Achieving effective renormalization scale and scheme independence via the Principle of Observable Effective Matching (POEM) - arXiv
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Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
Achieving effective renormalization scale and scheme
independence via the Principle of Observable Effective
Matching (POEM)
Farrukh A. Chishtie1,a
1
Department of Applied Mathematics, The University of Western Ontario, London, ON N6A 3K7, Canada
arXiv:2005.11783v8 [hep-ph] 24 Jan 2021
Received: date / Accepted: date
Abstract In this work, we explicate a new approach in its application for obtaining ETOs for predicting RSS
for eliminating renormalization scale and scheme (RSS) independent observables across domains of high energy
dependence in physical observables. We develop this theory and phenomenology, as well as other areas of
approach by matching RSS dependent observables to fundamental and applied physics, such as cosmology,
a theory which is independent of both these forms of statistical and condensed matter physics.
dependencies. We term the fundamental basis behind
Keywords Perturbation Theory, Resummation
this approach as the Principle of Observable Effective
Methods, Perturbative QCD
Matching (POEM), which entails matching of a scale
and scheme dependent observable with the fully physi-
cal and dynamical scale (PS) dependent theory at the 1 Introduction
tree and at higher loop orders at which RSS indepen-
dence is guaranteed. This is aimed towards achieving In perturbative Quantum Field Theory (QFT), infini-
so-called “effective” RSS independent expressions, as ties are encountered in theoretical expressions of physi-
the resulting dynamical dependence is derived from a cal observables which require renormalization techniques.
particular order in RSS dependent perturbation the- Rendering finiteness to such quantities, such as physi-
ory. With this matching at a PS at which the cou- cal cross sections and decay rates introduces renormal-
pling (and masses) are experimentally determined at ization scale and scheme (RSS) dependencies. In con-
this scale, and we obtain an “Effective Theoretical Ob- trast to such dependencies, it is surmised and observed
servable (ETO)”, a finite-order RSS independent ver- that at increasing order in the expansion of a small
sion of the RSS dependent observable. We illustrate expansion parameter, along with convergence such de-
our approach with a study of the cross section Re+ e− pendencies would be reduced and the penultimate ex-
for e+ e− → hadrons, which is demonstrated to achieve pression to be have full dependence only on dynami-
scale and scheme independence utilizing the 3- and 4- cal physical scales. The issue of RSS dependencies are
loop order M S scheme expression in QCD perturba- treated via the Principle of Minimal Sensitivity (PMS)
tion theory via matching at both one-loop and two- [1,2]. Other approaches to reducing and/or eliminating
loop orders for obtaining the ETO. At the tree-loop scheme dependence includes the approaches of Effec-
matching, we find the value of the RSS independent tive Charge [3], Brodsky-Lepage-Mackenzie (BLM)[4],
coupling a4Lef f (Λef f ) at the cutoff Λef f of the ETO Renormalization Group (RG) summation [5], RG sum-
observable, which is RSS independent. With two-loop mation with RS invariants [6-9], Complete Renormal-
matching, we obtain an ETO prediction of 11 3
Reef+fe− = ization Group Improvement (CORGI) [10,11], Principle
+0.0006
1.052431−0.0006 at Q = 31.6GeV , which is in excellent of Maximal Confomality (PMC) [12] and sequential ex-
agreement with the experimental value of 11 3
Reexp
+ e− = tended BLM (seBLM) [13]. In this work, we detail a
+0.005
1.0527−0.005 . Given its new conceptual basis, ease of use new and alternate approach towards achieving RSS in-
and performance, we contend that POEM be explored dependence, inspired by Effective Field Theory (EFT)
techniques. As such, it is based on what we term as the
a
e-mail: fachisht@uwo.ca “Principle of Observable Effective Matching” (POEM)
2
and our work is applicable for achieving RSS indepen- RSS independence holds true at the tree level when
dence only up to a fixed order in perturbation theory, no quantum corrections exist and this is also valid at
hence we use the term “effective” for the observable the some r-loop level at some PS (typically this is at
having only Physical Scale (PS) dependence which is one and two loop level). With these two properties in
also the scale at which matching occurs. We term the mind, we propose two conditions which can be applied
derived perturbative expression as an “Effective Theo- to achieve an effective RSS independence at an order-
retical Observable (ETO)”, which implies these caveats. by-order basis for truncated expressions of a physical
The paper is outlined as follows: we introduce POEM observable. These are:
next and then work out the RSS independent Quan-
tum Chromodynamics (QCD) cross section Re+ e− via (tree)
On(k) (aef f (Λef f ), mef f (Λef f ), Λef f , Q, M ) = Oef f (2)
POEM at the tree, one- and two-loop matching with a
RSS independent scheme in the following sections.
∗ ∗ ∗
On(k) (a(k) (k)
n (Q ), mn (Q ), Q , Q, M ) (3)
(r)
2 The Principle of Observable Effective = Oef f (aef f (Q∗ ), mef f (Q∗ ), Q∗ , Q, M )
Matching (POEM)
Here, Oef f is conceptualized in a finite-order per-
Physical observables computed to all orders in pertur- turbative physical representation termed as the “Effec-
bative QFT within various RSS approaches are expected tive Theoretical Observable” (ETO) which is truncated
to result in scale and scheme independence if results at at r-loop order in order to be consistent with RSS in-
all orders are computed, and hence must be the same dependence, and is dependent only on PSs. Since it is
in this limit. This is the starting point for POEM which known that RSS independence is guaranteed trivially at
can be expressed as: the tree and at higher r-loop levels at some PS, hence
this forms the basis of these equations. We can there-
fore achieve RSS independence by requiring in eq. 2,
lim On(i) (a(i) (i) (i) (i) (i)
n (µ ), mn (µ ), µ , Q, M ) (1) a dynamical scheme independent cutoff scale at renor-
n→∞
(j) (j) (j)
malization µ = Λef f for the physical observable in the
= lim Ol (al (µ(j) ), ml (µ(j) ), µ(j) , Q, M ) RSS dependent expression on the left hand side. This
l→∞
= Of ull (a(Q), m(M ), Q, M ) scale denotes the point at which quantum corrections to
the observable appear. As it is dependent on dynamical
Here O denotes perturbative contributions to a phys- scales, this RSS independent cutoff scale is indeed dis-
ical observable such as in a decay rate or a cross-section tinct from typical renormalization coupling and mass
computed in two schemes denoted by the superscript i cutoffs, which are scheme dependent (see [14] for anal-
and j at some loop-order n and l respectively, while a ysis on RS dependence of the strong coupling constant
and m denote coupling and mass respective tied to re- and its cutoff ΛQCD ). Equation 3 denotes another RSS
spective renormalization scales µi and µj . Q and m de- independence requirement whereby matching is done at
note dynamical PS. Eq. 1 can be further generalized by a PS Q∗ at r-loop, via an effective RSS independent dy-
including multiple couplings and masses as well as fac- namical coupling and mass depending on PS, which are
torization scales and momentum fractions. This equa- denoted as aef f and mef f respectively in both equa-
tion nevertheless implies that if O is computed at a tions; though in the RSS dependent expression these
certain large limit of perturbation theory, this results are computed via a truncated order, n, thereby denot-
in a RSS independent result with dependence only on ing the effective nature of these RSS independence con-
Q and M . While it is rather challenging to find expres- ditions. Scale independence is acheived via matching at
sions to all orders in perturbation theory, we consider a physical point Q∗ , whereby the couplings and masses
such to be a RSS independent theory or full theory are referenced to observed values which ultimately ren-
whereby all implicit and explicit renormalization scale ders the ETOs free from unphysical scale ambiguity.
within any scheme cancels, and the equalization across Scheme independence in aef f and mef f is done using
in eq. 1 implies an overall RSS independence leading to matching whereby explicit dependence is absorbed into
an independent observable termed as Of ull dependent these running parameters and implicit dependence is
only on scale independent coupling and mass, as well eliminated by restricting their running to r-loops, hence
as physical scales, Q and M . we use truncated RG functions at this order.
While this equation holds for a theory computed at As such, eq. 2 and 3 allow for a practical realization
all orders, for practical purposes, and as criteria to de- of eq. 1 at finite order of perturbation theory for our
rive an effective version of such an all-orders observable, proposed implementation of the POEM approach, and3
imply RSS independence in the ETO to hold both at an “effective dynamical renormalization” (EDR) of the
the matching scale Q∗ , and for dynamical degrees of RSS dependent physical observable via matching (and
freedom, Q and m, which can be explicitly stated as: resulting resummation) which renders it completely dy-
(tree) (tree) (r) (r)
namical, with the ETO tied to the physical relevance of
∂Oef f ∂Oef f ∂Oef f ∂Oef f the scale of matching with applicability to the underly-
= = = =0 (4)
∂µ ∂ci ∂µ ∂ci ing theory while preserving physical degrees of freedom.
where µ is the renormalization scale, while ci are renor- The advantage of using our approach over using fixed
malization scheme dependent RG coefficients. order perturbation theory is that both explicit and im-
plicit dependence on renormalization scale and scheme
We remark here that eq. 4 is due to application
is eliminated at an order-by-order basis via the POEM
of POEM and the resulting resummed expressions or,
(r) matching process and the derived ETOs are resummed
ETOs, Oef f . In contrast, the approach of PMS [1, 2] is
expressions, that are referenced at a physically relevant
applied to fixed order perturbative expressions to find
matching scale.
optimal scales. It is also to be remarked that for the
case of purely loop-mediated processes, eq. 2 will not Typically, EFT techniques are applied at the level of
hold, rather eq. 3 holds, as there are no tree-level con- operators instead of observables, and deal with physical
tributions. In equations 2 and 3, the k subscript denotes degrees of freedom, and dealing with RSS dependence
a particular scheme for a truncated observable O at a as auxiliary variables. Therefore, broadly speaking, our
loop order n (for example the M S scheme) which is approach via the POEM is to bridge conventional renor-
matched at the tree and r-loop level with an effective malization techniques with EFT techniques, with a first
version of the fully PS dependent version Oef f , whereby focus on observables in this work. Since POEM and the
r is to be chosen at the highest loop-order at which matching of observables to achieve RSS independence
scheme independence holds. The rationale behind this is unprecedented, we demonstrate its efficacy in this
requirement is that this allows matching at a perturba- approach first and we will explicate the construction
tive order at which this scheme independence require- of ETOs from the RSS independent Lagrangian of the
ment holds fully at the highest order of perturbation underlying EFT in a separate work [28].
theory, hence results derived are then more accurate We demonstrate the application of POEM concretely
than at lower orders where scheme independence may with the case of the cross section Re+ e− , for the e+ e− →
still hold. Typically, for most observables, r is either hadrons, where the matching scale Q∗ is chosen at the
at one-loop or two-loops of perturbation theory. In the Z-pole mass with five-active quarks, not only because
case study below, we study a QCD observable which the strong coupling constant is determined via experi-
is scheme independent at 2-loops, however, we do com- ment at this PS in the M S scheme, but also that at the
pute the results at both r = 1 and r = 2 loop orders center of mass energies, Q under consideration are well
and illustrate as to why matching at the latter order above the b-quark threshold mass and are within the
yields slightly different and better results. limits of applicability of the QCD as an EFT. It must
Since the POEM is based on EFT and focused di- be emphasized that in the case of QCD the strong cou-
rectly on the observable itself, it must also be noted that pling constant in M S is determined at the Z-pole mass
in this approach, the PS or reference matching scale Q∗ for the reason that at this physical scale it is most accu-
is not arbitrary, but is rather based on the physical ob- rate due to the EFT representing free quarks and glu-
servable, the relevant physical process and the EFT un- ons (rather than at lower energies), and hence a clean
der study. The value of the coupling (and masses) must determination of this strong coupling constant is most
be experimentally determined at this physical scale, and feasible here. With this choice for QCD, and the mo-
the range of applicability cannot exceed beyond the rel- tivation for POEM to yield physical RSS independent
evant physical degrees of freedom or go beyond pertur- results. Hence, more generally when a physical refer-
bative cutoff of the theory. Moreover, the resummation ence matching scale Q∗ and experimentally determined
that happens due to POEM is, in analogy, “integrates coupling is chosen and matching is done via POEM,
out” the unphysical RSS dependencies from the observ- this ends the µ dependence to the physical scales, Q
able to an independent ETO. In regards to the arbitrary and M . Hence, with this matching to relevant physical
renormalization scale µ dependence, by matching and scales there is no dependence of the ETO’s on Q∗ itself.
referencing the EFT at a physical scale Q∗ at which the This is expressed as,
coupling (and masses) are determined via experiment,
via POEM leads to scale independence in the observable
(tree) (r)
as the observable is now fully renormalized to the dy- ∂Oef f ∂Oef f
namical scale Q. As such, with POEM we also perform = =0 (5)
∂Q∗ ∂Q∗4
It follows that if there is a different matching scale and hence no relative scales exist for the ETO. After
chosen based from the initial physical matching refer- achieving RSS invariance, there are no issue of renor-
ence scale such that, malons [16, 17] encountered as well.
Q∗ → Q∗0 = kQ∗ , (6) 3 Attaining effective RSS independence: Re+ e−
then the dynamical scale must also re-scale by the Matching at Tree and One-Loop Order
same constant k as follows,
The form of the cross section Re+ e− is given by 3( i qi2 )(1+
P
R) where R at n-loop order has a perturbative contri-
Q → Q0 = kQ. (7) bution of order an+1 in
∞ ∞ X n
This is consistent with eq. 5, or in other words being
X X
R = Rpert = rn an+1 = Tn,m Lm an+1 (8)
consistent with the reference physical scale of Q∗ (and n=0 n=0 m=0
the underlying measurement of couplings and masses µ
at this scale). It maybe argued that Renormalization with L ≡ b ln( Q ), Q2
being the center of mass momen-
Group (RG) functions maybe used to evolve the cou- tum squared.
pling and masses using Q∗ to another matching scale, The explicit dependence of R on the renormaliza-
0
Q∗ , however, the ETO results would be invariant to tion scale parameter µ is compensated for by implicit
this shift as the reference matching scale has not changed dependence of the “running coupling” a(µ2 ) on µ,
(as it used as an initial value of the coupling evolu- ∂a
µ2 = β(a) = −ba2 1 + ca + c2 a2 + . . .
(9)
tion), and any rescaling of this physical scale would ∂µ2
then result in a compensatory rescaling in Q as eqs.
where a ≡ αs (µ2 )/π while αs is the QCD strong
5-7 indicate. The RSS independent ETO derived will
coupling constant.
be, of course, limited in terms of its range of applica-
For five active flavors of quark, we have,
bility given the reasons for the choice of Q∗ , and here
we explore this in detail with QCD as an underlying 3
R + − = 1 + Rpert . (10)
EFT with physical energies above the b-quark mass. 11 e e
As another example, if a physical observable is related
The choice of the number of flavors is based on the cen-
to lower energies is taken much below this threshold,
ter of mass energies Q (for which the ETO will be de-
then the matching scale Q∗ can be chosen at which the
rived) which namely for those above the b-quark mass.
strong coupling is experimentally measured at the tau
In the M S renormalization scheme [18] we have,
mass, and for 3 to 4 active quark flavors.
We also remark that our approach is distinct from
the Effective Charge (EC) approach [3], as in this method b = 23/12, c = 29/23, c2 = 9769/6624, (11)
there is matching done at one-loop and at a scale in
c3 = 9.835917120
which renormalization logarithms are set to zero, which
only allows renormalization scheme independence, while where the values of b and c are the same in any mass–
renormalization scale dependence remains in resulting independent renormalization scheme while the values of
expressions. Via POEM, and for deriving a resultant c2 and c3 in eqs. (11) are particular to the M S scheme.
ETO, the matching at the tree and at r-loop order at Furthermore, we find in ref. [19,20] that in the M S
a PS, we overcome these limitations, thereby achieving scheme
RSS independence simultaneously. In the case of the
QCD cross section ratio, Re+ e− studied here, we have T0,0 = 1, T1,0 = 1.4097, T1,1 = 2
an observable which is scheme-independent up to two- T2,0 = −12.76709, T2,1 = 8.160539, T2,2 = 4
loops, and we find that matching at this higher loop T3,0 = −80.0075, T3,1 = −66.54317, T3,2 = 29.525095
yields better results as compared to 1-loop matching, T3,3 = 8 (12)
hence, we demonstrate the advantages of matching at
a higher loop order with RSS independent results un- At the Z-pole mass (MZ = 91.1876 GeV ), we have
like the EC approach which are scale dependent, and [21] in the M S scheme, (with a(µ) defined by eq. (9)),
are fixed at one-loop order for all observables. More-
π aM S (MZ ) = 0.1179 ± 0.001. (13)
over, POEM based results are RSS independent, hence,
there are no “commensurate scale relations” [15] as the Utilizing eq. 2 and 3, we therefore find the following
observable is dynamical with respect to physical scales, POEM based relations:5
is a one-loop beta-function solution in the dynamical
∗
3 MS ETO with an initial value set at a1L ef f (µ0 = Q ) =
Re+ e− (aef f (Λef f ), Λef f ) 1L ∗
aef f (Q , Q), which in our case is given by eq. 16.
11
3 ef f Since we are at the one-loop matching, we find the
= 1 + Rpert (aef f (Λef f ), Λef f ) = R e+ e− = 1 one-loop solution to aef f (Q) which leads to the pre-
11
3 ef f +0.0007
(14) diction of 11 Re+ e− = 1.056943−0.0007 from eq. 15, at
Q = 31.6GeV , which is in excellent agreement with the
experimental value of of 11 3
Reexp +0.005
+ e− = 1.0527−0.005 [22].
ef f
3 MS We plot the Q dependence of R at 1-loop matched
Re+ e− (Q∗ , Q) = 1 + Rpert (Q∗ , Q) with 3- and 4-Loop M S in Figure 1, where the both
11
3 ef f results are similar. At the same time perturbative QCD
= Re+ e− (Q∗ , Q) = 1 + a1L ∗
ef f (Q , Q) (pQCD) predictions are found for renormalization scales,
11
(15) µ = MZ /2, µ = MZ and µ = 2MZ respectively, which
are all underestimates as compared to POEM based
From equation 14, at 3-loops in M S, we find a3L ef f (Λ ) =
ef f ETO results, and also indicate high uncertainty across
4L this range due to RSS dependence.
0.3404702, while for 4-loops, we obtain aef f (Λef f ) =
0.2094525. These values represent the value of the RSS
independent coupling aef f at a cutoff scale Λef f at
which no quantum corrections exist for the ETO, as
this is matching done at the tree loop level. The cut-
off scale (Λef f ), though scale and scheme-independent
cannot be determined from the value of the coupling
aef f (Λef f ).
With equation 15, which is matching done at the
one-loop level, we choose the matching point Q∗ =
MZ = 91.1876GeV and also subtract the only explicit
scheme dependent term, appearing in the expression
for T31 , which is 2c2 in the M S expression, rendering
the overall matching expression RSS independent. With
the central value of the strong coupling aM S = 0.1179 π
from eq. 12, which we substitute into eq. 15, we derive
a1L
ef f (MZ , Q) as follows:
MZ
a1L
ef f (M Z , Q) = 0.03868 + 0.0059614 ln
Q Fig. 1 The Q dependence of Renormalization Scale and
Scheme (RSS) independent Ref f at 1-loop matched with 3-
2 MZ 3 MZ
+0.0009918 ln + 0.0001117 ln and 4-Loop M S and compared to 4-Loop Perturbative QCD
Q Q
in M S scheme for Re+ e− referenced at renormalization scales,
(16) µ = MZ /2, µ = MZ and µ = 2MZ
The choice of Q∗ = MZ and the number of quarks,
nf = 5, is predicated on deriving the ETO for center
of mass energies Q above the b-quark mass (and also
below the t-quark threshold mass). 4 Two-Loop Matching
Since for the ETO we absorb higher order-loop con-
tributions from a RSS dependent scheme at one-loop Re+ e− is scheme-independent at 2-loops, hence this would
order for RSS independence, hence, Reef+fe− (Q) is ex- be applying POEM at the maximum loop order in this
pressed as, case. Utilizing eq. 4 at two-loop level, we find the fol-
3 ef f lowing scale and scheme independent POEM based re-
R + − (Q) = 1 + a1L
ef f (Q) (17) lation:
11 e e
where
3 MS 3 ef f
a1L ∗ R + − (Q∗ , Q) = 1 + Rpert (Q∗ , Q) = R + − (Q∗ , Q)
ef f (Q , Q) 11 e e 11 e e
a1L
ef f (Q) = Q ∗2 (18) ∗ ∗
1 − ba1L ∗
ef f (Q , Q) ln( Q2 ) = 1 + a2L 2L
ef f (Q , Q) + (T1,0 + T1,1 L)(aef f (Q , Q))
26
(19) show the behavior of RSS independent coupling aef f
at 1- and 2-Loop orders with respect to Q, in Figure
As in the previous section, we choose Q∗ = MZ , 3, which depict asymptotic freedom as is also seen in
eliminate the scheme dependent term, and find a2L ef f (MZ , Q)the ETO behavior in Figures 1 and 2, with both 3- and
from solving eq. 19 and obtain a RSS independent match- 4-Loop M S results close to each other, both at 1- and 2-
ing expression: loop ETO matched results, and within the experimental
4 M Z error bounds for the Q = 31.6GeV experimental value.
a2L
ef f (MZ , Q) = ±k1 [∓1 + {1.2181 + 0.0008939 ln
Q We note that the effective dynamical coupling aef f is
lower for 2-loops as compared to 1-loop matching as
MZ MZ
+0.008565 ln3 + 0.05328 ln2 depicted in Figure 3, which results in a better prediction
Q Q
3
of 11 Reef+fe− . In Figure 4, we show that at 2-loops r = 2
MZ 1/2 3
+0.3431 ln } ]. which is the highest loop order at which 11 Re+ e− is
Q
scheme independent, these ETO predictions are lower
(20)(and subsequently better) than those found at 1-loop.
h i−1 Both results derived here are within the experimental
Here, k1 = 2{1.4097 + 2 ln MQZ } . Since in the
errors with the 2-loop (r = 2) providing much better
ETO, namely, Reef+fe− we absorb higher order-loop con- agreement with data as compared to the 1-loop ETO.
tributions from a RSS dependent scheme at r = 2 (two- Based on these findings, we therefore surmise matching
loop orders) for RSS independence, hence, Reef+fe− (Q) is at the highest available order at which the observable
expressed as: is scheme-independent to yield best results.
With respect to considerations of higher order per-
3 ef f turbative corrections, we find that both the 1- and 2-
R + − (Q) = 1 + a2L 2L
ef f (Q) + T1,0 aef f (Q)
2
(21)
11 e e loop ETOs yield results which are nearly identical for
both 3- and 4-loop M S expressions. This is shown in
where
both Figures 1 and 2. For Q = 31.6GeV , for the central
1 value of the strong coupling constant, for the 2-loop
a2L
ef f (Q) = − h i (22)
c W−1 − exp(f
c
)
+ 1 ETO, we find for 3-loops 11 3
Reef+fe− = 1.057040 which
is 0.006% higher than the 4-loop prediction 1.052431
where (stated earlier). Using comparison of 3-loop with 4-
loop results for both 1 and 2 loop ETO indicate that
h i
∗ Q∗2
a2L
ef f (Q, Q ) b ln( Q2 ) + c ln(h) − c − 1
f= (23) the ETOs derived via POEM are highly convergent
ca2L ∗
ef f (Q, Q ) (since these are free from RSS dependence and also
and from renormalons). Hence higher order corrections in
∗ M S, as in 5-loop orders and beyond would contribute
ca2L
ef f (Q, Q ) − 1
h= . (24) negligibly for the highly convergent ETOs derived in
a2L ∗
ef f (Q, Q ) this work.
Eqs. 22-24 represent an exact closed form solution In comparison with our findings, Akrami and Mir-
of the two-loop beta-function for the ETO, which is jalili [25] present perturbative QCD, RG summation
expressed as a Lambert-W function. We utilize the ini- and RS invariants and CORGI approaches estimates of
tial value set at aef f (µ0 = Q∗ ) = aef f (Q∗ , Q), which R at Q = 31.6GeV to be 1.04617+0.0006 +0.00003
−0.0006 , 1.04711−0.00005
+0.0015
in our case is given by the positive root of eq. 20. At and 1.04615−0.0008 at 4-Loops respectively, which are
the two-loop ETO matching, we find that the predic- all underestimates, and fall outside of the experimen-
tion is nearly the same as the one-loop ETO, which is tal error bounds, in contrast to the results derived by
3 ef f +0.0006
11 Re+ e− = 1.052431−0.0006 from eq. 21 at Q = 31.6GeV , POEM. This pattern holds for other experimental mea-
which is in excellent agreement with the experimental sured values at Q = 42.5GeV and Q = 52.5GeV , which
value of 113
Reexp +0.005
+ e− = 1.0527−0.005 [22]. This is better are 1.0554 ± 0.2 [23] and 1.0745 ± 0.11 [24] respectively.
than our previously derived 1-loop matching ETO re- With POEM we find 1.047561+0.0005 +0.0005
−0.0005 and 1.044679−0.0005
sult. using the 2-Loop ETO matching. These predictions are
We plot the Q dependence of the ETO Ref f at 2- higher in value and more accurate than those reported
loop matching with 3- and 4-Loop M S in Figure 2 and for perturbative QCD, RG summation and RS invari-
find the same trends as 1-loop matching, with pQCD ants and CORGI approaches in [25]. However, our pre-
predictions lower than ETO results, though these are dictions are lower than the experimental values as they
slightly lower than the 1-loop ETO results. We also are based only on photon interactions, and a higher ac-7
curacy is expected when electroweak contributions aris-
ing from electron-positron annihilation to Z boson are
taken into account. We will address this in a future
work, which will also additionally address other pro-
cesses [27].
Fig. 4 The Q dependence of Renormalization Scale and
Scheme (RSS) independent 11 3
Reef+ fe− derived via matching
ETO at 1- and 2-Loop orders. Both ETO results are within
experimental error bounds with 2-Loop result yielding better
results.
Fig. 2 The Q dependence of Renormalization Scale and 5 Conclusions
Scheme (RSS) independent Ref f at 2-loop matched with 3-
and 4-Loop MS and compared to 4-Loop perturbative QCD
in M S scheme for Re+ e− referenced at renormalization scales,
In this work, we have introduced a new approach for
µ = MZ /2, µ = MZ and µ = 2MZ (dashed lines) achieving RSS independence via a principle termed as
POEM, which is based on equivalence of physical ob-
servables across both physical scale and scheme depen-
dencies in their perturbative content under certain lim-
its. Inspired by EFT techniques, which involve match-
ing at a physical scale, this combined and renormal-
ization approach, via EDR provides results as RSS in-
dependent ETO, which contain only physical scales at
fixed order of perturbation theory. In this work, we
demonstrate that POEM provides excellent results for
the QCD cross section ratio Reef+fe− and agreement with
experiment is better than other comparable estimates
based on perturbative QCD, RG summation and RS in-
variants and CORGI approaches [25]. Moreover, both
the 1- and 2-loop ETO’s show remarkable convergence
as both 3- and 4-loop provide results which are nearly
identical, which indicates the high convergence rate of
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Fig. 3 The Q dependence of Renormalization Scale and bution based approach [27]. In this work, we have only
Scheme (RSS) independent coupling aef f at 1- and 2-Loop
orders.
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