Higher-Loop Calculations of the Ultraviolet to Infrared Evolution of a Vectorial Gauge Theory in the Limit , with Fixed
We consider an asymptotically free vectorial SU() gauge theory with fermions in the fundamental representation and analyze higher-loop contributions to the evolution of the theory from the ultraviolet to the infrared in the limit where and with a fixed, finite constant. We focus on the case where the -loop beta function has an infrared zero, at , where . We give results on , the anomalous dimension of the fermion bilinear evaluated at , denoted , and certain structural properties of the beta function, . The approach to this limit is investigated, and it is shown that the leading correction terms are strongly suppressed, by the factor . This provides an understanding of a type of approximate universality in calculations for moderate values of and , namely that , , and structural properties of the beta function are similar in theories with different values of and provided that they have similar values of . We give results up to four loops for nonsupersymmetric theories and up to three loops for supersymmetric theories.
The evolution of an asymptotically free gauge theory from the ultraviolet (UV) to the infrared (IR) is of fundamental field-theoretic interest. The UV to IR evolution of the gauge coupling as a function of the Euclidean momentum scale, , is determined by the function beta ()
where and . Here we consider this evolution for a vectorial gauge theory with gauge group and massless fermions , , transforming according to the fundamental representation of fm (). We also point out some contrasts with results for fermions in higher-dimensional representations. We focus on the case where the -loop function has an infrared zero, at a value . The condition of asymptotic freedom requires that be bounded above by a value where the one-loop coefficient in vanishes b1 (). For large enough (less than ), the two-loop function has an infrared zero at a certain value of , denoted b2 (); bz (). The desire to understand better both the behavior of the running coupling in quantum chromodynamics (QCD) and the properties of an IR zero that occurs for sufficiently large have motivated calculations of higher-loop terms in b3 (); b4 () and higher-loop corrections to the two-loop result for the IR zero gkgg ()-bc (). In bvh (); ps (), calculations of the IR zero in , and the associated anomalous dimension of the (gauge-invariant) fermion bilinear, , were done to four-loop order for an asymptotically free vectorial gauge theory with gauge group and fermions in an arbitrary representation , with explicit results for equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. Further generalizations and results on higher-loop structural properties of the function were given in bc ().
An interesting and important property that one notices in these calculations for an SU() theory with massless fermions in the fundamental representation is that the values of the -loop , evaluated at , denoted , and of the product , are similar for theories with different values of and , provided that these theories have similar values of the ratio . Indeed, the computations in bc () show that this is also true for other structural quantities describing the UV to IR evolution, including the derivative of the -loop beta function, evaluated at and the products and , where denotes the value of where is a minimum, and is the value of at this minimum. These observations show that there is an underlying approximate universality in the form of the quantities that control the UV to IR evolution of these theories. This motivates a more detailed study to elucidate this phenomenon. We carry out this study in the present work.
where is a constant and is a function depending only on . We will use the symbol for this limit, where “LNN” stands for “large and ” (with the constraints in Eq. (LABEL:lnn) imposed). The reasons for these two constraints in Eq. (LABEL:lnn) (that the ratio and the product are fixed and finite) are that these constitute the necessary and sufficient conditions in order that (i) fermions give nonvanishing contributions to the function, anomalous dimension , and other quantities, and (ii) scattering amplitudes remain finite, in the limit , respectively. More generally, if the fermions are nonsinglets under other gauge groups with squared couplings , one also requires that the products be finite as lnc ().
As we will show in detail below, a study of the LNN limit (LABEL:lnn) and the approach to it provides an explanation of the approximate universality in the UV to IR evolution of theories with different values of and but the same, or similar, values of that is exhibited in explicit calculations (with appropriate scalings understood for certain quantities, such as multiplying by ). A crucial property of the LNN limit is that it reduces the number of variables on which the UV to IR evolution depends. Thus, for finite and , this evolution and the function that describes it, depend on three variables: (and thus, parametrically, ), , and , while in the LNN limit, they only depend on the two variables and .
Since the rational numbers are dense in the real numbers , it follows that in the LNN limit, one can choose values of and so that the rational number is arbitrarily close to any non-negative real number. Therefore, henceforth, to arbitrarily good accuracy, we may simply treat as a real number, and we will do so. As is well-known, in the limit and also in the LNN limit, the gauge group SU( is effectively equivalent to U(). The use of a large- limit, where is the number of components in a spin or field, has been valuable in the past partly because it allowed one to obtain exact results for statistical mechanical models stanley () and quantum field theories thooftlargeN ()-lnc (), earlylargeN (); largeNreview (). Our purpose in using it here is somewhat different, namely to gain further insight into the above-mentioned approximate universality that is exhibited by calculations of the UV to IR evolution of theories with different and but equal or similar values of .
As part of our analysis, we will briefly contrast the properties of theories with fermions in the fundamental representation with properties of theories with fermions in higher-dimensional representations. In the case where fermions are in a two-index representation (including the adjoint, and symmetric and antisymmetric rank-2 tensor representations), the condition that is necessary and sufficient to construct a finite limit is to set equal to a (non-negative, integer) constant. This is a result of the fact that the quadratic Casimir invariants and casimir () that enter into the coefficients of the beta function grow like (a constant times) as . If the fermions are in a representation involving three or more indices, then generically for fixed finite , the fermion contributions dominate over the gauge field contributions to the function by powers of as . For example, for the symmetric rank-3 tensor representation, for large , so that the fermion contribution to the leading function coefficient dominates over the gauge-field contribution, which is , spoiling the asymptotic freedom of the theory. Hence, aside from our primary focus on the case of fermions in the fundamental representation, we will restrict our discussion of other representations to the adjoint, and symmetric and antisymmetric rank-2 tensors.
By taking near to its maximum value allowed by asymptotic freedom, one can arrange that the zero of occurs at an arbitrarily small value of , and one may conclude that in the infrared the theory is in a deconfined non-Abelian Coulomb phase without any spontaneous chiral symmetry breaking. In this case, the IR zero of at is an exact fixed point of the renormalization group for the theory. In contrast, as decreases, increases, and studies with finite and lead to the conclusion that for less than a critical value, , as decreases though a value denoted , the interaction strength exceeds a critical value, , to produce spontaneous chiral symmetry breaking and associated dynamical mass generation for the fermions. For a given , the theory may thus be considered to undergo a (zero-temperature) chiral phase transition as passes through this value, ap (), and there has been an intensive research program using lattice gauge simulations to determine for values such as and conf (). Correspondingly, in the LNN limit considered here, the theory undergoes a chiral phase transition as passes through , where , with the UV to IR evolution leading to a chirally symmetric phase for and a phase with spontaneous chiral symmetry breaking for . If , then in the effective low-energy field theory below , one integrates out the fermions (which have dynamically generated masses of order ), and the function changes to become that of a pure non-Abelian gauge theory, which does not have a (perturbative) zero. In this case, is only an approximate fixed point.
This paper is organized as follows. In Section II we define an appropriately scaled beta function, called , that is a finite function of in the LNN limit and in this section, and in Sections III and IV we investigate its structure up to four-loop order. Our results include an analysis of the behavior of the coefficients as functions of , the LNN limits for the -loop IR zero, , the value of where is a minimum, the value of at this minimum, and the derivative evaluated at . In Section V we carry out a similar analysis of the coefficients in the anomalous dimension of the fermion bilinear, , and its value at , again up to four-loop order. In Section VI we calculate correction terms to the LNN limits for various quantities and give a general analytic explanation for the rapidity with which this limit is approached, namely that these correction terms are strongly suppressed, by the factor . Section VIII is devoted to a corresponding study of the LNN limit of a supersymmetric gauge theory. Our conclusions are contained in a final section.
Ii Function and Some General Properties in the LNN Limit
In this section we analyze the function in the limit (LABEL:lnn). It will be convenient to define and
(The argument will often be suppressed in the notation.) In terms of , or equivalently, , the beta function has the series expansion
where denotes the loop order and . Thus, the -loop beta function is given by Eq. (8) with replaced by as the upper limit on the summation over loop order, . The coefficients for are independent of the scheme used for the regularization and renormalization of the theory and were calculated in b1 () and b2 (). The with are scheme-dependent and have been calculated up to -loop order b3 (); b4 () in the modified minimal subtraction ms () () scheme msbar (). The usefulness of the scheme has been demonstrated, e.g., by the fact that inclusion of three-loop and four-loop corrections in the running of in QCD significantly improves the fit to experimental data bethke (). The scheme-dependence of the higher-loop IR zero of the beta function was recently studied in sch ().
For our present analysis of the theory in the LNN limit, the first step is to construct a beta function that has a finite, nontrivial LNN limit. We do this by multiplying both sides of (8) by and then taking the LNN limit. The result is a function of and can be expressed as
This function has the expansion
Thus, similarly to the relation between and ,
As with Eq. (11), it is understood here and below that all expressions have been evaluated in the LNN limit (LABEL:lnn). The function, calculated to -loop () order, is denoted by and is given by Eq. (10) with replaced by as the upper limit on the sum over .
To analyze the zeros of , aside from the double zero at , we extract an overall factor of and calculate the zeros of the reduced polynomial
As is clear from Eq. (13), the zeros of away from the origin depend only on ratios of coefficients, which can be taken as for . Although Eq. (13) is an algebraic equation of degree , with roots, only one of these is physically relevant as the IR zero of . We denote this as .
ii.2 Behavior of Coefficients in as Functions of
In the scheme, from the expression for b3 (), we obtain
and from b4 (), we obtain
to the indicated numerical floating-point accuracy, where is the Riemann function. For some purposes it is more convenient to deal with the , since they are free of factors of , while for numerical purposes it is often more convenient to use the , since the range of values of as functions of is somewhat smaller than the range for the . In Table 1 we list values of for as functions of in the interval .
In bvh (), was defined as the value or set of values of where and thus, for our present analysis, we define
From Eq. (14), we have
As is evident from Eqs. (14) and (15), the coefficients and are both monotonically (and linearly) decreasing functions of . The coefficient decreases from 11/3 to 0 as increases from 0 to . We require that the theory be asymptotically free, i.e.,
The coefficient decreases from 34/3 at and passes through zero to negative values as increases through the value
As , reaches the value
Therefore, in the LNN limit, the interval in , denoted , where the two-loop function has an IR zero, is given by
(i.e., ). With , we will, correspondingly, focus on the UV to IR evolution in the interval
The coefficient vanishes at two values of , denoted
where here and below, the floating-point values are given to the indicated accuracy. This coefficient is monotonically decreasing in the interval , decreasing from at and passing through zero to negative values as increases through in Eq. (29). As increases from , continues to decrease and passes through the value
at the lower end of the interval . As increases throughout the interval , decreases further, and as increases to its maximum, , at the upper end of this interval, reaches the value
it follows that in the scheme,
Given this result and the fact that the quantity will appear in later formulas, it will be convenient to denote
which is positive for .
For completeness, we note that reaches a minimum at , and, for larger , it increases, passing through zero again at the value in Eq. (30). Since these values of lie above , they are not of direct interest for our present study.
As increases through the range , the coefficient in the scheme decreases from the value
(i.e., ), passes through zero with negative slope at
reaches a minimum of at , and then increases. At the lower end of the interval , at ,
As increases in the interval , passes through zero again, at
this time with positive slope, and attains the value
(i.e., ) at the upper end of the interval . Some special values, expressed in terms of the coefficients, are listed in Table 2.
Concerning the sign of in the range ,
That is, numerically, if or , and if . The zero of at lies below the lower end of the interval (at ), while the zero at lies in the interior of the interval . Hence, restricting to ,
i.e., numerically, for , if and if . Since is a cubic polynomial in , there is a third value of where it vanishes, but this is at the negative, and hence unphysical, value and hence is of no direct relevance here. Although our analysis here presumes the LNN limit, a remark is in order for finite and . The interval where is negative for is not present for sufficiently small and . This is evident from the explicit values listed in Table I of our Ref. bvh () for and . This interval of negative values is present for .
Iii IR Zero of
Combining the results from the previous section, we exhibit the explicit four-loop function. For this purpose, it is simplest to use the variable defined in Eq. (7). We have
iii.1 Two-Loop Level
At the two-loop level, if , then has an IR zero at
Since this is obtained from a perturbative calculation, it is only reliable if is not too large. As , , and hence in the upper end of the interval , one may plausibly expect that this two-loop expression becomes a progressively more and more accurate approximation to the IR zero of the exact function. As at the lower end of the interval , grows too large for this perturbative calculation to be applicable. It will be useful here and below to give values of various quantities at an illustrative value of .
iii.2 Three-Loop Level
At the three-loop level, the IR zero of is given by the physical (smallest positive) root of the quadratic equation
This equation has, formally, two solutions, namely
Since we have shown that for for a general scheme that preserves the existence of the IR zero in the (scheme-independent) at the three-loop level, we can rewrite (61) as
with equality only at , where . In bc () we pointed out that the corresponding inequality applies more generally than just in the scheme, and the same is true of the inequality (64). We recall the reasoning for this. Since the existence of an IR zero in the two-loop function, , is a scheme-independent property of the theory, a reasonable scheme should maintain the existence of this IR zero (albeit with a shifted value) at higher-loop order. Now in order for a scheme to maintain this zero, a necessary and sufficient condition is that , so that the square root in Eq. (61) is real. But the lower end of the interval is defined by the condition that as . Given that so has an IR zero, this means that a reasonable scheme, which preserves the existence of this zero at the three-loop level, should have for . From this, by the same type of proof as was given in bc () for this class of schemes, the inequality (64) follows.
Substituting the relevant expressions for the in (63), we have, in the scheme, the explicit result
where was defined above in Eq. (36), and it is convenient to define the shorthand notation
The polynomial has only one real zero, at (to the indicated accuracy) and is positive for , and hence for all . The polynomial vanishes at the lower end of the interval and is negative for , but it is smaller than , so for , as is necessary for it to be physical.
In Table 3 we list numerical values of and for . As is evident in this table, and decrease monotonically as a function of throughout this interval (as does , to be discussed below). At the lower end of this interval,
and at the upper end,
The ratio increases monotonically from 0 as increases from the value at the lower end of the interval , and this ratio approaches 1 from below as approaches the upper end of the interval at . It is useful here and below to give illustrative values of various quantities and ratios at an illustrative value of . For this purpose, we choose an approximately in the middle of the , namely . We have
This ratio provides an illustrative measure of the decrease in the value of the IR zero of when one calculates it at three-loop order, as compared with two-loop order.
iii.3 Four-Loop Level
At the four-loop level, the IR zero of is the (smallest positive) root of the cubic equation
Now for , and we recall our discussion above, that for in the scheme and other schemes that maintain the existence of the IR zero in at the three-loop level. We can therefore write Eq. (72) as
iii.4 Shift of IR Zero From -Loop to -Loop Level
In bc (), a general result was derived concerning the sign of the shift of the IR zero of going from the -loop level to the -loop level. Provided the scheme has the property that with are such as to maintain the existence of the zero in the two-loop function, then if if and if if . The same proof can be applied here to deduce that, provided that the scheme has the property that with are such as to maintain the existence of the zero in the two-loop function, then
We may apply this inequality for the comparisons of with and with . Since for , this result provides another way of deducing the inequality (64) for the two-loop versus three-loop comparison. Applying the general inequality for the three-loop versus four-loop comparison, we infer that
These inequalities are evident in Table 3. For example, at , , while for , . One sees that the magnitude of the fractional difference
is reasonably small. This is in agreement with one’s general expectation that if a perturbative calculation is reliable, then as one calculates this quantity to progressively higher-loop order, the magnitudes of the fractional differences between the values at the ’th and ’th orders should decrease.
iii.5 Summary of Results on IR Zero of
We summarize our findings concerning as follows. As one goes from the (scheme-independent) two-loop level to the three-loop level, the value of the IR zero of decreases. For in the lower part of the interval where the two-loop function has an IR zero, this reduction in the value of the IR zero is rather substantial. For example, for , near the lower end of the interval , , while for , near the upper end of , . Going from three-loop to four-loop order, the change in the value of the IR zero is smaller in magnitude and can be of either sign, depending on the value of . In general, both and are smaller than .
Iv Some Structural Properties of
For theories which exhibit an IR zero, , in the two-loop beta function, , there are several structural properties of interest in addition to higher-loop values of this IR zero. These include
the value of at which reaches a minimum in the interval , denoted , where the subscript denotes minimum
the value of at this minimum, denoted
the derivative of at , denoted
Note that because and , the factor of divides out in the derivative , so that
As was discussed in bc (), higher-loop calculations of the derivative are of interest because this enters into estimates of a dilaton mass in a quasiconformal gauge theory. In turn, this also provides one motivation for studying the LNN limit of this derivative, .
iv.1 Position of Minimum in
Concerning the position of the minimum in for , we calculate that at the two-loop level,
At the three-loop level in the scheme, we find
where we define the shorthand notation
This may be compared with the corresponding ratio of two-loop quantities . In contrast to the latter ratio, which is a constant, independent of , the ratio (93) is a monotonically decreasing function of . At the lower end of this interval,
As approaches the upper end of the at , the ratio (93) approaches the limit
Note that both and individually approach zero as , although their ratio in Eq. (95) approaches a constant.
Illustrative values in the LNN limit for are
so that for this value, , in addition to the ratio