A Feature of the Off-Shell Renormalization Schemes in Quantum Field Theory

1. Introduction

Renormalization schemes for subtraction of ultraviolet divergences in Quantum Field Theory can be divided in two essentially different classes: the off-shell schemes and the on-shell schemes. The most known off-shell schemes are the Minimal Subtraction scheme - the MS-scheme [1], its commonly used modification the $\overline{MS}$-scheme [2], and the Momentum Subtraction scheme - the MOM-scheme, see e.g. [3] and references therein. The essential point of the off-shell schemes is that one introduces an arbitrary renormalization parameter (usually denoted as $\mu$) within these approaches during subtractions of ultraviolet divergences of Green functions. (Of course off-shell schemes can in principle contain more than one arbitrary renormalization paramerers.) But physical quantities should be independent on this unphysical renormalization parameter $\mu$. The coupling constants in the off-shell renormalization schemes are the so called running coupling constants depending on the free renormalization parameter $\mu$ of the schemes.

The on-shell renormalization schemes do not have free renormalization parameters and contain only physical parameters. The famous example of the application of the on-shell scheme is the anomalous magnetic moment of the electron. The agreement between the theory and the experiment in this case within at least ten decimal points convinces us that Quantum Field Theory and in particular the on-shell renormalization scheme is a valid approach. The coupling constants in the on-shell schemes do not depend on free parameters and are the fixed physical constants.

The purpose of the present paper is to show that the off-shell renormalization schemes produce zero for sums of perturbative corrections to physical quantities when all perturbation orders are taken into account. That is the off-shell renormalization schemes in Quantum Field Theory are in this sense unphysical. In this connection it seems to be desirable to develop on-shell renormalization schemes, in particular for Quantum Chromodynamics and Quantum Gravity.

2. Main Part

We will consider the famous R-ratio within perturbative Quantum Chromodynamics (QCD) in the MS-scheme as a typical illustrative example in view of its importance for phenomenological applications:

$$\widetilde{R\left(s\right)}={\displaystyle \frac{{\sigma }_{total}(e^{+}{e}^{-}\to hadrons)}{{\sigma }_{tree}(e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})}}.$$

(1)

I.e. $\tilde{R}$ is the total cross-section of the electron-positron annihilation into hadrons normalized by the tree level cross-section of the electron-positron annihilation into the muon-antimuon pairs. The momentum transfer squared $q^2$ of this reaction will be denoted as s: $s\equiv q^2$.

Ignoring masses one can write

$$\widetilde{R\left(s\right)}=3\sum _i^{n_f}{q}_i^2(1+R\left(s\right)),$$

(2)

where $q_i$ are quark electric charges in the units of the electron charge and $n_f$ is the number of active quarks.

Here R has the following form within perturbative QCD:

$$R\left(s\right)=a(1+\sum _{k=1}^{\infty }{r}_k{a}^k)=a(1+r_{1}a+r_2{a}^2+...),$$

(3)

where $a\equiv a\left({\mu }^2\right)={\displaystyle \frac{g^2}{16{\pi }^2}}$ is the renormalized QCD running coupling constant depending on the arbitrary renormalization parameter $\mu$ of the MS-scheme. g is the strong coupling constant in the QCD Lagrangian.

The perturbative coefficients $r_k\equiv r_k(s,{\mu }^2)$ are calculable in perturbation theory and depend both on the momentum transfer squared s and ${\mu }^2$.

The physical quantity R must be independent on the arbitrary parameter $\mu$ according to the basic principle of renormalization group invariance of physical quantities in Quantum Field Theory. This requirement generates the renormalization group equation:

$${\displaystyle \frac{dR\left(s\right)}{dln\left({\mu }^2\right)}}={\displaystyle \frac{\partial R}{\partial ln\left({\mu }^2\right)}}+{\displaystyle \frac{\partial R}{\partial a}}{\displaystyle \frac{\partial a}{\partial ln\left({\mu }^2\right)}}=0.$$

(4)

Usually one deals with the truncated perturbative series for R to compare R with experiments:

$$R_N=a(1+\sum _{k=1}^{N-1}{r}_k{a}^k)=a+r_1{a}^2+...+r_{N-1}{a}^N.$$

(5)

The approximation $R_3$ was analitically calculated in [4] and the approximation $R_4$ in [5].

The truncated perturbative series $R_N$ does depend on ${\mu }^2$, but the derivative of the $R_N$ in ${\mu }^2$ is of the order of $a^{N+1}$ only [6] :

$${\displaystyle \frac{d{R}_N}{dln\left({\mu }^2\right)}}=O\left(a^{N+1}\right).$$

(6)

This equation is the reminiscence of the $\mu$-independence (4) of R.

The perturbative coefficients $r_k$ have the well-known form:

$$r_k=\sum _{i=0}^k{r}_{k,i}l{n}^i({\mu }^2/s),$$

(7)

where the powers of the logarithms are in accordance with the equation (6). Here $r_{k,i}$ are constants calculable in perturbative QCD.

The running coupling constant $a\left({\mu }^2\right)$ obeys the standard renormalization group equation:

$${\displaystyle \frac{da\left({\mu }^2\right)}{dln\left({\mu }^2\right)}}=\beta \left(a\right).$$

(8)

The renormalization group $\beta$-function has the well known form:

$$\beta \left(a\right)=-\sum _{n=0}^{\infty }{\beta }_n{a}^{n+2}=-{\beta }_0{a}^2-{\beta }_1{a}^3-{\beta }_2{a}^4-...,$$

(9)

where the constants ${\beta }_n$ are calculable in perturbative QCD.

In particular, the famous result [7,8,9]

$${\beta }_0=11-{\displaystyle \frac{2}{3}}{n}_f$$

(10)

has led to the discovery of the asymptotic freedom in QCD.

The equation (8) can be solved by iterations in the region of the applicability of the method with the known result:

$$a\left({\mu }^2\right)={\displaystyle \frac{1}{{\beta }_{0}ln({\mu }^2/{\Lambda }^2)}}-{\displaystyle \frac{{\beta }_{1}lnln({\mu }^2/{\Lambda }^2)}{{\beta }_0^{3}l{n}^2({\mu }^2/{\Lambda }^2)}}+O\left({\displaystyle \frac{1}{l{n}^3({\mu }^2/{\Lambda }^2)}}\right),$$

(11)

where $\Lambda$ is the known fundamental parameter of QCD with the dimension of the mass.

Substituting the coefficients $r_k$ from the equation (7) and the expression for $a\left({\mu }^2\right)$ from (11) into $R\left(s\right)$ in eq. (3) one gets the standard expression for $R\left(s\right)$ in terms of the logarithms $ln({\mu }^2/s)$ and of the logarithms $ln({\mu }^2/{\Lambda }^2)$:

$$R\left(s\right)=a+(r_{1,0}+r_{1,1}ln\left({\displaystyle \frac{{\mu }^2}{s}}\right))a^2+(r_{2,0}+r_{2,1}ln\left({\displaystyle \frac{{\mu }^2}{s}}\right)+r_{2,2}l{n}^2\left({\displaystyle \frac{{\mu }^2}{s}}\right))a^3+...,$$

(12)

where a is given in (11). Thus R in the above equarion (12) is the well defined function of $\mu$.

One can note that in each perturbative order in the r.h.s. of (12) the minimal power of $ln\left({\mu }^2\right)$ in the denominator is one more (due to the eqs. (11)) than the maximal power of $ln\left({\mu }^2\right)$ in the numerator. Thus one can find that the limit of each perturbative order in (12) at $\mu \to \infty$ is zero. Hence the limit of the whole $R\left(s\right)$ at $\mu \to \infty$ is also zero within perturbation theory:

$$\underset{\mu \to \infty }{lim}R\left(s\right)=0.$$

(13)

Thus from one side $R\left(s\right)$ is independent on $\mu$ (see eq.(4)) and from another side the limit of $R\left(s\right)$ at $\mu \to \infty$ is zero (see eq.(13)). Thus one can conclude that $R\left(s\right)$ is identically zero for any $\mu$ within the region of applicability of perturbation theory. The inclusion of the non-sero masses will not change the result.

It is well known that perturbative series in Quantum Field Theory are asymptotic series. But the limit $\mu \to \infty$ is well established for the full asymptotic series in particular for QCD. See e.g. the known paper [10], where the famous exact relation between $\Lambda$-parameters in different renormalization schemes is obtained by means of this limit for the full asymptotic series.

Some remarks are in order. In principle one could obtain the exact perturbative expression for $R\left(s\right)$ if one has enough mathematics. (In particular the functional integral of the generating functional of Green functions is already the exact solution of the theory only not in the enough apparent form.) The expansion of this exact function is our asymptotic series for R. It is clear that this function (in other words the sum of the asymptotic series) can not be non-zero if the N-th term of the series is zero for any finite N, as it is the case in the limit $\mu \to \infty$.

In fact it is not of principal importance what is the exact value of the limit in (13). The principal point is that this value is a constant which does not depend on s since s enters $R\left(s\right)$ only via the ratioo ${\mu }^2/s$. Hence the off-shell renormalization scheme produces for $R\left(s\right)$ this constant value for any s which is unphysical.

3. Conclusions

We have demonstrated above that the off-shell renormalization schemes for subtraction of ultraviolet divergences in Quantum Field Theory produce zero for sums of perturbative corrections to physical quantities when all perturbation orders are taken into account. Thus it seems to be desirable to develop on-shell renormalization schenes, in particular for Quantum Chromodynamics and also for Gravity which was quite recently quantized for the relativistic $R^2$-model [11,12]. It should be noted that the on-shell renormalization scheme for Quantum Chromodynamics does not mean the introduction of on-shell quarks. It means in particular that quark masses should be taken as perturbative pole masses which do not depenr on unphysical parameters.