Anomalous dimensions at large charge in d=4 O(N) theory
LLTH1252February 24, 2021
Anomalous dimensions at large charge in d = 4 O ( N ) theory I. Jack and D.R.T. Jones Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK
Abstract
Recently it was shown that the scaling dimension of the operator φ n in λ ( ¯ φφ ) theorymay be computed semiclassically at the Wilson-Fisher fixed point in d = 4 − (cid:15) , for genericvalues of λn , and this was verified to two loop order in perturbation theory at leading andsubleading n . In subsequent work, this result was generalised to operators of fixed charge¯ Q in O ( N ) theory and verified up to three loops in perturbation theory at leading andsubleading ¯ Q . Here we extend this verification to four loops in O ( N ) theory, once againat leading and subleading ¯ Q . We also investigate the strong-coupling regime. [email protected] [email protected] a r X i v : . [ h e p - t h ] F e b Introduction
Renormalizable theories with scale invariant scalar self-interactions have been subjects of en-during interest. In particular, the study of theories with quartic ( φ ) interactions in d = 4 − (cid:15) dimensions has played a central role in the development of the theory of critical phenomena,since the pioneering work of Wilson [1, 2] and Wilson and Fisher [3] in 1971. Study of therenormalisation group flow of the coupling or couplings of the theories facilitates the determi-nation of the order of phase transitions and the associated critical indices. For example, thetheory with a single scalar field exhibits a Wilson-Fisher fixed point (FP) where the couplingconstant λ is O ( (cid:15) ), and this infra-red (IR) attractive FP is associated with a second orderphase transition.Historically, the majority of work in renormalisable quantum field theories has involvedthe weak coupling expansion, in other words the Feynman diagram loop expansion. Howeverthis expansion fails or becomes ponderous at either strong coupling or (less obviously) for φ n amplitudes at large n . The latter has obviously developed in importance as colliderenergies have increased. Remarkable progress [4–12] here came with the use of a semi-classicalexpansion in the path integral formulation of the theory .In Ref [8] the anomalous dimension of the φ n operator was considered in the O ( N )-invariant g ( φ ) theory with an N -dimensional scalar multiplet φ , for large n and fixed gn .In Ref. [9] the scaling dimension of the same operator in the U (1)-invariant λ ( ¯ φφ ) theory(corresponding to the special case N = 2) was computed at the Wilson-Fisher fixed point λ ∗ as a semiclassical expansion in λ ∗ , for fixed λ ∗ n . Subsequently this was generalised in Ref. [10]to the case of an operator of charge ¯ Q in the O ( N )-invariant theory. In Ref. [9], the U (1)result was compared with perturbation theory up to two loops, and in Ref. [10] the check wasperformed for the O ( N ) theory up to three loops. Here we proceed directly with the O ( N )case, since, at least for our purposes, many salient features of the analysis are very similarin both cases; and the results for U (1) may be recovered from those for O ( N ), essentially bysetting N = 2. We extend the comparison with perturbation theory up to four loops, andalso discuss the large ( g ¯ Q ) case, generalising the large λn analysis of Ref. [9].The paper is organised as follows: In Section 2 we describe the semiclassical calculationin the O ( N ) case, following Ref. [10]. Then in Section 3 we compare the result of thiscalculation with perturbative calculations up to and including 4 loops. This represents asignificant extension of previous calculations. In Section 4 we address the large ( g ¯ Q ) limitand compare in detail with earlier work. O ( N ) case In the O ( N ) case we have a multiplet of fields φ i , i = 1 . . . N , and the Lagrangian is L = 12 ∂ µ φ i ∂ µ φ i + g
4! ( φ i φ i ) . (2.1)The β -function for this theory is well-known [18]16 π β ( g ) = − (cid:15)g + g N + 8) − g N + 14) + O ( g ) , (2.2) An analogous analysis was pursued for φ theories for d = 3 − (cid:15) and φ theories for d = 6 − (cid:15) in Refs. [13–15] g ∗ = 3 (cid:15)N + 8 + 9(3 N + 14)( N + 8) (cid:15) + O ( (cid:15) ) . (2.3)As shown in Ref. [10], the fixed-charge operator of charge ¯ Q may be taken to be T ¯ Q = T i i ...i ¯ Q φ i φ i . . . φ i ¯ Q , (2.4)where T i i ...i ¯ Q is symmetric, and traceless on any pair of indices. The scaling dimension ∆ T ¯ Q is expanded as ∆ T ¯ Q = ¯ Q (cid:18) d − (cid:19) + γ T ¯ Q = (cid:88) κ = − g κ ∆ κ ( g ¯ Q ) . (2.5)We initially work in general d . The semiclassical computation of ∆ − and ∆ is performed bymapping the theory via a Weyl transformation to a cylinder R × S d − , where S d − is a sphereof radius R ; where the R φ ∗ φ term ( R being the Ricci curvature) generates an effective m φ ∗ φ mass term with m = d − R . This mapping process along with other technical simplifications [9]relies on conformal invariance and therefore we now assume that we are at the conformalfixed point in Eq. (2.3). It was shown in Ref. [9] that stationary configurations of the actionare characterised by a chemical potential µ , related to the cylinder radius R by Rµ ∗ = 3 + (cid:104) g ∗ ¯ Q + (cid:112) g ∗ ¯ Q ) − (cid:105) [6 g ∗ ¯ Q + (cid:112) g ∗ ¯ Q ) − (2.6)The computation of the leading contribution ∆ − is entirely analogous to the U (1) case andis given by 4∆ − ( g ∗ ¯ Q ) g ∗ ¯ Q = 3 [ x + √ x − + [ x + √ x − + 3 { + [ x + √ x − } [ x + √ x − , (2.7)where x = 6 g ∗ ¯ Q . Its expansion for small g ∗ ¯ Q takes the form∆ − ( g ∗ ¯ Q ) g ∗ = ¯ Q (cid:20) g ∗ ¯ Q −
29 ( g ∗ ¯ Q ) + 827 ( g ∗ ¯ Q ) − g ∗ ¯ Q ) + O (cid:8) ( g ∗ ¯ Q ) (cid:9)(cid:21) . (2.8)As in the U (1) case, for simplicity we give in Eq. (2.6) the result for d = 3. The non-leadingcorrections ∆ are once more given by the determinant of small fluctuations. There are twomodes corresponding to those in the abelian case, with the dispersion relation ω ± ( l ) = J l + 3 µ − m ± (cid:113) J l µ + (3 µ − m ) (2.9)where J l = l ( l + d − R (2.10)is the eigenvalue of the Laplacian on the sphere. In addition there are N − N − ω ±± ( l ) = (cid:113) J l + µ ± µ, (2.11)2ith J l as defined in Eq. (2.10). We then find that ∆ is given by∆ ( g ∗ ¯ Q ) = 12 ∞ (cid:88) l =0 σ l (2.12)where σ l = Rn l (cid:26) ω ∗ + ( l ) + ω ∗− ( l ) + (cid:18) N − (cid:19) [ ω ∗ ++ ( l ) + ω ∗−− ( l )] (cid:27) . (2.13)Here n l = (2 l + d − l + d − l + 1)Γ( d −
1) (2.14)is the multiplicity of the laplacian on the d -dimensional sphere, and ω ∗± , ω ∗ ++ , ω ∗−− are definedas in Eqs. (2.9), (2.11) respectively, evaluated at the fixed point with R , µ ∗ related by Eq. (2.6).For the small ( g ∗ ¯ Q ) computation, we need to isolate the divergent contribution in the sum inEq. (2.12). We use the large- l expansion of σ l , σ l = ∞ (cid:88) n =1 c n l d − n (2.15)with c = N, c =3 N,c = 12 [5 N − N + 2)( Rµ ∗ ) ] ,c = 12 [ N − N + 2)( Rµ ∗ ) ] ,c = N + 88 ( R µ ∗ − (cid:20) − (cid:18) γ − (cid:19) (cid:15) (cid:21) − R µ ∗ − (cid:15) − (cid:18) R µ ∗ − (cid:19) N (cid:15). (2.16)We can write ∆ ( g ∗ ¯ Q ) = − µ ∗ R + 6 µ ∗ R −
516 + 12 ∞ (cid:88) l =1 σ l + (cid:114) µ ∗ R − − (cid:18) N − (cid:19) [7 − Rµ ∗ + 6 R µ ∗ + 3 R µ ∗ ] , (2.17)where σ l = σ l − c l − c l − c l − c − c l . (2.18)Here the divergent parts have been isolated and the sums over l performed, as explained inRefs. [9] and [10]. The sum over l d − n for n = 5 leads to a pole in (cid:15) which cancels against thepole in the bare coupling. The sum over σ l is then finite and setting d = 4 and expanding insmall g ∗ ¯ Q can be performed analytically. We obtain∆ = −
16 (10 + N ) g ∗ ¯ Q + 118 (6 − N )( g ∗ ¯ Q ) N −
36 + 2(14 + N ) ζ ]( g ∗ ¯ Q ) −
181 [4( N −
73) + 2(6 N + 65) ζ + 5( N + 30) ζ ]( g ∗ ¯ Q ) + . . . (2.19)Adding Eqs. (2.8) and (2.19), we find [10]∆ − ( g ∗ ¯ Q ) g ∗ + ∆ ( g ∗ ¯ Q ) = ¯ Q + 16 [2 ¯ Q − ( N + 10)] g ∗ ¯ Q −
118 [4 ¯ Q + ( N − g ∗ ¯ Q ) + 127 [8 ¯ Q + N −
36 + 2( N + 14) ζ ]( g ∗ ¯ Q ) + (cid:110) − Q −
181 [4( N −
73) + 2(6 N + 65) ζ + 5( N + 30) ζ ] (cid:111) ( g ∗ ¯ Q ) + . . . (2.20) In this section we carry out the perturbative calculation to confirm the semiclassical resultat leading and next-to-leading order in ¯ Q ) up to four-loop level, as displayed in Eq. (2.20). (a) (b) (c) Figure 1: One- and two-loop diagrams for γ T ¯ Q contributing at leading n The one-loop contribution to γ T ¯ Q comes solely from the diagram depicted in Fig. 1(a) andis given by γ (1) T ¯ Q = − g ¯ Q (1 − ¯ Q ) . (3.1)As mentioned before, the derivation of the semiclassical result relied on working at the con-formal fixed point g ∗ . However, surprisingly, at two, three and four loops we will see thatthe functional forms of the semiclassical and perturbative results agree for general g and notjust on substitution of g = g ∗ with g ∗ as given in Eq. (2.3). It is only at one loop where theagreement only holds at the fixed point. Specifically, the leading terms ¯ Q (cid:0) d − (cid:1) + γ (1) T ¯ Q onthe left-hand side of Eq. (2.5) (as given in Eq. (3.1)) only agree with the O ( g ) and O ( g )terms in ∆ − ( g ¯ Q ) g + ∆ ( g ¯ Q ) on the right-hand side of Eq. (2.5) (as obtained from Eq. (2.20))after substituting g = g ∗ ≈ (cid:15)N +8 . In this case, specialising to the fixed point has induced amixing between the classical and one-loop O ( ¯ Q ) terms.The leading O ( ¯ Q ) two-loop contribution to γ T ¯ Q comes purely from the diagram depictedin Fig. 1(b) (with three lines emerging from the T ¯ Q vertex), while the next-to-leading O ( ¯ Q )contributions are generated by this diagram together with those in Fig. 1(c) (with two lines4merging from the T ¯ Q vertex). The contributions are given by γ (2)( b ) = − g ¯ Q ( ¯ Q − Q − , (3.2) γ (2)( c ) = − g (cid:18) N (cid:19) ¯ Q ( ¯ Q − , (3.3)producing leading and next-to-leading terms given by γ (2) T ¯ Q = −
118 ( g ¯ Q ) (4 ¯ Q − N ) , (3.4)in accord with the semiclassical results in Eq. (2.20). As emphasised earlier, this agreementholds for general g and not just at the conformal fixed point. This is because at two andhigher loops, in contrast to what we saw at one loop, specialising to the fixed point g = g ∗ as given in Eq. (2.3) does not induce any mixing between leading or next-to-leading termsat different loop orders. Therefore if Eq. (2.5) holds at the fixed point, it must also holdin general. In fact the agreement was already checked at the fixed point in Ref. [10] in thegeneral O ( N ) case, and in the U (1) case in Ref. [9]. (a) (b) (c) Figure 2: Three-loop diagrams for γ T ¯ Q contributing at leading n (a) (b) (c)(d) (e) Figure 3: Three-loop diagrams for γ T ¯ Q contributing at next-to-leading n The leading O ( ¯ Q ) three-loop contributions to γ T ¯ Q come purely from the diagrams de-picted in Fig. 2 (with four lines emerging from the T ¯ Q vertex), while the next-to-leading5raph Symmetry Factor Simple Pole2(a)
154 ¯ Q !( ¯ Q − −
227 ¯ Q !( ¯ Q −
154 ¯ Q !( ¯ Q −
127 ¯ Q !( ¯ Q − −
427 ¯ Q !( ¯ Q − A −
227 ¯ Q !( ¯ Q −
427 ¯ Q !( ¯ Q − A
827 ¯ Q !( ¯ Q − B ζ Table 1: Three-loop results from Figs. 2, 3 O ( ¯ Q ) contributions are generated by these diagrams together with those in Fig. 3 (withthree lines emerging from the T ¯ Q vertex).The simple pole contributions from individual three loop diagrams may be extracted fromRef. [17] and are listed in Table 1, together with the corresponding symmetry factor. A factorof g is understood in each case. The N -dependent factors A and B are given by A = 18 ( N + 6) , B = 116 ( N + 14) . (3.5)When added and multiplied by a loop factor of 3, the leading and non-leading three-loopcontributions to γ T ¯ Q are found to be γ (3) T ¯ Q = 127 ( g ¯ Q ) [8 ¯ Q + N −
36 + 2(14 + N ) ζ ] , (3.6)6nce again in accord with the semiclassical results in Eqs. (2.20), for general g . Equivalently,this agreement was already checked at the fixed point in Ref. [10].The leading O ( ¯ Q ) four-loop contributions to γ T ¯ Q come purely from the diagrams depictedin Fig. 4 (with five lines emerging from the T ¯ Q vertex), while the next-to-leading O ( ¯ Q )contributions are generated by these diagrams together with those in Fig. 5 (with four linesemerging from the T ¯ Q vertex). (a) (b) (c) (d)(e) (f) Figure 4: Four-loop diagrams for γ T ¯ Q contributing at leading n The simple pole contributions from the four-loop diagrams in Fig. 4 were readily evaluatedusing standard techniques (see for instance Ref. [18]). Those from Figs. 5, 6 may be extractedfrom Ref. [17]. The contributions from each four-loop diagram are listed in Tables 2, 3respectively, together with the corresponding symmetry factor. A factor of g is understoodin each case, and the N -dependent factor C is given by C = 132 ( N + 30) . (3.7)When added and multiplied by a loop factor of 4, the leading and non-leading three-loopcontributions to γ T ¯ Q are found to be γ (4) T ¯ Q = −
181 ( g ¯ Q ) [42 ¯ Q + 4( N −
73) + 2(6 N + 65) ζ + 5( N + 30) ζ ] , (3.8)once again in accord with the semiclassical results in Eqs. (2.20), for general g . g ∗ ¯ Q calculation In this section we discuss the large g ∗ ¯ Q limit of ∆ T ¯ Q . The large g ∗ ¯ Q limit of ∆ − as given byEq. (2.7) is readily obtained as∆ − g ∗ = 34 g ∗ (cid:34) (cid:18) g ∗ ¯ Q (cid:19) + 12 (cid:18) g ∗ ¯ Q (cid:19) + O (1) (cid:35) . (4.1)7raph Symmetry Factor Simple Pole4(a)
481 ¯ Q !( ¯ Q −
281 ¯ Q !( ¯ Q − −
181 ¯ Q !( ¯ Q − −
181 ¯ Q !( ¯ Q −
281 ¯ Q !( ¯ Q −
181 ¯ Q !( ¯ Q − − Table 2: Four-loop results from Fig. 48raph Symmetry Factor Simple Pole5(a)
281 ¯ Q !( ¯ Q − (11 − ζ )5(b)
281 ¯ Q !( ¯ Q − A (11 − ζ )5(c)
481 ¯ Q !( ¯ Q − A − Q !( ¯ Q − C ζ
881 ¯ Q !( ¯ Q − B (2 ζ − ζ )5(f)
881 ¯ Q !( ¯ Q − A −
281 ¯ Q !( ¯ Q − A (1 − ζ )5(h) Q !( ¯ Q − − − ζ )5(i) Q !( ¯ Q − − − ζ )5(j)
281 ¯ Q !( ¯ Q − − (1 − ζ )5(k)
181 ¯ Q !( ¯ Q − (1 − ζ )Table 3: Four-loop results from Fig. 59raph Symmetry Factor Simple Pole6(a)
481 ¯ Q !( ¯ Q − A − (5 − ζ )6(b)
881 ¯ Q !( ¯ Q − B (2 ζ + ζ )6(c)
481 ¯ Q !( ¯ Q − A −
281 ¯ Q !( ¯ Q − −
481 ¯ Q !( ¯ Q − A −
281 ¯ Q !( ¯ Q − − (5 − ζ )6(g)
281 ¯ Q !( ¯ Q − − (5 − ζ )6(h)
481 ¯ Q !( ¯ Q − − (5 − ζ )6(i)
481 ¯ Q !( ¯ Q − − (5 − ζ )6(j)
881 ¯ Q !( ¯ Q − A −
481 ¯ Q !( ¯ Q − − Table 4: Four-loop results from Fig. 610 a) (b) (c) (d)(e) (f) (g) (h)(i) (j) (k)
Figure 5: Four-loop diagrams for γ T ¯ Q contributing at next-to-leading n We follow the procedure described in Ref. [9] for evaluating ∆ by means of an approximationto the sum over l followed by a numerical fit. The procedure involves selecting integers N , N and picking A ≥ ARµ ∗ is an integer (this represents a cut-off in the summation,beyond which we approximate it by an integral). The accuracy may be made as great asdesired by increasing N , N and A . We obtain∆ = N + 816 ( R µ ∗ − ln( ARµ ∗ ) + F ( Rµ ∗ ) , (4.2)where F ( Rµ ∗ ) = f N ,A ( Rµ ∗ ) − σ ARµ ∗ + 12 ARµ ∗ (cid:88) l =0 σ l − N (cid:88) k =1 B k (2 k )! σ (2 k − ARµ ∗ , (4.3)and here f N ,A ( Rµ ∗ ) = 12 ( ARµ ∗ ) N (cid:88) n =1 ,n (cid:54) =5 c n ( ARµ ∗ ) n − ( n − a) (b) (c) (d)(e) (f) (g) (h)(i) (j) (k) Figure 6: Four-loop diagrams for γ T ¯ Q contributing at next-to-leading n (continued)+ N + 816 ( R µ ∗ − (cid:18) γ − (cid:19) − R µ ∗ − − (cid:18) R µ ∗ − (cid:19) N. (4.4)With some help from one of the authors [19] we have corrected some typos in the correspondingequations in Ref. [9], which were not reflected in their final results. The function f N ,A ( Rµ ∗ )derives from replacing the sum over l for l ≥ ARµ ∗ in Eq. (2.12) by an integral over l . It is thenappropriate to use the large l expansion in Eq. (2.15). The integral over l (cid:15) correspondingto the c term leads to a pole term in (cid:15) . The potential pole in ∆ is cancelled by the polein the bare coupling, but the O ( (cid:15) ) term in c in Eq. (2.16) leads to the terms in the last lineof Eq. (4.4). The details of the procedure may be found in Ref. [9]. In Eq. (4.3), we canset d = 4. We now evaluate F ( Rµ ∗ ) in Eq. (4.3) numerically. We take N = 4, N = 10and A = 10, using the same numbers as Ref. [9] for comparison purposes. The result is thenfitted with an expansion in ( Rµ ∗ ) − , starting from ( Rµ ∗ ) , with 4 parameters. We find that F ( Rµ ∗ ) is given by F ( Rµ ∗ ) ∼ − (1 . . N )( Rµ ∗ ) + (1 . . N )( Rµ ∗ ) − (0 . . N ) + O (( Rµ ∗ ) − ) , (4.5)12nd this may be inserted into Eq. (4.2) to give the full result for ∆ . Expanding Rµ ∗ as givenby Eq. (2.6) in terms of large g ∗ ¯ Q , we find Rµ ∗ = (cid:18) g ∗ ¯ Q (cid:19) + 13 (cid:18) g ∗ ¯ Q (cid:19) − + . . . (4.6)and then we obtain from Eq. (4.2)∆ = (cid:20) α + N + 848 ln (cid:18) g ∗ ¯ Q (cid:19)(cid:21) (cid:18) g ∗ ¯ Q (cid:19) + (cid:20) β − N + 872 ln (cid:18) g ∗ ¯ Q (cid:19)(cid:21) (cid:18) g ∗ ¯ Q (cid:19) + O (1) , (4.7)where α = − . − . N,β = − . − . N. (4.8)The results for U (1) should be recovered by setting N = 2; and indeed for N = 2 we findEqs. (4.5), (4.7), (4.8) agree with the corresponding results given in Ref. [9].Following Ref. [9] and combining Eqs. (2.3), (2.5), (4.1) and (4.7), we may write the fullscaling dimension in the form∆ T ¯ Q = 1 (cid:15) (cid:18) (cid:15) ¯ QN + 8 (cid:19) dd − (cid:20) N + 8)16 + (cid:15) (cid:18) α + 3(3 N + 14)16( N + 8) (cid:19) + O ( (cid:15) ) (cid:21) + 1 (cid:15) (cid:18) (cid:15) ¯ QN + 8 (cid:19) d − d − (cid:20) N + 88 + (cid:15) (cid:18) β − N + 148( N + 8) (cid:19) + O ( (cid:15) ) (cid:21) + O [( (cid:15) ¯ Q ) ] (4.9)In Ref. [15], we found that we could reproduce the coefficients in the large Rµ ∗ expansion ofthe N -dependent part of ∆ (the terms involving ω ∗ ++ and ω ∗−− in Eq. (2.13)) by an analyticcomputation. This fails to work here; an analytic large- Rµ ∗ expansion of ω ∗ ++ and ω ∗−− asgiven by Eq. (2.11) leads to odd negative powers of Rµ ∗ , whereas our numeric computation inEq. (4.5) only contains even powers of Rµ ∗ . It appears that the simple properties of ω ∗ ++ and ω ∗−− identified in Ref. [15], in particular their expansion in powers of J l R µ ∗ , are not enoughfor our analytic computation to work in the d = 4 case. A little trial and error indicates thatthe fact that in d = 3, n l ∝ ddl J l , may also be crucial; but further insight is required. Approaches that extend the reach of (or even transcend the need for) perturbation theoryhave always been challenging, and are all the more interesting now because of the increasedimportance attached to multi-leg amplitudes, which can present formidable calculational ob-stacles at higher loop orders. In this paper we have followed Refs. [8–10] in the applicationof semi-classical methods to the calculation of φ n amplitudes in d = 4 renormalisable scalartheories with quartic interactions. Ref [10] generalises this calculation of Ref. [9] from U (1)to an O ( N ) invariant interaction. Another motivation for studying this class of theories is13heir (classical) scale invariance (CSI). As remarked in Ref [10], the Standard Model (SM) is“almost” CSI. Indeed, in 1973, Coleman and Weinberg (CW) [20] had hoped to argue that theSM might indeed be viable with the omission of the Higgs (wrong-sign) (mass) term. Thisattractive idea failed. Neglecting Yukawa couplings (which seemed reasonable at the time)led to a Higgs mass prediction which was too small; but including the top quark Yukawacoupling destabilised the Higgs vacuum altogether . CW introduced the idea of dimensionaltransmutation as a means of generating a physical mass scale in a CSI theory. The samephenomenon has been pursued [22–24] in the CSI form of quantum gravity [25–30].Our purpose here has been to compare the results of Ref [10] with straightforward (albeitintricate) perturbation theory. Generally the results have supported the validity of the semi-classical approximation, in its domain of validity.Future work might include the application of the semi-classical methods and perturbativemethods used here to the remaining class of CSI theories with scalar self-interactions; that is φ theories in d = 6; or even perhaps the case of CSI quantum gravity mentioned above. Acknowledgements
We are grateful to Gabriel Cuomo for helpful correspondence. DRTJ thanks the LeverhulmeTrust for the award of an Emeritus Fellowship. This research was supported by the Lever-hulme Trust, STFC and by the University of Liverpool.
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