- Email: [email protected]
Renormalization scale setting for some QCD observable, based on the PMC and CORGI approaches
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Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98 www.elsevier.com/locate/nppp
Renormalization scale setting for some QCD observable, based on the PMC and CORGI approaches M. Akramia , A. Mirjalilia a Physics
Department, Yazd University, 89195-741,Yazd, Iran
Abstract In this paper we present the principle of Maximum conformality (PMC) and Complete Renormalization Group Improvement (CORGI) approaches of scale setting which are based essentially on Refs.[1–4]. The PMC establishes a systematic method to eliminate the renormalization scheme and renormalization scale dependencies of conventional perterbative QCD predictions. This approach, using the renormalization group equation, exposes a pattern of βi − terms in the perterbative series. Then by absorbing all βi − terms into the running coupling constant, one can obtain a result that is independent of renormalization scale and scheme. On the other hand the CORGI approach improves the precision of calculations. In this approach the standard perturbative series of QCD observable reconstructed in terms of scheme-invariant quantities. Here, we will examine the PMC and CORGI approaches by comparing its predictions for the electron-positron annihilation to hadrons and the Higgs decay width to gluon. The two results are compared with each other and with the available experimental data. At the end we may conclude which approach has more advantage in considering the perturbative series. Keywords: Conformal and nonconformal terms, Renormalization Group, Scale and scheme invariants.
1. Introduction All physical predictions in QCD should be invariant under any choice of renormalization scale and renormalization scheme. Thus, a key problem in making proper perterbative QCD (pQCD) predictions is how to set the renormalization scale of the running coupling constant at each order. Here we pay attention two approaches, the PMC and CORGI ones. In the PMC approach after proper scale setting , all non-conformal βi -terms in the perturbative expansion are absorbed into the running coupling constant. So the remaining terms in the perturbative series are identical to that of a conformal theory, ∗ Talk given at 21th International Conference in Quantum Chromodynamics (QCD 18, 33th anniversary), 3–6 july 2018, Montpellier FR Email addresses: [email protected] (M. Akrami), [email protected] (A. Mirjalili)
https://doi.org/10.1016/j.nuclphysbps.2018.12.017 2405-6014/© 2018 Elsevier B.V. All rights reserved.
i.e., the theory with βi = 0. To improve and increase the ability of calculations, we also try to use the CORGI approach. In this approach the standard perturbative series of QCD observable is reconstructed in terms of schemeinvariant quantities. So it is expected to get more reliable results with respect to what we obtain in conventional QCD approach. 2. An overview of the principle of maximum conformality The main step of this approach is to use an arbitrary dimensional renormalization scheme such as Rδ scheme. Here an arbitrary constant −δ is subtracted in addition to the standard subtraction ln 4π − γE that used in MS scheme [1]. Using the RG-equation and its degeneracy relations, one can identify which terms in the perturbative series are associated with the QCD
94
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
β-function and which terms remain in β = 0 , i.e, conformal terms. The optimal renormalization scales then are determined by absorbing all βi terms into the running coupling constant at each order of perturbation theory. The final resummed result is a conformal series. On the other words when the renormalization scales at each perturbative order are set properly, all nonconformal terms in a perturbative expansion that arising from renormalization are summed into the running coupling. The PMC scales for different Rδ -schemes(Rδ1 and δ1 −δ2 Rδ2 ) differ only by a factor e 2 . Therefore the β- function in any Rδ -scheme is the same as in MS [1]. Using the displacement relation between couplings in any Rδ -
scheme, one finds [1]: a(μ0 ) = a(μδ ) +
∞ 1 dn a(μ) | (−δ)n , 2 n μ=μδ n! (d ln μ ) 0 n=1
(1)
where ln μ20 /μ2δ = −δ. Now in order to employ practically the PMC approach, an observable in pQCD is considered as it follows: ρ0 (Q2 ) = a(μ0 )n
∞
rk+1 (Q2 /μ20 ) a(μ0 )k ,
(2)
k=0
where μ0 is an initial renormalization scale. By expanding Eq.(1) up to k + 3 and substituting it in Eq.(2) for the observable ρ, the following result is obtained [1]:
n(n + 1) 2 2 β0 r1 δ1 ]a3 (μδ )n+2 + ρδ (Q2 ) = r1 a1 (μδ )n + r2 + nβ0 r1 δ1 a2 (μδ )n+1 + [r3 + nβ1 r1 δ1 + (n + 1)β0 r2 δ2 + 2 n(3 + 2n) (n + 1)(n + 1) 2 2 [r4 + nβ2 r1 δ1 + (n + 1)β1 r2 δ2 + (n + 2)β0 r3 δ3 + β0 β1 r1 δ21 + β0 r2 δ2 2 2 n(n + 1)(n + 2) 3 2 + β0 r1 δ1 ]a4 (μδ )n+3 . 3! (3)
This equation consist of a pattern of {βi }-terms in the coefficients at each order. Here one can be concluded
that some of the coefficients of the {βi } terms are degenerate. Thus the observable can be written as follow:
n(n + 1) 2 β0 r3,2 ]a(Q)n+2 2 n(3 + 2n) (n + 1)(n + 2) 2 β0 β1 r3,2 + β0 r4,2 +[r4,0 + nβ2 r2,1 + (n + 1)β1 r3,1 + (n + 2)β0 r4,1 + 2 2 n(n + 1)(n + 2) 3 β0 r4,3 ]a(Q)n+3 + ... , + 3!
ρ(Q)2 = r1,0 a(Q)n + [r2,0 + nβ0 r2,1 ]a(Q)n+1 + [r3,0 + nβ1 r2,1 + (n + 1)β0 r3,1 +
(4)
where ri,0 are conformal and ri, j for j 0 are nonconformal parts of the perturbative coefficients, i.e. ri = ri,0 + O({βi }). One can resumes all ri,1 terms to all orders by defining new scale Qi at each order as follows [1]: rk,0 a(Qk )k = rk,0 a(Q)k − ka(Q)k−1 β(a)rk+1,1 .
(5)
By expanding Eq.(1) one finds the following relation:
a(Qk )k = a(Q)k + ka(Q)k−1 β(a) ln [β
Q2k k + a(Q)k−2 Q2 2
Q2 ∂β a(Q) + (k − 1)β(a)2 ]ln2 k2 + ... . ∂a Q (6)
Substituting Eq.(6) into Eq.(5), one can find the fol-
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
lowing scale Qi at each order. The results for the leading logarithm order (LLO), Next-to-NLO (NLLO) and finally Next-to-NLO (NLLO) are respectively as following [2]: ln
ln
Q2k,LLO Q2
ln
Q2k,LLO Q2
=
95
−rk+1,1 /rk,0 1+
1 ∂β 2 [ ∂a
+ (k − 1) βa ](−
rk+1,1 rk,0 )
, (8)
rk+1,1 =− , rk,0
Q2k,NNLLO Q2
=
(7)
−rk+1,1 /rk,0 1 + 12 [ ∂β ∂a + (k −
r 1) βa ](− k+1,1 rk,0 )
+
∂2 β 1 3! [β ∂a2
By substituting Eq.(5) in Eq.(4) and replacing new scales Qi at each order the non-conformal parts are disappeared and one can obtain a scheme-
2
− 12 ( ∂β ∂a ) −
2 r (k−1)(k+1) β2 ](− k+1,1 2 rk,0 ) a2
.
(9)
independent conformal series [2] as it follows
ρ(Q2 ) = r1,0 a(Qn,NNLLO )n + r2,0 a(Qn+1,NLLO )n+1 + r3,0 a(Qn+2,LLO )n+2 + r4,0 a(Q)n+3 + ... .
(10)
2.1. Electron-positron annihilation in PMC approach The annihilation of an electron and positron into hadrons, Re+ e− , is defined as [1]:
Re+ e− (Q) = γ0 + γ1 a(Q) + [γ2 + β0 Π1 ]a(Q)2 + [γ3 + β1 Π1 + 2β0 Π2 − β20
π2 γ1 ]a(Q)3 3
π2 γ1 π2 γ2 5 − 3β20 − β30 π2 Π1 ]a(Q)4 , +[γ4 + β2 Π1 + 2β1 Π2 + 3β0 Π3 − β0 β1 2 3 3 (11)
3 where Re+ e− (Q) = 11 Re+ e− (Q). This equation is based on anomalous dimension ,γi , and vacuum polarization function, Πi [5–7]. If the following identifications for the coefficients are used then Eq.(11) can be written in the form of Eq.(4):
ri,0 = γi
ri,1 = Πi−1
π γi−2 3 = −π2 Πi−3
ri,2 = − ri,3
, 2
i ≥ 2,
i ≥ 3,
By using Eqs.(7,8,9) we get the following numerical results for Qi which are closed with the numerical results, reported in [1]: Q1 = 44.72 GeV , Q2 = 37.86 GeV , Q3 = 168.68 GeV , Q4 = 31.6 GeV . (13) √ Taking s = 31.6 GeV and Λ MS = 419 MeV with five active quark flavors, the following numerical result for Re+ e− observable is obtained: 3 Re+ e− (Q) = 1 + α(Q1 ) 11 +1.84α(Q2 )2 − 1.00α(Q3 )3 − 10.82α(Q4 )4 . Re+ e− (Q) =
i≥4. (12)
(14)
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
96
In Eq.(14) the numerical coefficient of fifth term is different with respect to [1] Using the Qi scales, represented by Eq.(13) and substituting them in Eq.(14), we get Re+ e− (Q) =
Γ(H → gg)
3 Re+ e− = 1.060 , 11
=
(15)
=
2.2. Higgs decay to gluon in the PMC approach The pQCD prediction for decay width of the Higgs decay to a pair of gulon can be written as [9]:
4G F MH3 c1,0 a2s (MH ) + (c2,0 + c2,1 n f ) a3s (MH ) + (c3,0 + c3,1 n f + c3,2 n2f ) a4s (MH ) √ 9 2π +(c4,0 + c4,1 n f + c4,2 n2f + c4,3 n3f ) a5s (MH ) + O(a6s ) ,
where G F is Fermi constant and MH is Higgs boson mass. The coefficients ci, j are given in [9]. Considering the conformal and non-conformal
Γ(H → gg)
which can be compared with the experimental value 3 +0.005 + − 11 Re e = 1.052−0.005 [8].
(16)
coefficients and following similar process as for Re+ e− to absorb all non-conformal {βi }-terms into the coupling constant, the attained result is:
4G F MH3 1 6.333 α2s (Q1 ) + 9.068 α3s (Q2 ) − 79.962 α4s (Q3 ) − 68.804 α5s (Q4 ) + O(α6s ) . √ 9 2π 1000 (17)
Using Eqs.(7,8,9) the following PMC scales Qi are obtained: Q1 = 23.10 GeV , Q3 = 31.46 GeV .
Q2 = 10.49 GeV
, (18)
What is numerically gotten for this decay is: Γ(H → gg) = 0.364 MeV which is comparable with the result of the conventional scale setting: 0.373 MeV [2]. 3. Overview on the CORGI approach In another approach so called Compleat Renormalization Group Improvement (CORGI), all perturbative terms are scheme independent. So there is no dependency on the renormalization (μ) or factorization scale (M) for these terms. We can assume an observable R(Q) in a standard approach has a perturbative series like:
Using the self-consistency requirement we have the following expressions for the partial derivatives of the rn with respect to the scheme parameters [10]: ∂r2 ∂r2 ∂r2 = 2r1 + c, = −1, = 0, . . . . ∂r1 ∂c2 ∂c3
(20)
By integrating these terms simultaneously, one finds [3]: r2 (r1 , c2 )
=
r3 (r1 , c2 , c3 )
=
r1 2 + cr1 + X2 − c2 , 5 r1 3 + cr1 2 + (3X2 − 2c2 )r1 + 2 1 (21) X3 − c3 , 2
where X2 , X3 are constants of integration and renormalization scheme (RS) invariants. By substituting Eq.(21) in Eq.(19) one finds:
R(Q) = a + r1 a2 + r2 a3 + . . . + rn an+1 + . . . , (19) 5 1 R(Q) = a + r1 a2 + (r12 + cr1 + X2 − c2 )a3 + (r13 + cr12 + (3X2 − 2c2 )r1 + X3 − c3 )a4 + . . . , 2 2
(22)
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
where a≡a(r1 , c2 , c3 , . . .) [3]. Here r1 at NLO approximation is known but X2 , X3 , . . . are unknown. All known terms in Eq.(22) at the NLO approximation can be summed to yield the following completed subset [3]:
97
Here a = απs . It is possible to have a relation such as Eq.(26) in Minkowski space [12]: ˜ dn an ) . (27) D(s) = a(1 + n>0
a0 ≡a + r1 a2 + (r12 + cr1 − c2 )a3 + (r13 + 5 2 1 cr1 − 2c2 r1 − c3 )a4 + . . . . (23) 2 2 By using the t’Hooft scheme (r1 = r2 = ... = rn = 0, c1 = c2 = ... = cn = 0) one finds a0 = a. The a0 can be written in terms of the Lambert W-function [11]. By doing the required calculations at higher orders, one arrives at [3, 4]: R(Q) = a0 +X2 a0 3 +X3 a0 4 +. . .+Xn a0 n+1 +. . . ,(24) Each term in Eq.(24) involves a resummation of infinite terms at specified order. For instance, the first term, a0 , is a representation of resummation over NLO contribution of all terms in Eq.(22) and the second term, X2 a30 , as a representation of resummation for the NNLO contribution .
The required coefficients di have been calculated in [13, ˜ is the Adler D-function and in the CORGI 14]. The D(s) approach it has the following expansion [4]: = a0 + X2 a3 + X3 a40 + ... + Xn an+1 D(s) . 0 0
The relation between di and ri coefficients is given by analytic continuation [15]. Numerical result for this √observable at the NNLO approximation is: 3 + − 11 Re e ( s = 31.6 GeV) = 1.051 which is very close to its experimental value: 1.052+0.005 −0.005 [8] and even better than the PMC result, given by Eq.(15).
3.2. Higgs decay to gg in CORGI approach The higgs decay width to a gluon-gluon is generally given by: Γ(H → gg) =
3.1. e+ e− annihilation in CORGI approach The concerned observable Re+ e− in S U(N) perturbtion theory is defined by [12]: ⎛⎜ ⎞⎟2 ⎟⎟ ⎜⎜ 3 2 ˜ ¯ ,(25) R(s) = N Q f 1 + C F R(s) +⎜⎜⎝⎜ Q f ⎟⎟⎠⎟ R(s) 4 f
f
where Q f denoting quark charges and C F is color factor.R˜ denotes the perturbative corrections to the parton model result and has the perturbative expansion : ˜ = a(1 + rn an ) , (26) R(s)
(28)
4G F MH3 R(MH ) √ 9 2π
(29)
where MH is Higgs mass and G F is Fermi constant. In perturbative theory R(MH ) has a perturbative expansion R = a2 + r1 a3 + r2 a4 + r3 a5 + ... .
(30)
The coefficients r1 , r2 , r3 are known [16]. Using the self consistency principle for the first two terms in Eq.(30), we will obtain: ∂r1 = 2 ⇒ r1 − 2τ = ρ1 , ∂τ
(31)
where ρ1 is RS invariant and τ = b ln Λμ¯ . Considering the third term in the Eq.(30) we will arrive at:
n>0
∂r2 ∂r2 3 3 = −2, = c + r1 ⇒ r2 = cr1 + r12 − 2c2 + X2 . ∂c2 ∂r1 2 4
where the partial derivative of coupling constant with respect to ci scheme parameters are used [10]. Following the same strategy by considering the forth term of
(32)
the series expansion, we get:
r3 =
r13 7 2 + cr − 4c2 r1 + 2X2 r1 − c3 + X3 , 2 4 1
(33)
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
98
where X2 and X3 are constants of integrations which are scheme invariants. By substituting the results for ri coefficients, given by Eqs.(32,33), in Eq.(30) and using t’hooft scheme we will get the following expression for the desired decay in the CORGI approach: Γ(H → gg) =
4G F MH3 2 [a0 + X2 a40 + X3 a50 + ...] .(34) √ 9 2π
Like the one for Re+ e− , we need to convert the computations from Euclidean to Minkowski space. Then up to NNLO approximation, we will get: d1 = r1
,
d2 = r2 + , 7 d3 = r3 + 2π2 r1 β20 + β0 π2 β1 . 3
in terms of the ri−1 and the unknown invariant constants. Then it is possible to do a resummation at any specified order and to reconstruct the perturbative series such that the final result is scheme invariant and depends just on the Q physical scale. Dose these two approaches are related to each other?. Which one has more advantages in considering the perturbative series?. These are the questions which we are looking for them and we have to report them in future.
References
β20 π2
(35)
The numerical result for this decay is 0.382 MeV where the PMC result and the conventional ones are respectively 0.373 MeV and 0.359MeV. 4. Conclusion Here two different approaches were used to avoid the scale and scheme ambiguities in perturbative QCD. The first one is the PMC approach in which the perturbative series are divided to two conformal and non-conformal parts. The non conformal parts are absorbed to the running coupling constant. Using the scale displacement equation for coupling constant, the PMC scale setting are determined and finally a conformal series with coefficient that are independent of the renormalization scheme are obtained in the PMC approach. As a second approach (CORGI), using the self-consistency of perturbation theory, it is possible to write the ri coefficient
[1] S.J. Brodsky, M. Mojaza, X.G. Wu, Phys. Rev. D 89 (2014) 014027 . [2] X.G.Wu,Y.Ma, S.Q. Wang, S.J. Brodsky and M. Mojaza Rep.Prog.Phys.78 (2015) 126201. [3] C.J.Maxwell, Nucl.Phys.Proc.Suppl.86 (2000) 74. [4] C.J.Maxell, A.Mirjalili, Nucl.Phys.B 299 (2000) 289. [5] P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn and J. Rittinger,JHEP 1207 (2012) 017. [6] P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn and J. Rittinger,Phys. Lett. B 714 (2012) 62. [7] T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett. B 400 (1997) 379. [8] R. Marshall, Z. Phys. C 43 (1989) 595 . [9] Sheng.Quan .Wang, Xing.Gang. Wu, Xu.Chang .Zheng, Jian.Ming. Shen and Qiong.Lian .Zhang, Eur.Phys.J. C 74 (2014) 2825. [10] P. M. Stevenson,Phys. Rev. D 23 (1981) 2916 . [11] Einan Gardi, Georges Grunberg and Marek Karliner, JHEP 07 (1998) 007. [12] C.J.Maxell, A.Mirjalili, Nucl.Phys.B 611 (2001) 423. [13] M. Davier, S. Descotes-Genon, A. Hocker, B. Malaescu and Z. Zhang, Eur. Phys. J. C 56 (2008) 305. [14] S. Menke, arXiv:hep-ph/0904.1796. [15] A. Mirjalili, M. Akrami, Int.J.Mod.Phys. A 26 (2011) 2047. [16] P.A. Baikov and K.G. Chetyrkin, Phys. Rev. Lett. 97 (2006) 061803.
Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98 www.elsevier.com/locate/nppp
Renormalization scale setting for some QCD observable, based on the PMC and CORGI approaches M. Akramia , A. Mirjalilia a Physics
Department, Yazd University, 89195-741,Yazd, Iran
Abstract In this paper we present the principle of Maximum conformality (PMC) and Complete Renormalization Group Improvement (CORGI) approaches of scale setting which are based essentially on Refs.[1–4]. The PMC establishes a systematic method to eliminate the renormalization scheme and renormalization scale dependencies of conventional perterbative QCD predictions. This approach, using the renormalization group equation, exposes a pattern of βi − terms in the perterbative series. Then by absorbing all βi − terms into the running coupling constant, one can obtain a result that is independent of renormalization scale and scheme. On the other hand the CORGI approach improves the precision of calculations. In this approach the standard perturbative series of QCD observable reconstructed in terms of scheme-invariant quantities. Here, we will examine the PMC and CORGI approaches by comparing its predictions for the electron-positron annihilation to hadrons and the Higgs decay width to gluon. The two results are compared with each other and with the available experimental data. At the end we may conclude which approach has more advantage in considering the perturbative series. Keywords: Conformal and nonconformal terms, Renormalization Group, Scale and scheme invariants.
1. Introduction All physical predictions in QCD should be invariant under any choice of renormalization scale and renormalization scheme. Thus, a key problem in making proper perterbative QCD (pQCD) predictions is how to set the renormalization scale of the running coupling constant at each order. Here we pay attention two approaches, the PMC and CORGI ones. In the PMC approach after proper scale setting , all non-conformal βi -terms in the perturbative expansion are absorbed into the running coupling constant. So the remaining terms in the perturbative series are identical to that of a conformal theory, ∗ Talk given at 21th International Conference in Quantum Chromodynamics (QCD 18, 33th anniversary), 3–6 july 2018, Montpellier FR Email addresses: [email protected] (M. Akrami), [email protected] (A. Mirjalili)
https://doi.org/10.1016/j.nuclphysbps.2018.12.017 2405-6014/© 2018 Elsevier B.V. All rights reserved.
i.e., the theory with βi = 0. To improve and increase the ability of calculations, we also try to use the CORGI approach. In this approach the standard perturbative series of QCD observable is reconstructed in terms of schemeinvariant quantities. So it is expected to get more reliable results with respect to what we obtain in conventional QCD approach. 2. An overview of the principle of maximum conformality The main step of this approach is to use an arbitrary dimensional renormalization scheme such as Rδ scheme. Here an arbitrary constant −δ is subtracted in addition to the standard subtraction ln 4π − γE that used in MS scheme [1]. Using the RG-equation and its degeneracy relations, one can identify which terms in the perturbative series are associated with the QCD
94
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
β-function and which terms remain in β = 0 , i.e, conformal terms. The optimal renormalization scales then are determined by absorbing all βi terms into the running coupling constant at each order of perturbation theory. The final resummed result is a conformal series. On the other words when the renormalization scales at each perturbative order are set properly, all nonconformal terms in a perturbative expansion that arising from renormalization are summed into the running coupling. The PMC scales for different Rδ -schemes(Rδ1 and δ1 −δ2 Rδ2 ) differ only by a factor e 2 . Therefore the β- function in any Rδ -scheme is the same as in MS [1]. Using the displacement relation between couplings in any Rδ -
scheme, one finds [1]: a(μ0 ) = a(μδ ) +
∞ 1 dn a(μ) | (−δ)n , 2 n μ=μδ n! (d ln μ ) 0 n=1
(1)
where ln μ20 /μ2δ = −δ. Now in order to employ practically the PMC approach, an observable in pQCD is considered as it follows: ρ0 (Q2 ) = a(μ0 )n
∞
rk+1 (Q2 /μ20 ) a(μ0 )k ,
(2)
k=0
where μ0 is an initial renormalization scale. By expanding Eq.(1) up to k + 3 and substituting it in Eq.(2) for the observable ρ, the following result is obtained [1]:
n(n + 1) 2 2 β0 r1 δ1 ]a3 (μδ )n+2 + ρδ (Q2 ) = r1 a1 (μδ )n + r2 + nβ0 r1 δ1 a2 (μδ )n+1 + [r3 + nβ1 r1 δ1 + (n + 1)β0 r2 δ2 + 2 n(3 + 2n) (n + 1)(n + 1) 2 2 [r4 + nβ2 r1 δ1 + (n + 1)β1 r2 δ2 + (n + 2)β0 r3 δ3 + β0 β1 r1 δ21 + β0 r2 δ2 2 2 n(n + 1)(n + 2) 3 2 + β0 r1 δ1 ]a4 (μδ )n+3 . 3! (3)
This equation consist of a pattern of {βi }-terms in the coefficients at each order. Here one can be concluded
that some of the coefficients of the {βi } terms are degenerate. Thus the observable can be written as follow:
n(n + 1) 2 β0 r3,2 ]a(Q)n+2 2 n(3 + 2n) (n + 1)(n + 2) 2 β0 β1 r3,2 + β0 r4,2 +[r4,0 + nβ2 r2,1 + (n + 1)β1 r3,1 + (n + 2)β0 r4,1 + 2 2 n(n + 1)(n + 2) 3 β0 r4,3 ]a(Q)n+3 + ... , + 3!
ρ(Q)2 = r1,0 a(Q)n + [r2,0 + nβ0 r2,1 ]a(Q)n+1 + [r3,0 + nβ1 r2,1 + (n + 1)β0 r3,1 +
(4)
where ri,0 are conformal and ri, j for j 0 are nonconformal parts of the perturbative coefficients, i.e. ri = ri,0 + O({βi }). One can resumes all ri,1 terms to all orders by defining new scale Qi at each order as follows [1]: rk,0 a(Qk )k = rk,0 a(Q)k − ka(Q)k−1 β(a)rk+1,1 .
(5)
By expanding Eq.(1) one finds the following relation:
a(Qk )k = a(Q)k + ka(Q)k−1 β(a) ln [β
Q2k k + a(Q)k−2 Q2 2
Q2 ∂β a(Q) + (k − 1)β(a)2 ]ln2 k2 + ... . ∂a Q (6)
Substituting Eq.(6) into Eq.(5), one can find the fol-
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
lowing scale Qi at each order. The results for the leading logarithm order (LLO), Next-to-NLO (NLLO) and finally Next-to-NLO (NLLO) are respectively as following [2]: ln
ln
Q2k,LLO Q2
ln
Q2k,LLO Q2
=
95
−rk+1,1 /rk,0 1+
1 ∂β 2 [ ∂a
+ (k − 1) βa ](−
rk+1,1 rk,0 )
, (8)
rk+1,1 =− , rk,0
Q2k,NNLLO Q2
=
(7)
−rk+1,1 /rk,0 1 + 12 [ ∂β ∂a + (k −
r 1) βa ](− k+1,1 rk,0 )
+
∂2 β 1 3! [β ∂a2
By substituting Eq.(5) in Eq.(4) and replacing new scales Qi at each order the non-conformal parts are disappeared and one can obtain a scheme-
2
− 12 ( ∂β ∂a ) −
2 r (k−1)(k+1) β2 ](− k+1,1 2 rk,0 ) a2
.
(9)
independent conformal series [2] as it follows
ρ(Q2 ) = r1,0 a(Qn,NNLLO )n + r2,0 a(Qn+1,NLLO )n+1 + r3,0 a(Qn+2,LLO )n+2 + r4,0 a(Q)n+3 + ... .
(10)
2.1. Electron-positron annihilation in PMC approach The annihilation of an electron and positron into hadrons, Re+ e− , is defined as [1]:
Re+ e− (Q) = γ0 + γ1 a(Q) + [γ2 + β0 Π1 ]a(Q)2 + [γ3 + β1 Π1 + 2β0 Π2 − β20
π2 γ1 ]a(Q)3 3
π2 γ1 π2 γ2 5 − 3β20 − β30 π2 Π1 ]a(Q)4 , +[γ4 + β2 Π1 + 2β1 Π2 + 3β0 Π3 − β0 β1 2 3 3 (11)
3 where Re+ e− (Q) = 11 Re+ e− (Q). This equation is based on anomalous dimension ,γi , and vacuum polarization function, Πi [5–7]. If the following identifications for the coefficients are used then Eq.(11) can be written in the form of Eq.(4):
ri,0 = γi
ri,1 = Πi−1
π γi−2 3 = −π2 Πi−3
ri,2 = − ri,3
, 2
i ≥ 2,
i ≥ 3,
By using Eqs.(7,8,9) we get the following numerical results for Qi which are closed with the numerical results, reported in [1]: Q1 = 44.72 GeV , Q2 = 37.86 GeV , Q3 = 168.68 GeV , Q4 = 31.6 GeV . (13) √ Taking s = 31.6 GeV and Λ MS = 419 MeV with five active quark flavors, the following numerical result for Re+ e− observable is obtained: 3 Re+ e− (Q) = 1 + α(Q1 ) 11 +1.84α(Q2 )2 − 1.00α(Q3 )3 − 10.82α(Q4 )4 . Re+ e− (Q) =
i≥4. (12)
(14)
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In Eq.(14) the numerical coefficient of fifth term is different with respect to [1] Using the Qi scales, represented by Eq.(13) and substituting them in Eq.(14), we get Re+ e− (Q) =
Γ(H → gg)
3 Re+ e− = 1.060 , 11
=
(15)
=
2.2. Higgs decay to gluon in the PMC approach The pQCD prediction for decay width of the Higgs decay to a pair of gulon can be written as [9]:
4G F MH3 c1,0 a2s (MH ) + (c2,0 + c2,1 n f ) a3s (MH ) + (c3,0 + c3,1 n f + c3,2 n2f ) a4s (MH ) √ 9 2π +(c4,0 + c4,1 n f + c4,2 n2f + c4,3 n3f ) a5s (MH ) + O(a6s ) ,
where G F is Fermi constant and MH is Higgs boson mass. The coefficients ci, j are given in [9]. Considering the conformal and non-conformal
Γ(H → gg)
which can be compared with the experimental value 3 +0.005 + − 11 Re e = 1.052−0.005 [8].
(16)
coefficients and following similar process as for Re+ e− to absorb all non-conformal {βi }-terms into the coupling constant, the attained result is:
4G F MH3 1 6.333 α2s (Q1 ) + 9.068 α3s (Q2 ) − 79.962 α4s (Q3 ) − 68.804 α5s (Q4 ) + O(α6s ) . √ 9 2π 1000 (17)
Using Eqs.(7,8,9) the following PMC scales Qi are obtained: Q1 = 23.10 GeV , Q3 = 31.46 GeV .
Q2 = 10.49 GeV
, (18)
What is numerically gotten for this decay is: Γ(H → gg) = 0.364 MeV which is comparable with the result of the conventional scale setting: 0.373 MeV [2]. 3. Overview on the CORGI approach In another approach so called Compleat Renormalization Group Improvement (CORGI), all perturbative terms are scheme independent. So there is no dependency on the renormalization (μ) or factorization scale (M) for these terms. We can assume an observable R(Q) in a standard approach has a perturbative series like:
Using the self-consistency requirement we have the following expressions for the partial derivatives of the rn with respect to the scheme parameters [10]: ∂r2 ∂r2 ∂r2 = 2r1 + c, = −1, = 0, . . . . ∂r1 ∂c2 ∂c3
(20)
By integrating these terms simultaneously, one finds [3]: r2 (r1 , c2 )
=
r3 (r1 , c2 , c3 )
=
r1 2 + cr1 + X2 − c2 , 5 r1 3 + cr1 2 + (3X2 − 2c2 )r1 + 2 1 (21) X3 − c3 , 2
where X2 , X3 are constants of integration and renormalization scheme (RS) invariants. By substituting Eq.(21) in Eq.(19) one finds:
R(Q) = a + r1 a2 + r2 a3 + . . . + rn an+1 + . . . , (19) 5 1 R(Q) = a + r1 a2 + (r12 + cr1 + X2 − c2 )a3 + (r13 + cr12 + (3X2 − 2c2 )r1 + X3 − c3 )a4 + . . . , 2 2
(22)
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
where a≡a(r1 , c2 , c3 , . . .) [3]. Here r1 at NLO approximation is known but X2 , X3 , . . . are unknown. All known terms in Eq.(22) at the NLO approximation can be summed to yield the following completed subset [3]:
97
Here a = απs . It is possible to have a relation such as Eq.(26) in Minkowski space [12]: ˜ dn an ) . (27) D(s) = a(1 + n>0
a0 ≡a + r1 a2 + (r12 + cr1 − c2 )a3 + (r13 + 5 2 1 cr1 − 2c2 r1 − c3 )a4 + . . . . (23) 2 2 By using the t’Hooft scheme (r1 = r2 = ... = rn = 0, c1 = c2 = ... = cn = 0) one finds a0 = a. The a0 can be written in terms of the Lambert W-function [11]. By doing the required calculations at higher orders, one arrives at [3, 4]: R(Q) = a0 +X2 a0 3 +X3 a0 4 +. . .+Xn a0 n+1 +. . . ,(24) Each term in Eq.(24) involves a resummation of infinite terms at specified order. For instance, the first term, a0 , is a representation of resummation over NLO contribution of all terms in Eq.(22) and the second term, X2 a30 , as a representation of resummation for the NNLO contribution .
The required coefficients di have been calculated in [13, ˜ is the Adler D-function and in the CORGI 14]. The D(s) approach it has the following expansion [4]: = a0 + X2 a3 + X3 a40 + ... + Xn an+1 D(s) . 0 0
The relation between di and ri coefficients is given by analytic continuation [15]. Numerical result for this √observable at the NNLO approximation is: 3 + − 11 Re e ( s = 31.6 GeV) = 1.051 which is very close to its experimental value: 1.052+0.005 −0.005 [8] and even better than the PMC result, given by Eq.(15).
3.2. Higgs decay to gg in CORGI approach The higgs decay width to a gluon-gluon is generally given by: Γ(H → gg) =
3.1. e+ e− annihilation in CORGI approach The concerned observable Re+ e− in S U(N) perturbtion theory is defined by [12]: ⎛⎜ ⎞⎟2 ⎟⎟ ⎜⎜ 3 2 ˜ ¯ ,(25) R(s) = N Q f 1 + C F R(s) +⎜⎜⎝⎜ Q f ⎟⎟⎠⎟ R(s) 4 f
f
where Q f denoting quark charges and C F is color factor.R˜ denotes the perturbative corrections to the parton model result and has the perturbative expansion : ˜ = a(1 + rn an ) , (26) R(s)
(28)
4G F MH3 R(MH ) √ 9 2π
(29)
where MH is Higgs mass and G F is Fermi constant. In perturbative theory R(MH ) has a perturbative expansion R = a2 + r1 a3 + r2 a4 + r3 a5 + ... .
(30)
The coefficients r1 , r2 , r3 are known [16]. Using the self consistency principle for the first two terms in Eq.(30), we will obtain: ∂r1 = 2 ⇒ r1 − 2τ = ρ1 , ∂τ
(31)
where ρ1 is RS invariant and τ = b ln Λμ¯ . Considering the third term in the Eq.(30) we will arrive at:
n>0
∂r2 ∂r2 3 3 = −2, = c + r1 ⇒ r2 = cr1 + r12 − 2c2 + X2 . ∂c2 ∂r1 2 4
where the partial derivative of coupling constant with respect to ci scheme parameters are used [10]. Following the same strategy by considering the forth term of
(32)
the series expansion, we get:
r3 =
r13 7 2 + cr − 4c2 r1 + 2X2 r1 − c3 + X3 , 2 4 1
(33)
M. Akrami, A. Mirjalili / Nuclear and Particle Physics Proceedings 300–302 (2018) 93–98
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where X2 and X3 are constants of integrations which are scheme invariants. By substituting the results for ri coefficients, given by Eqs.(32,33), in Eq.(30) and using t’hooft scheme we will get the following expression for the desired decay in the CORGI approach: Γ(H → gg) =
4G F MH3 2 [a0 + X2 a40 + X3 a50 + ...] .(34) √ 9 2π
Like the one for Re+ e− , we need to convert the computations from Euclidean to Minkowski space. Then up to NNLO approximation, we will get: d1 = r1
,
d2 = r2 + , 7 d3 = r3 + 2π2 r1 β20 + β0 π2 β1 . 3
in terms of the ri−1 and the unknown invariant constants. Then it is possible to do a resummation at any specified order and to reconstruct the perturbative series such that the final result is scheme invariant and depends just on the Q physical scale. Dose these two approaches are related to each other?. Which one has more advantages in considering the perturbative series?. These are the questions which we are looking for them and we have to report them in future.
References
β20 π2
(35)
The numerical result for this decay is 0.382 MeV where the PMC result and the conventional ones are respectively 0.373 MeV and 0.359MeV. 4. Conclusion Here two different approaches were used to avoid the scale and scheme ambiguities in perturbative QCD. The first one is the PMC approach in which the perturbative series are divided to two conformal and non-conformal parts. The non conformal parts are absorbed to the running coupling constant. Using the scale displacement equation for coupling constant, the PMC scale setting are determined and finally a conformal series with coefficient that are independent of the renormalization scheme are obtained in the PMC approach. As a second approach (CORGI), using the self-consistency of perturbation theory, it is possible to write the ri coefficient
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