys. Rev. D 51 (1995) the leading, 1/Q power corrections in QCD, Phys. Lett.
5924–5941, [hep-ph/9412265]. B 357 (1995) 646–652, [hep-ph/9504248].
[47] Y. L. Dokshitzer, G. Marchesini and B. R. Webber, [65] P. Nason and M. H. Seymour, Infrared renormalons and
Dispersive approach to power behaved contributions in power suppressed effects in e+ e- jet events, Nucl. Phys.
QCD hard processes, Nucl. Phys. B 469 (1996) 93–142, B 454 (1995) 291–312, [hep-ph/9506317].
[hep-ph/9512336]. [66] M. Beneke and V. M. Braun, Heavy quark effective
[48] G. Cvetic and C. Valenzuela, An Approach for theory beyond perturbation theory: Renormalons, the
evaluation of observables in analytic versions of QCD, pole mass and the residual mass term, Nucl. Phys. B
J. Phys. G 32 (2006) L27, [hep-ph/0601050]. 426 (1994) 301–343, [hep-ph/9402364].
[49] G. Cvetic and C. Valenzuela, Various versions of [67] I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev and A. I.
analytic QCD and skeleton-motivated evaluation of Vainshtein, The Pole mass of the heavy quark.
observables, Phys. Rev. D 74 (2006) 114030, Perturbation theory and beyond, Phys. Rev. D 50
[hep-ph/0608256]. (1994) 2234–2246, [hep-ph/9402360].
[50] G. Cvetic, Techniques of evaluation of QCD low-energy [68] U. Aglietti and Z. Ligeti, Renormalons and confinement,
physical quantities with running coupling with infrared Phys. Lett. B 364 (1995) 75, [hep-ph/9503209].
fixed point, Phys. Rev. D 89 (2014) 036003, [1309.1696].You can also read
