LLarge N flavor β -functions: a recap B. Holdom ∗ Department of Physics, University of TorontoToronto ON Canada M5S1A7
Abstract β -functions for abelian and non-abelian gauge theories are studied in the regimewhere the large N flavor expansion is applicable. The first nontrivial order in the 1 /N expansion is known for any value of N α , and there are also various indications as tothe nature of higher order effects. The singularity structure as a function of
N α hasimplications for the existence of nontrivial fixed points.
For a sufficiently large number of flavors a non-abelian gauge theory will loose asymptoticfreedom and will in this way resemble an abelian gauge theory. We shall thus focus ourattention on the abelian case and then later extend the discussion to the nonabelian case.It is generally believed that a U (1) gauge theory with N charged fermions has a runningcoupling that grows monotonically towards the ultraviolet, and thus suffers from a Landaupole. This is the indication from the one-loop β -function. But there is much more knownabout the perturbative β -function and there have been recent calculations that have extendedour knowledge to 5-loops. We also have complete knowledge of the first nontrivial order inthe 1 /N expansion for any value of N α , and so this makes the large N expansion a usefulway to organize the perturbative expansion. The question is whether any of this allows usto glean anything further about the possible existence of nontrivial fixed points. Althoughany perturbative approach will introduce a renormalization scheme dependence, it can stillbe hoped that the existence or nonexistence of a fixed point will leave some mark on theperturbative results.According to a lattice result [1] there is no nontrivial fixed point in a U (1) gauge theoryfor N = 4. We shall be concerned with larger N where the large N expansion suggests otherpossibilities. This provides some motivation to study larger N values on the lattice as well,for sufficiently large values of N α . Extending the current lattice result to N = 8, 12, 16,...would appear to be relatively straightforward. ∗ [email protected] a r X i v : . [ h e p - ph ] O c t he U (1) β -function is defined as β ( α ) = ∂ ln α∂ ln µ . (1)The one loop result is β ( α ) = 2 A/ A ≡ N α/π . We may write an expansion in 1 /N asfollows, 32 β ( α ) A = 1 + ∞ (cid:88) i =1 F i ( A ) N i . (2)The “1” corresponds to the one loop result and we shall refer to it as the zeroth order term inthe 1 /N expansion. Each F i ( A ) represents a class of diagrams having the same dependenceon N when A is held fixed, and such diagrams exist to all orders in A . If the functions | F i ( A ) | were bounded then for sufficiently large N one could conclude that the zeroth order termdominates and that the Landau pole is unavoidable. But singularities in the F i ( A ) will keepus from drawing this conclusion.We collect together what is known about the F i ( A )’s in the MS renormalization scheme. F ( A ) = (cid:90) A I ( x ) dx (3) I ( x ) = (1 + x ) (2 x − (2 x − sin ( π x ) Γ ( x − Γ ( − x )( x − π (4) F ( A ) = − A + (cid:18) − ζ (3) (cid:19) A + (cid:18) π − ζ (3)144 (cid:19) A + ... (5) F ( A ) = − A + ... (6) F ( A ) = (cid:18) ζ (3) (cid:19) A + ... (7)Important for our study is the fact that F ( A ) is known completely [2]. We have expressed theintegrand I ( x ) in a form that makes more clear the location of its zeros and poles. The A terms in F ( A ) and F ( A ) were calculated in [3], the A term in F ( A ) in [4], and the F ( A )term in [5]. The latter two results are 5-loop calculations.One way to express the results in (3-7) is to plot their sum and ignore what is not known.The result for the 3 β ( α ) / A for various N is displayed in Fig. (1). A zero would indicate anontrivial fixed point, but the zeros are occurring at values of A that are too high to ignorehigher order terms. Thus we cannot deduce much from this plot, except to notice sensitivityof the β -function to N .The 2-loop contribution to β ( α ) involves one fermion loop and one internal photon andit gives rise to the first term in the expansion of F ( A ), which is A . The higher orderterms in F ( A ) correspond to the insertion of the appropriate number of fermion loops on thephoton line, and these bubble chains have been summed up to produce the result (3) [2]. An2 = 1 N = 2 N = 8 N = 4 A –10–8–6–4–20246810 1 2 3 4 5 6 β ( α ) / A Figure 1: The sum of the known contributions to 3 β ( α ) / A in (3-7).important feature of F ( A ) is that its expansion displays a finite radius of convergence, whichis nonvanishing due to the slower than factorial growth in the number of diagrams. We expectthis to be true of the other F i ( A )’s as well.Simple poles of alternating sign appear in I ( x ) at x = + n for integer n ≥
0. Theintegration can be handled with a Cauchy principal value prescription and the result is shownin Fig. (2). Clearly F ( A ) only changes logarithmically as the singular points are approached.And these singularities become weaker for larger n . Close to the first singularity at A = 15 / F ( A ) ≈ π (cid:60) (log(1 − A )) + 0 . . (8)Nevertheless even this weak singularity can cause the β -function at this order to vanish at afixed value of N . As Fig. (2) indicates there will be two nearly coincident zeros of 1 + F ( A ) /N at A = 152 ± . e − π N/ . (9)The lower (upper) one is an ultraviolet (infrared) fixed point. In either case the runningcoupling achieves its fixed point value at some finite scale µ ∗ , and at this scale the running ofthe coupling abruptly stops.It is useful to compare the β -function to the γ m -function defined as γ m ( α ) = − ∂ ln m∂ ln µ = ∞ (cid:88) i =1 G i ( A ) N i . (10) G ( A ) is also known to all orders in A , and in fact it is directly related to F ( A ). From the3 AF ( A ) Figure 2: F ( A ) as defined in (3).results of [2] one can deduce that dF ( A ) dA = 12 A (1 − A A G ( A ) . (11)Thus the singularities in these two functions occur at the same locations. This could beexpected since the bubble chain re-summation is intrinsic to both functions. The strength ofthe singularities are also related; the logarithmic singularities in F ( A ) correspond to simplepoles in G ( A ).It could be argued that the Cauchy principal value prescription used in the evaluation of F ( A ) is not unique. Rather than approaching the pole equally closely from the two sides,another prescription would be to approach the pole unequally from the two sides. This wouldshift the β -function on the right of a singularity by an additive constant, as allowed by (11).But this ambiguity does not alter the appearance of the fixed points at first order in 1 /N .It is important to know how large N has to be for the 1 /N expansion to be under control.We can require that the known expansion terms of the higher F i ( A )’s be sufficiently small for A as large as the radius of convergence of F ( A ), at A = 15 /
2. If we rescale A = 15 / A and N = 16 ˜ N then the expansion (2) numerically reads1 + 1˜ N ( . A − . A − .
567 ˜ A + 5 .
342 ˜ A + 1 .
60 ˜ A − . A + ... )+ 1˜ N ( − . A − .
602 ˜ A − .
244 ˜ A + ... ) (12)+ 1˜ N ( − . A + ... ) + 1˜ N ( . A + ... )4e see that ˜ N (cid:38) / ˜ N to beunder control. One can in particular compare the leading terms at each order in 1 / ˜ N , i.e. the˜ A i term at order 1 / ˜ N i . These terms correspond to the one fermion loop diagrams, and theyhave a special significance in that they are renormalization scheme independent [3].Thus for sufficiently large N the higher orders in 1 /N are under control in the usualsense of an asymptotic series. But the presence of singularities in the F i ( A ) indicates thatthe 1 /N expansion needs to be reconsidered for A close to these singularities. Although thesingularities of F ( A ) are logarithmic, we do not expect this to continue for the higher F i ( A ).The appearance of poles in G ( A ) reinforces this view. In particular if F ( A ) has a simplepole at A = 15 / F ( A ),as the exponentially small spacing in (9) makes clear.It is interesting that the first three coefficients in the expansion of F ( ˜ A ) are negative.This is certainly consistent with a pole, since the expansion of the expression˜ A .
771 ˜ A + 0 . A − .
494 (13)has the same first three terms. But we would argue that this does little to prove the existenceof a pole. In the F ( A ) expansion we note that the 2 fermion loop A term in F ( A ) receivescontributions from graphs with topology different from the 1 fermion loop A term. That is,unlike the case of F ( A ), dressing photon lines of the A graphs does not give all A graphs.In fact the graphs with the new topology (the light-by-light scattering contributions) givethe dominant contribution to the A term [3]. In this situation we do not expect a Padeapproximant to be very predictive until more orders in the expansion are known.An example of the appearance of a pole in an exact β -function is provided by SU ( N c )SUSY pure gluodynamics where [6, 7] β ( a ) = 3 a a − /N c , a ≡ α c π . (14)A pole of this sign, as in (13), means that the coupling will evolve towards the singularity inthe infrared, and it does so from both the weak and strong coupling sides of the pole. Thecoupling reaches the singularity at some minimum finite renormalization scale µ ∗ , and thegauge theory ceases to provide a description below this scale. This suggests that the theorydevelops a mass gap and/or some fundamentally different description is needed for energiesbelow µ ∗ . The authors of [8] argue that there is evidence of such a pole in the QCD β -function from the study of Pade approximants. A pole of opposite sign would instead producean ultraviolet cutoff on the gauge theory description.Beyond the A term in F ( A ), including the known A term, the diagrams are obtainedby simply inserting fermion loops into photon lines, thus building up bubble chains. This istrue for sufficiently high powers of A in the expansion of any F i ( A ); one ends up only dressing5 A UV β /A ( a ) AA IR A A UV ( b ) A A A UV ( c ) AA IR A ( d ) Figure 3: Schematic examples of the behavior of 3 β/ A when poles are present. The arrowsshow the infrared flow close to the poles.photons lines with fermion loops in graphs that belong to a basic set of topologies. It is forthis reason that all these functions should have a nonvanishing radius of convergence, and thebubble chains should generate singularities in the higher F i ( A ) just as they did for F ( A ) and G ( A ). We expect that the set of locations of singularities in the higher F i ( A ) will includethe locations of singularities in F ( A ).Let us consider a set of poles in F ( A ) occurring at the same locations at the singularitiesin F ( A ) (we shall provide more evidence for this shortly). Poles in F ( A ) at A n = + 3 n need to be considered along with the zeroth order term in (2). Depending on the sign of thesepoles, the result is the existence of various nontrivial infrared and/or ultraviolet fixed points.For example an infrared pole at A = 15 /
2, i.e. having the sign of 1 / ( A − A ) as in (13) and(14), implies an ultraviolet fixed point at some A UV < A (see Fig. (3a)). When the couplingis below A UV the theory flows to the known weakly coupled behavior in the infrared, but inthe ultraviolet the Landau pole has been eliminated in favor of a fixed point. Possibilities fornontrivial fixed points in the regions between successive poles at A i and A i +1 are shown inFigs. (3b,c,d).We now describe some evidence that is relevant not only to the existence of poles in F ( A ),but also to their location and sign. The first input comes from the γ m -function, since it turnsout that G ( A ) is also known to all orders in A . This comes from an impressive calculationin QCD [9], from which the corresponding QED result can be extracted. The surprisingly6imple result is that the singularities in G ( A ) occur at the same locations as in G ( A ), andeach simple pole in G ( A ) has been replaced by a double pole in G ( A ).Further input comes from examples of theories where the analogs of G ( A ) and F (cid:48) ( A )have the same singularity structure; that is the set of poles of these functions have the samelocation, sign and order of pole. In particular in N flavor ( φ ) theory in d = 4 − ε dimensionsthe critical exponents λ ( ε ) and ω ( ε ) have a correspondence to 1 − γ m and 2 β (cid:48) . In [10] thesecritical exponents are encoded in the corresponding critical exponents of the large N σ -modelin d = 2 − ε dimensions, where the functions appearing at order 1 /N i are labeled λ i ( ε ) and ω i ( ε ). These functions are the analogs of − G i ( A ) and F (cid:48) i ( A ). From the results in [10] thefunctions − λ ( ε ) and ω ( ε ) have singularities that match in location, sign and order of pole,as do − λ ( ε ) and ω ( ε ). Thus it would not be surprising for QED to display similar behavior, so that G ( A ) and F (cid:48) ( A ) exhibit a common singularity structure just as G ( A ) and F (cid:48) ( A ) do. Such behaviorcould be expected due to the simple relationship between the graphs contributing to F i ( A ) and G i ( A ), suggesting that the behavior also extends to higher orders. Cutting out an externalgauge field vertex from any graph contributing to F i ( A ) (vacuum polarization graphs) andreplacing the other external vertex with a mass insertion gives a graph contributing to G i ( A )(mass renormalization graphs). This is not true in QCD due to the gluon self-interactions,and so there is no corresponding behavior for QCD even at first order in 1 /N .Thus supposing that F (cid:48) ( A ) and G ( A ) have the same singularity structure, the signs ofthe poles in F ( A ) at A n = + 3 n are determined by the known results [9] for G ( A ). Thiswould imply that the poles in F ( A ) all have the same sign and that they are all ultravioletpoles, i.e. opposite in sign to (13) and (14). Thus of the various possibilities displayed inFig. (3), the available evidence suggests that only Fig. (3d) is correct for the interval betweenany two adjacent singularities.If this is correct then at this order in the 1 /N expansion there is an infinite number ofnew theories each flowing towards an associated nontrivial infrared fixed point. There isstill the weakly coupled theory that flows to the trivial infrared fixed point. All the theoriesare cut off in the ultraviolet by the UV poles. We also note that for large N some of thenontrivial infrared fixed points can occur at values of A smaller than the critical value forchiral symmetry breaking, A crit = N/ /N expansion close to thesingularities. As a singularity is approached the β -function experiences a significant change,and a change of 3 β/ A of order unity would be an indication that the 1 /N expansion isbreaking down for these values of A . The higher F i ( A )’s must be considered in these regions.For these functions to overcome the suppressions from large N and loop factors, they would ω ( ε ) has simple poles, while ω ( ε ) has an additional pole and all its poles are fourth order. F i ( A ) function has ( i − G ( A ) and G ( A ) functions.A 1 /N expansion with poles of ever increasing order would indicate the existence of a set ofessential singularities in the β -function. But the implications for fixed points remain similar toour discussion of simple poles. Whenever an essential singularity drives 3 β/ A by an amountof order unity or more in the right direction then a nontrivial fixed point can result.We may speculate further about the nature of the higher order poles based on an emergingpattern for poles of even or odd order. The adjacent logarithmic (even order) singularitiesin F ( A ) are alternating in sign while the simple (odd order) poles in F ( A ) are all of thesame sign. Let us consider the continuation of this pattern (alternating even order poles andsame sign odd order poles) to higher orders, where we treat the overall sign at any order asunknown. It means that on a given interval between adjacent singularities, and at any givenorder, 3 β/ A diverges with opposite signs on the two ends of the interval. The divergencesof the all-orders-summed result (assuming that the summed result diverges) can then alsobe expected to come with opposite signs on the two ends. This would imply at least onenontrivial ultraviolet or infrared fixed point on each such interval. Also, the sum of the oddorder poles produces a divergence pattern that is the same on every interval, while for the sumof the even order poles the pattern flips in sign on adjacent intervals. Thus the existence ofinfrared fixed points in the total result, at least on every second interval, is made more likelyif the odd poles are always of the ultraviolet variety as in Fig. (3d). Due to cancellations itis also conceivable that 3 β/ A approaches finite or even vanishing values at the endpoints ofsome intervals. Such behavior that depends on the side of approach can also be consistentwith essential singularities.We now turn to a SU ( N c ) gauge theory, where results are expressed in terms of theCasimirs, C G = N c , C R = ( N c − / N c , T R = 1 / N still denotes the number of flavors.The term in the β -function of zeroth order in the 1 /N expansion is as before, β ( α ) = 2 A/ ... ,if we choose a new definition for A ≡ N T R α/π . Similarly the 1 /N expansion is32 β ( α ) A = 1 + ∞ (cid:88) i =1 H i ( A ) N i . (15)From the results in [11] we can deduce that H ( A ) = − C G T R + (cid:90) A/ I ( x ) I ( x ) dx, (16) I ( x ) = C R T R + (20 − x + 32 x − x + 4 x )4 (2 x −
1) (2 x −
3) (1 − x ) C G T R . (17) This result is presented somewhat more explicitly than in [11], but its series expansion is in agreement. H ( A ) –16–14–12–10–8–6 0 1 2 3 4 5 6 7 Figure 4: H ( A ) as defined in (16) for N c = 3. I ( x ) is defined in (4) and so up to the definition of A and the C R /T R factor, the C R term isjust the QED result for F ( A ). By inspection of I ( x ) and I ( x ) one can see that the C G termbrings in a new pole in the integrand at x = 1 ( A = 3). This pole is of the same sign as thepole at A = 15 /
2. We show a plot of H ( A ) in Fig. (4) for N c = 3. Compared to the F ( A )function of QED, H ( A ) is negative and its first singularity is occurring at a smaller value of A (and α ).As in the QED case, a key question is how large N has to be for the 1 /N expansion to beunder control, for A at least as large as 3. It is apparent that higher N will be needed, sincefactors of N must now compete with factors of N c . If we rescale A = 3 ˜ A and N = 32 ˜ N thenthe expansion (15) for N c = 3 numerically reads1 + 1˜ N ( − . . A − . A − . A + . A − . A − . A + ... )+ 1˜ N ( − . A + 0 . A − .
003 ˜ A + ... ) (18)+ 1˜ N ( − . A + 1 .
073 ˜ A + ... ) + 1˜ N ( − . A + ... )The leading terms of the expansions of H ( A ), H ( A ), H ( A ) have been obtained from the4-loop results in [12]. Due to the pure glue contributions, each order begins with a lower powerof A as compared to QED. The main observation here is that large values of N are needed, N (cid:38)
32 or more, for the 1 /N expansion to be under control.For such large values of N , asymptotic freedom has been lost, and the implications of9 A + H ( A ) / N A crit Figure 5: 3 β ( α ) / A at first order in 1 /N where a zero occurs at the critical coupling for chiralsymmetry breaking ( N = 8 .
88 and N c = 3).the singularities in H ( A ) and the higher H i ( A )’s will resemble that of QED. The large N required also indicates that the large N expansion lacks quantitative control for the studyof other interesting phenomena that occur for values of N (cid:28) One such question isthe lowest value of N at which the Banks-Zaks fixed point survives before chiral symmetrybreaking occurs.For completeness we give the answer to this question while only keeping the 1 + H ( A ) /N terms in (15). We can find the N such that the zero of 1 + H ( A ) /N occurs at the usualestimate for the critical coupling for chiral symmetry breaking, or A crit = N T R / C R . For N c = 3 this gives N = 8 .
88 as the lower bound on the conformal window. This may becompared with the conventional result of N = 11 . β ( α ). We plot 1 + H ( A ) /N for N = 8 .
88 in Fig. (5).At order 1 /N , the singularity structure of the QCD β -function is significantly more com-plicated. And unlike QED, the singularity structures of the γ m and β -functions do not match.For example H (cid:48) ( A ) has the additional singularity at A = 3 when compared to the QCD γ m -function at order 1 /N , which is essentially the QED result G ( A ). As mentioned before, thiscan be understood diagrammatically in terms of additional contributions to the β -functiondue to the gluon self-coupling. Nevertheless there will be contributions to singularities in H (cid:48) ( A ) that can be associated with the singularities that appear at order 1 /N in the QCD γ m -function. Since the latter is known [9] we can deduce that the H ( A ) singularity struc- Of course if the study involved values of A significantly less than A = 3 then such large values of N wouldnot be needed. A = 3 /
2, a positive logsingularity at A = 3, a negative log singularity at A = 9 /
2, and IR poles of third order at A = 15 / n . Some of these singularities could change sign or be replaced by even strongersingularities in the full result for H ( A ). The UV pole is interesting since it would imply aninfrared fixed point just above A = 3 / N flavor expansion of the β -functions of QED and QCD. The logarithmic singularities thatappear at first order in 1 /N are the first signs of simple and higher order poles that willappear at higher orders in 1 /N . In the case of QED these singularities are expected to berelated to those of the γ m -function, for which the singularity structure is surprisingly simpleat order 1 /N . This gives information about the signs and locations of a set of poles in the β -function at this order. This in turn implies an infinite set of nontrivial infrared fixed points.The poles are likely to turn into essential singularities when all orders of the 1 /N expansionare considered, but the existence of nontrivial fixed points will persist. For QCD we haveindicated an even richer singularity structure for which we have less information. Acknowledgments
I am grateful to J. Gracey for bringing to my attention his results in [11], which extends the1 /N expansion to the QCD β -function. This work was supported in part by the NaturalScience and Engineering Research Council of Canada. References [1] J. B. Kogut, C. G. Strouthos, Phys. Rev. D71 (2005) 094012, arXiv:hep-lat/0501003.[2] A. Palanques-Mestre and P. Pascual: Comm. Math. Phys. 95 (1984) 277.[3] S. G. Gorishny, A. L. Kataev, S. A. Larin and L. R. Surguladze, Phys. Lett. B 256 (1991)81.[4] P. A. Baikov, K. G. Chetyrkin, J. H. Khn, Phys. Rev. Lett. 88 (2002) 012001, hep-ph/0108197.[5] P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, Proceedings of 8th International Sympo-sium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenol-ogy, 1-6 October 2007, Florence, Italy PoS (RADCOR 2007) 023, arXiv:0810.4048. This would be subdominant to the negative log singularity appearing in H ( A ).).