Higher-Loop Structural Properties of the Function in
Vectorial Gauge Theories
We investigate some higher-loop structural properties of the function in asymptotically free vectorial gauge theories. Our main focus is on theories with fermion contents that lead to an infrared (IR) zero in . We present analytic and numerical calculations of the value of the gauge coupling where reaches a minimum, the value of at this minimum, and the slope of at the IR zero, at two-, three-, and four-loop order. The slope of at the IR zero is relevant for estimates of a dilaton mass in quasiconformal gauge theories. Some inequalities are derived concerning the dependence of the above quantities on loop order. A general inequality is derived concerning the dependence of the shift of the IR zero of , from the -loop to the -loop order, on the sign of the -loop coefficient in . Some results are also given for gauge theories with supersymmetry.
The evolution of an asymptotically free gauge theory from high Euclidean momentum scales in the deep ultraviolet (UV) to small scales in the infrared is of fundamental field-theoretic interest. This evolution is described by the function of the theory. Following the pioneering calculations of the function at one-loop b1 () and two-loop b2 () order, this function was subsequently calculated to three-loop b3 () and four-loop b4 () order in the modified minimal ms () subtraction () scheme msbar (). The anomalous dimension of the (gauge-invariant) fermion bilinear operator, , has also been calculated up to four-loop order in this scheme gamma4 ().
Here we consider the UV to IR evolution of an asymptotically free vectorial gauge theory with gauge group and massless fermions transforming according to a representation of mf (). An interesting property of this type of theory is that, for sufficiently large , the two-loop function has an IR zero b2 (); bz (). If is near to the maximum allowed by the property of asymptotic freedom, then this IR zero occurs at a small value, but, as decreases, it increases to stronger coupling. This motivates the calculation of the IR zero of at higher-loop order gk98 (). Calculations of this IR zero, and the associated anomalous dimension of the (gauge-invariant) fermion bilinear, , have recently been 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 bvh (); ps (). A corresponding analysis was carried out for an asymptotically free vectorial gauge theory with supersymmetry in bfs (). Although the terms in the function at three- and higher-loop order, and the terms in at two- and higher-loop order are dependent on the scheme used for regularization and renormalization of the theory, these higher-loop calculations are valuable because they give a quantitative measure of the accuracy and stability of the lowest-order calculations of and . A study of the effect of scheme transformations on results for was performed in sch ().
In this paper we will present calculations at the -loop level, where , of several important quantities that provide a detailed description of the UV to IR evolution of a theory with an IR zero in its function. Our general results apply for an arbitrary (non-Abelian) gauge group . We denote the running gauge coupling at a scale as , and define . (The argument will often be suppressed in the notation.) The loop order to which a quantity is calculated is indicated explicitly via the subscript , standing for -loop, so that the -loop function and its IR zero are denoted and . Given the asymptotic freedom of the theory, the UV to IR evolution, as described by , occurs in the interval
In addition to , the three structural properties of that we study are (i) the value of where reaches its minimum in the interval (1), denoted , (ii) the minimum value of on this interval, , and (iii) the slope of at , denoted . The importance of the first two quantities for the UV to IR evolution of the theory is clear. One would like to know where the rate of running, , has maximum magnitude, as a function of , and hence, as a function of . Further, one is interested in what this maximum magnitude in the rate of running, i.e., (since ), the minimum value of is in the interval . The third quantity, the slope of the function at , is of interest because it describes how rapidly approaches zero as approaches . A knowledge of this slope is also valuable because it is relevant for estimates of a dilaton mass in gauge theories that exhibit approximate scale-invariance associated with an IR zero of at a value, , that is sufficiently large that this approximate dilatation symmetry is broken by the formation of a fermion condensate. For each of these structural quantities, one would like to see how higher-loop calculations compare with the two-loop computation. As part of our work, we derive some inequalities concerning the relative values of each of these quantities at the two- and three-loop order.
We also generalize some results that were obtained in bvh () concerning . In bvh () it was shown that in a theory with a given , , and for which the two-loop function has an IR zero, the three-loop zero satisfies the inequality in the minimal subtraction () scheme used there alfdif34 (). The reduction in the value of the IR zero going from two-loop to three-loop level is typically substantial; for example, for and , , while , and for and , , while . A natural question that arises from the analysis in bvh () is how general this inequality is and, specifically, whether it also holds for other schemes. We address and answer this question here. We prove that for an asymptotically free theory with a given , , and for which has an IR zero, the inequality holds in any scheme which has the property that the sign of the three-loop coefficient in is opposite to that of the one-loop coefficient for , which thus preserves for the existence of an IR zero that was true of . This preservation of the two-loop IR zero in is physically desirable, since is scheme-independent, so if it exhibits an IR zero, then a reasonable scheme should maintain the existence of this zero at higher-loop level. More generally, we will derive a result that shows how shifts, upward or downward, to , when it is calculated to the next higher-loop order.
For a given gauge group , the infrared properties of the theory depend on the fermion representation and the number of fermions, . For a sufficiently large number, , of fermions in a given representation (as bounded above by the requirement of asymptotic freedom), the IR zero in occurs at a relatively small value of and the theory evolves from the UV to the IR without any spontaneous chiral symmetry breaking (SSB). In this case, the IR zero of is an exact infrared fixed point of the renormalization group. Thus, the infrared behavior of the theory exhibits scale-invariance (actually conformal invariance dilconf ()) in a non-Abelian Coulomb phase. For small , as the theory evolves from the UV to the IR, and the reference scale decreases below a scale which may be denoted , the gauge interaction becomes strong enough to confine and produce bilinear fermion condensates, with the associated spontaneous chiral symmetry breaking and dynamical generation of fermion masses of order . As decreases below , and one constructs the effective low-energy field theory applicable in this region, one thus integrates out these now-massive fermions, and the function changes to that of a pure gauge theory, which does not have any perturbative IR zero. Hence, in this case the infrared zero of is an approximate, but not exact, fixed point of the renormalization group.
If is only slightly less than the critical value for spontaneous chiral symmetry breaking, so that is only slightly greater than the critical value, (depending on and ) for fermion condensation, then the UV to IR evolution exhibits approximate scale (dilatation) invariance for an extended logarithmic interval, because as increases toward , while less than , approaches zero, i.e., the rate of change of as a function of approaches zero. Thus, is large, of O(1), but slowly running (“walking”). This is quite different from the behavior of in quantum chromodynamics (QCD). This approximate scale invariance at strong coupling plays an important role in models with dynamical electroweak symmetry breaking otherwtc (); chipt (), and occurs naturally in models with an approximate infrared fixed point chipt (). Since and is a power series in , there is an enhancement of in such models, which, in turn, is useful for generating sufficiently large Standard Model fermion masses. Approximate calculations of hadron masses and related quantities have been performed using continuum field theoretic methods for these theories sg (). Recently, an intensive effort has been made using lattice methods to study the properties of SU() gauge theories with various fermion contents, in particular, theories that exhibit quasi-scale-invariant behavior associated witn an exact or approximate IR zero of the respective functions. For example, for SU(3) with fermions in the fundamental representation, measurements of have been reported in lgt (). In theories where , , and are such that is only slightly greater than , so this approximate scale invariance associated with an IR zero of at strong coupling holds, the spontaneous breaking of this symmetry by the formation of a fermion condensate, may lead to a light state which is an approximate Nambu-Goldstone boson (NGB), the dilaton dil () (see also sg ()). The mass of the dilaton depends on several quantities, including the effective value of the function at the relevant scale where the SSB takes place. The desire to study the quasi-scale-invariant behavior of such a theory is an important motivation for obtaining more detailed information about the structure of the function, as contained in the structural quantities (i)-(iii) discussed above.
Ii Beta Function
The UV to IR evolution of the theory is described by the function
where . This has the series expansion
where denotes the number of loops involved in the calculation of , , and
As noted above, the one- and two-loop coefficients and , which are scheme-independent, were calculated in b1 (); b2 () (see Appendix A). The for are scheme-dependent; in the commonly used scheme, the have been calculated up to four-loop order b3 (); b4 (). For analytical purposes it is more convenient to deal with the , since they are free of factors of . However, for numerical purposes, it is usually more convenient to use the , since, as is evident from Table I of bvh (), the range of values of is smaller than the range for the . We will use both interchangeably. We denote the function calculated to -loop order as
Some explicit examples of four-loop functions are given in Appendix B.
With our sign conventions, the restriction to an asymptotically free theory means that . This is equivalent to the condition
( stands for zero). For the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations of , the upper bound (6) allows the following ranges of : (i) for fundamental, (ii) for adjoint, (iii) for symmetric (antisymmetric) rank-2 tensor. In the case of a sufficiently large representation , this upper bound may forbid even the value . For example, for the rank-3 symmetric tensor representation of , the upper bound is , and the right-hand side of this bound is larger than 1 only for (analytically continued to nonnegative real numbers) in the interval , i.e., (to the indicated floating-point accuracy). Hence, if is equal to 2, 3, or 4, the bound allows only the single value , and if , then the bound does not allow any nonzero (integer) value of . For , with a representation labeled by the integer or half-integer , the inequality (6) is
This bound is: (i) if ; (ii) if ; (iii) if . The right-hand side of (8) decreases through 1 as (continued to real numbers) increases through 1.562, so that the upper bound (8) does not allow a nonzero number of fermions in a representation of SU(2) with othergg ().
To analyze the zeros of the -loop function, , aside from the double zero at , one extracts the overall factor of and calculates the zeros of the reduced polynomial
or equivalently, . As is clear from Eq. (9), the zeros of away from the origin depend only on ratios of coefficients, which can be taken as for . Although Eq. (9) is an algebraic equation of degree , with roots, only one of these is physically relevant as the IR zero of . We denote this as . In analyzing how the -loop function describes the UV to IR evolution of the theory, we will focus on the interval (1).
To investigate how changes when one calculates it to higher-loop order, it is useful to characterize the full set of zeros of . In general, if one has a polynomial of degree , , and one denotes the set of roots of the equation as , then the discriminant of this equation is defined as disc ()
Since is a symmetric polynomial in the roots of the equation , the symmetric function theorem implies that it can be expressed as a polynomial in the coefficients of symfun (). We will sometimes indicate this dependence explicitly, writing . The discriminant is a homogeneous polynomial of degree in the roots . For our present purpose, to analyze the zeros of away from the origin, given by the roots of Eq. (9), of degree , we will thus use the discriminant , or equivalently, . Note that, because of the homogeneity properties,
Some further details on discriminants are given in Appendix C.
Although we focus on the behavior of in the physical interval (1), in characterizing the zeros of , we will make use of some formal mathematical properties of as an abstract function of . For large , since , it follows that if is large and positive, then , while for large negative , . Thus, for large positive is equal to for large negative . Since is negative in the vicinity of the origin, it follows that for (both even and odd) ,
Furthermore, again because for large , a consequence is that for ,
Of course, the behavior of at negative values of is not directly physical, and the behavior at large positive is beyond the range of validity of the perturbative calculation, but these mathematical properties will be useful in characterizing the total set of zeros of at higher-loop order.
Given that , so that has an IR zero, we can track how this zero changes as the loop order increases. One general result is as follows. As at the upper end of the interval , . This is a result of the fact that in this limit, , so that reduces to , which has a root at . Starting at the -loop level and tracking the physical IR zero at three- and higher-loop order, one can infer that generically is the root of that moves toward zero in this limit .
Because is a polynomial in and hence a continuous function, and because at the two ends of the interval (1), at and , and is negative for small (positive) , it follows that reaches a minimum in this interval (1). This occurs at a point where , which we label (where the subscript stands for “minimum in ”), and we denote
From Eq. (5), one calculates , with the result
The equation for the critical points, where , is thus an algebraic equation of degree , with formal roots, one of which is . Assuming that , i.e., , so that the two-loop function has an IR zero (and also, in higher-loop calculations, that the scheme preserves the existence of this IR zero), it follows that, among the remaining roots, one is real and positive and yields the minimum value of for in the relevant interval (1), and this root is the above-mentioned .
Given that has an IR zero at and is analytic at this point, one may expand it in a Taylor series about . This involves the slope of the function at . For compact notation, we denote
and, for the -loop quantities,
With , the expansion of for near to is
Here we have written this expansion for the full function; a corresponding equation applies for .
ii.2 IR Zero of at the Two-Loop Level
We next review some background on the two-loop function that is relevant for present work. The two-loop function is . This has an IR zero at
which is physical if and only if . The coefficient is a linear, monotonically decreasing function of , which is positive for zero and small and passes through zero, reversing sign, as increases through , where
For arbitrary and , , as is proved by the fact that
Hence, there is always an interval of values for which the two-loop function has an IR zero, namely
For example, in the case of fermions in the fundamental representation, denoted ,
so that, for , the interval is ; for , is ; and as , approaches .
Since we are primarily interested in studying the IR zero of and since the presence or absence of an IR zero of the two-loop function, , is a scheme-independent property, we focus on , where this IR zero of is present. A general result is that for a given gauge group and fermion representation and , is a monotonically decreasing function of . As decreases from , increases from 0. As decreases through a value labeled , increases through a critical value, , where fermion condensation takes place. Thus,
The value of is of fundamental importance in the study of a non-Abelian gauge theory, since it separates two different regimes of IR behavior, viz., an IR conformal phase with no SSB for and an IR phase with SSB for . As approaches at the lower end of the interval , becomes too large for Eq. (25) to be reliable.
Because of the strong-coupling nature of the physics at an approximate IR fixed point with , there are significant higher-order corrections to results obtained from the two-loop function, which motivated the calculation of the location of the IR zero in , and the resultant value of evaluated at this IR zero, to higher-loop order for a general , , and bvh (); ps ().
ii.3 Function and Dilaton Mass
Here we focus on a theory in which the IR zero of , , is slightly greater than , so that, in the UV to IR flow, there is an extended interval in over which is approaching from below, but is still less than . In this interval, , but is small, and hence the theory is approximately scale-invariant. As decreases through , increases through , the fermion condensate forms, and the fermions gain dynamical masses, this approximate scale invariance is broken spontaneously. In terms of the (symmetric) energy-momentum tensor , the dilatation current is , and one has , where is the field-strength tensor for the gauge field. When taking matrix elements, the deviation of this divergence from zero, i.e., the nonconservation of the dilatation current, thus arises from two sources, namely the facts that is not exactly equal to zero and the nonzero value of the matrix element of , defined appropriately at the scale . An analysis of the matrix element of between the vacuum and the dilaton state , in conjunction with a dimensional estimate of the gluon matrix element, and the Taylor series expansion (24) evaluated with yields the resulting estimate for the dilaton mass dil ()
In terms of -loop level quantities, the right-hand side of Eq. (31) . The importance of the slope at , , and the -loop calculation of this slope, , in estimating a dilaton mass in a quasiconformal theory is evident from Eq. (31). As is the case with , because of the strong-coupling nature of the physics, it is valuable to compute higher-loop corrections to the two-loop result, . Below, we will present two- and higher-loop analytic and numerical calculations of . Other effects on have been discussed in the literature dil (), including the effect of dynamical fermion mass generation associated with the spontaneous chiral symmetry breaking as descends through the value . Owing to this and other nonperturbative effects on , we restrict ourselves here to presenting one input to this calculation, namely , for which we can give definite analytic and numerical results.
ii.4 IR Zero of at the Three-Loop Level
Let us assume that , so that has an IR zero. Here we analyze how this IR changes as one calculates the function to three-loop order, extending our results in bvh () to (an infinite set of) schemes more general than the scheme used in that paper. Since the existence of of the IR zero in the two-loop function is a scheme-independent property of the theory, it is reasonable to restrict to schemes that preserve this IR zero of at the three-loop level. We first determine a condition for this to hold.
The three-loop function is , so, aside from the double zero at (the UV fixed point), vanishes at the two roots of the factor , namely,
where . The analysis of the IR zero of requires an consideration of the sign of . The condition that have an IR zero requires, in particular, that its two zeros away from the origin be real, i.e., that . For a given , , and , so that and are fixed, this condition amounts to an upper bound on , namely . Now, at the lower end of the interval , so that, insofar as one considers the analytic continuation of from positive integers to positive real numbers, the above bound generically requires that for . This is also required if , and one studies the theory in the limit and with fixed, since in this case there are discrete pairs of values that enable one to approach arbitrarily close to the lower end of the interval at where . In order to preserve the existence of the two-loop IR zero at the three-loop level, one is thus motivated to restrict to schemes in which for , and we will do so here. (The marginal case is not generic, since varies as a function of and , so we will not consider it further.)
Before proceeding, it is worthwhile to recall how the property for arises in the scheme. In this scheme, is a quadratic function of with positive coefficients of the term and the term independent of . This coefficient vanishes, with sign reversal, at two values of , denoted and , given as Eq. (3.16) in bvh (), with for (and for and . In bvh () it was shown that in this scheme, for all of the representations considered there, namely, the fundamental ( ), adjoint, and rank-2 symmetric () and antisymmetric () tensor representations, and , so that for all . For example, for fermions in the representation, (i) for , and ; (ii) for , , and ; (iii) as , while and , while . In Table 2 we list the values of with calculated in the scheme, for the illustrative cases and in the respective intervals. Since for in this scheme, it follows that all of the entries in this table have .
Given that for , Eq. (34) can be rewritten as . The solution with a sign in front of the square root is negative and hence unphysical; the other is positive and is , i.e.,
In bvh () it was shown that in the scheme, for all , . Here we demonstrate that this result holds more generally than just in the scheme. We prove that for arbitrary gauge group , fermion representation , and , in any scheme in which for (which is thus guaranteed to preserve the IR zero present at the two-loop level), it follows that . To prove this, we consider the difference
The expression in square brackets is positive if and only if
This difference is equal to the nonnegative quantity , which proves the inequality. Note that, since is nonzero for asymptotic freedom, this difference vanishes if and only if , in which case . We have therefore proved that
As noted above, is a monotonically decreasing function of . With for , this monotonicity property is also true of . As increases from to in the interval , decreases from
to zero as at the upper end of this interval, vanishing like
as and .
ii.5 IR Zero of at the Four-Loop Level
The four-loop function is , so has three zeros away from the origin, at the roots of the cubic equation
(where was given in Eq. (9)). These zeros were analyzed for the scheme in bvh (); ps (). Here we extend this analysis to a more general class of schemes that have for , and hence maintain at the three-loop level the IR zero of the scheme-independent two-loop function.
The nature of the roots of Eq. (41) is determined by the sign of the discriminant , or equivalently,
The following properties of are relevant here: (i) if , then all of the roots of Eq. (41) are real; (ii) if , then Eq. (41) has one real root and a complex-conjugate pair of roots; (iii) if , then at least two of the roots of Eq. (41) coincide. Given the scheme-independent properties and (i.e., ), and provided that for , we can write this discriminant as
If were zero, then the zeros of would coincide with those of , and the property that these are all real is in accord with the reduction
which is positive.
Now consider nonzero . First, assume that the scheme has the property that . Then we can write Eq. (41) as
From an application of the Descartes theorem on roots of algebraic equations, it follows that there are at most two (real) positive roots of this equation and at most one negative root. Moreover, from Eq. (19) we can deduce that in this case with , in addition to the double zero at , has a zero at a negative value of , so the upper bound on negative zeros from the Descartes theorem is saturated. Furthermore, since is negative at large positive , there are then two possibilities: either the two remaining zeros of Eq. (41) are a complex-conjugate pair, or else they are both real and positive.
If, on the other hand, the scheme is such that , then we can write Eq. (41) as
From a similar application of the Descartes theorem, we infer that there is at most one positive real root and at most two negative real roots of Eq. (50). From Eq. (14) we deduce that has a zero at a positive real value of . Depending on , the other two roots of Eq. (50) may be real and negative or may form a complex-conjugate pair.
Combining the information from both the Descartes theorem and the discriminant , we derive the following conclusions about the roots of Eq. (41) and hence the zeros of aside from the double zero at . As before, we assume that (i.e., ) so that has an IR zero, and also that the scheme is such that for , guaranteeing that this IR zero is maintained at the three-loop level. Then,
If and , then Eq. (41) has one negative and two positive real roots,
If and , then Eq. (41) has one negative root and a complex-conjugate pair of roots,
If and , then Eq. (41) has one positive root and two negative roots,
If and , then Eq. (41) has one positive root and a complex-conjugate pair of roots.
For a particular pair , the marginal case might occur, and would mean that two of the roots of Eq. (41) are degenerate. Since this equation is a cubic, it would follow that all of the roots are real. If and , then Eq. (41) has one negative root and a positive root with multiplicity 2, while if , then (41) has one positive root and a negative root with multiplicity 2.
It is reasonable to avoid schemes that lead to the outcome (2) above, with no real positive root of Eq. (41), since these fail to preserve the IR zero of the scheme-independent two-loop function. Although the positivity of is not a necessary condition for this preserving of the IR zero, it is a sufficient condition. We thus investigate the conditions under which is positive. As shown via Eq. (48), if , then . By continuity, for small , remains positive, and there is only a small shift in the two zeros that were present in , together with the appearance of a new zero. Since the highest-degree term in involving , namely is negative-definite, it follows that, other things being equal, for sufficiently large , will decrease through zero and become negative. The two values at which