Higher-Order Scheme-Independent Calculations of Physical Quantities in the Conformal Phase of a Gauge Theory

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by  Thomas A. Ryttov

Higher-Order Scheme-Independent Calculations of Physical Quantities in the
Conformal Phase of a Gauge Theory
Thomas A. Ryttova and Robert Shrockb
(a) CP3-Origins and Danish Institute for Advanced Study
University of Southern Denmark, Campusvej 55, Odense, Denmark and
(b) C. N. Yang Institute for Theoretical Physics
Stony Brook University, Stony Brook, NY 11794, USA
WeconsideranasymptoticallyfreevectorialSU(Nc)gaugetheorywithNf masslessfermionsina
representationR,havinganinfrared fixedpoint(IRFP)oftherenormalization group atαIR inthe
conformal non-Abelian Coulomb phase. The cases with R equal to the fundamental, adjoint, and
symmetricrank-2tensorrepresentationareconsidered. Wepresentscheme-independentcalculations
7 oftheanomalous dimension γψ¯ψ,IR toO(∆4f) and βI′R toO(∆5f)at thisIRFP,where ∆f is an Nf-
1 dependentexpansion parameter. Comparisons are made with conventional n-loop calculations and
0 latticemeasurements. Asatestoftheaccuracyofthe∆f expansion,wecalculateγψ¯ψ,IR toO(∆3f)
2 in N =1 SU(Nc) supersymmetric quantum chromodynamics and find complete agreement, to this
order, with the exactly known expression. The ∆ expansion also avoids a problem in which an
n f
IRFPmay not bemanifest as an IR zero of a higher n-loop beta function.
a
J

1 A fundamental problem in quantum field theory con- series expansions of γ¯ and β′ are [8]
2 ψψ,IR IR
cerns the properties at a conformal fixed point of the
h] rsievneorcmurarleiznattiionnvegsrtoiguapt.ioAn csopnecceifirncsqtuheestipornopuenrdtieersionfteann- γψ¯ψ,IR = ∞ κj∆jf (2)
-t asymptotically free (AF) non-Abelian Yang-Mills vecto- Xj=1
p
rialgaugetheory(ind=4spacetimedimensions)witha and [9]
e
set of massless fermions at an IRFP of the renormaliza-
h
∞
[ tainodncgornoufoprminatlhinevCaroiualnocmeb[1p,h2a].seH, werheewreeitcoenxshidibeirtsastchaele- βI′R = dj∆jf , (3)
1 ory of this type, with gauge group G = SU(N ) and N Xj=1
v c f
masslessfermionsψ , 1≤j ≤N , inarepresentationR,
3 j f where d1 =0 for all G and R. For general G and R, the
where R is the fundamental (F), adjoint (adj), or sym-
8 κ were calculated to order j = 3 in [8] and the d to
j j
0 metric rank-2 tensor (S). The dependence of the gauge orderj =4in[9], andforG=SU(3)andR=F, κ4 was
6 coupling g = g(µ) on the Euclidean momentum scale computed in [10] and d5 in [9].
0 µ is described by the beta function, β = dα/dt, where
Here we report our calculations of these scheme-
1. α(µ) = g(µ)2/(4π) and dt =dlnµ. The IRFP occurs at independentexpansionsofγ¯ andβ′ tothehighest
ψψ,IR IR
0 an IR zero of β at αIR. At this fixed point, an operator orders yet achieved, presenting κ4 and d5 for an asymp-
7 O for a physical quantity exhibits scaling behavior with
totically free SU(N ) gauge theory with a conformal IR
1 a dimension DO = DO,free−γO, where DO,free is the fixed point, for R =c F, adj, S. We also report our cal-
:
v free-field dimension and γO is the anomalous dimension. culation of κ3 for supersymmetric quantum chromody-
i Twoimportantquantitiesthatcharacterizetheproper- namics (SQCD). We believe that our new results are a
X
ar tγiψe¯ψs,aItRthaendIRβFI′PR.αIHReraer,eβγIψ′¯Rψ [i3s]eaqnudivβa′le≡ntdtβo/dtαhe,daennoomtead- sthuebosrtya.ntiOaluardrveasnucltesinhatvheetkhneowadlevdagnetaogfecoonffsocrhmeamlefieinld-
lous dimension of Fa,µνFaµν, where Faµν is the (rescaled) dependence at each order in ∆f, in contrast to scheme-
field-strength tensor [4]. As physical quantities, γψ¯ψ,IR dependent (SD) series expansions of γψ¯ψ,IR and βI′R in
and β′ are scheme-independent (SI) [5]. However,con- powers of α [11]-[16] and they complement other ap-
IR
ventional series expansions of these quantities in powers proaches to understanding conformal and superconfor-
of α, calculated to a finite order, do not maintain this mal field theory, such as the bootstrap [17] and lattice
scheme-independence beyond the lowest orders. Clearly, simulations [18].
it is very valuable to calculate and analyze series expan- Theconventionalpower-seriesexpansionsofβ andγ¯
ψψ
sionsforγ¯ andβ′ thatarescheme-independentat are
ψψ,IR IR
each order. Some early work was in [6, 7]. A natural ∞
α ℓ
expansion variable is β =−2α b (4)
ℓ
4π
Xℓ=1 (cid:16) (cid:17)
∆ =N −N , (1)
f u f and
∞
α ℓ
wNhfearlel,owfoerdabgyivaesnymNpctaontidcRfr,eeNduomis.thSechuepmpee-rin(due)pliemnditentot γψ¯ψ =Xℓ=1cℓ(cid:16)4π(cid:17) , (5)
2
wherebℓ andcℓ aretheℓ-loopcoefficients;b1 [19],b2 [20], byanelectrictheorywithdivergentαIR. Hence,another
and c1 =6Cf are scheme-independent, while the bℓ with important finding here is that the complete agreement
ℓ ≥ 3 and the c with ℓ ≥ 2 are scheme-dependent, i.e. that we obtain in SQCD to O(∆3) between Eq. (2) and
ℓ f
they depend on the scheme used for regularization and the exact Eq. (6) holds for arbitrarily strong α . Even
IR
renormalization [5]. We denote the n-loop (nℓ) β and apart from the issue of scheme dependence in Eq. (5),
γψ¯ψ as βnℓ and γψ¯ψ,nℓ and the IR zero of βnℓ as αnℓ. this agreement could not be achieved with the conven-
The calculation of κj requires,as inputs, the values of tional expansion (5) of γψ¯ψ,IR in powers of α.
the b for 1 ≤ ℓ ≤ j +1 and the c for 1 ≤ ℓ ≤ j. The
ℓ ℓ
The ∆ expansion also avoids a problem in which an
calculation of d requires, as inputs, the values of the b f
j ℓ
IRFPmaynotbemanifestasaphysicalIRzeroofthen-
for1≤ℓ≤j. Thus,importantly,κ doesnotreceiveany
j
loopbetafunctionforsomen. Indeed,althoughβ hasa
corrections from b with ℓ > j+1 or c with ℓ > j, and nℓ
ℓ ℓ
physicalα inSQCDforn=2, 3loops[27],wehave
similarly, d doesnotreceiveanycorrectionsfromanyb IR,nℓ
j ℓ
with ℓ>j. analyzedβ4ℓ (in the DR scheme), and we find that for a
range of N in the NACP, it does not exhibit a physical
The coefficients κ were calculated in [8] for an (AF f
j
vectorial) supersymmetric gauge theory (SGT) with αIR,4ℓ. This is analogous to the situation that we found
gauge group G and N pairs of chiral superfields in the for αIR,5ℓ in the non-supersymmetric gauge theory [16].
f
R and R¯ representation, for j = 1, 2. Complete agree- In both cases,the ∆f expansions (2) and(3) circumvent
thisproblemofapossibleunphysicalα thatonemay
mentwasfound,totheordercalculated,withtheexactly IR,nℓ
encounterinusingtheconventionexpansions(4)and(5).
known result in the conformal non-Abelian Coulomb
phase (NACP) [21–23] We next present our results for κ4 and d5 for a (non-
supersymmetric)SU(N )gaugetheory,makinguseofthe
γ = 32CTfA∆f . (6) impressiverecentcompuctationofb5 in[28]. (Wehaveac-
IR,SGT 1− 2Tf ∆ tuallycalculatedκ4 andd5 forgeneralGandR[29], but
3CA f only present results here for R = F, adj, S.) The two-
loop beta function has an IR zero (IRZ) in the interval
In this theory, N = 3C /(2T ), and the conformal
u A f
I : N <N <N , with upper and lower(ℓ) ends at
NACP is the interval N <N <N , where N =N /2, IRZ ℓ f u
ℓ f u ℓ u N =11N /(4T )andN =17C2/[2T (5C +3C )][24].
so that ∆f varies from 0 to a maximum of (∆f)max = u c f ℓ A f A f
The non-Abelian Coulomb phase extends downward in
3C /(4T ) in the NACP [24]. Hence, γ increases
A f IR,SGT
I from N to a lower value denoted N [30]. Since
monotonically from 0 to 1 as N decreases from N to IRZ u f,cr
f u
chiral symmetry is exact in the NACP, one can classify
Nℓ, saturating the upper bound γψ¯ψ,IR,SGT < 1 from the bilinear fermion operators according to their flavor
conformal invariance in this SGT [25].
transformation properties. These operators include the
Asatestoftheaccuracyofthe∆ expansion,wehave
f flavor-singlet ψ¯ψ and the flavor-adjoint ψ¯T ψ, where T
now calculated κ3 for SQCD with R = F, using inputs a a
from [26]. We find κ3 =1/(3Nc)3, in perfect agreement, is a generator of SU(Nf). These have the same anoma-
to this order, with the exact result, Eq. (6). This agree- lous dimension [31], which we write simply as γψ¯ψ. For
generalGandR,thecoefficientsb werecomputedupto
ment explicitly illustrates the scheme independence of ℓ
loop order ℓ=4 [32] (checked in [33]) and the c also up
the κ , since our calculations in [8] and here used inputs ℓ
j
to loop order ℓ = 4 [34], in the widely used MS scheme
computedintheDRscheme,while(6)wasderivedinthe
[35]. These results were usedin [8] to calculate the κ to
NSVZscheme[21]. Ournewresulthasafar-reachingim- j
order j =3 and in [9] to calculate d to order j =4. For
plication: itstronglysuggeststhatκ =[2T /(3C )]j for j
all j, so that the expansion (2) for jthis supfersymAmetric Nc = 3 and R = F, b5 was computed in [36], and this
gaugetheory,calculatedtoorderO(∆p), agreeswiththe wasusedtocalculateκ4 in[10]andd5 in[9]forthis case
f (see also [16]).
exact result to the given order for all p.
Because of electric-magnetic duality [22], as Nf →Nℓ We first report our results for κ4 and d5 for R = F,
intheNACP,thephysicsisdescribedbyamagneticthe- usingb5 from[28]. WedenotetheRiemannzetafunction
orywithcouplingstrengthgoingtozero,orequivalently, as ζ = ∞ n−s. We obtain
s n=1
P
4(N2−1)
κ4,F = 34Nc4(25cNc2−11)7 (cid:20)(cid:16)263345440Nc12−673169750Nc10+256923326Nc8
− 290027700N6+557945201N4−208345544N2+6644352
c c c
(cid:17)
+ 384(25Nc2−11) 4400Nc10−123201Nc8+480349Nc6−486126Nc4+84051Nc2+1089 ζ3
(cid:16) (cid:17)
+ 211200Nc2(25Nc2−11)2(Nc6+3Nc4−16Nc2+22)ζ5 (cid:21) (7)
3
and
25
d5,F = 36Nc3(25Nc2−11)7 (cid:20)Nc12(cid:16)−298194551−423300000ζ3+528000000ζ5(cid:17)
+ Nc10 414681770+1541114400ζ3−821040000ζ5 +Nc8 80227411−4170620256ζ3+2052652800ζ5
(cid:16) (cid:17) (cid:16) (cid:17)
+ Nc6 210598856+5101712352ζ3−4268183040ζ5 +Nc4 −442678324−2250221952ζ3+2744628480ζ5
(cid:16) (cid:17) (cid:16) (cid:17)
+ Nc2 129261880+304571520ζ3−534103680ζ5 +3716152+1022208ζ3(cid:21) , (8)
(cid:16) (cid:17)
wherethesimplefactorizationsofthedenominatorshave
beenindicated. ForthisR=F case,wefindthatκ4 >0,
as was also true of κj with 1 ≤ j ≤ 3 (indeed, κ1 and
κ2 are manifestly positive for any G and R). We also
find the same positivity results for R = adj and R =
S. The property that for all of these representations R,
κ >0for1≤j ≤4andforallN impliestwoimportant
j c
monotonicityresults. First,fortheseR,andwithafixed
p in the interval 1≤p≤4, γψ¯ψ,IR,∆p is a monotonically
f
increasing function of ∆ for N ∈ I . Second, for
f f IRZ
these R, and with a fixed Nf ∈ IIRZ, γψ¯ψ,IR,∆p is a
f
monotonically increasing function of p in the range 1 ≤
p ≤ 4. In addition to the manifestly positive κ1 and
κ2, a plausible conjecture is that, for these R, κj > 0
for all j ≥ 3. Note that the exact result (6) for the
supersymmetric gauge theory shows that in that theory,
κ >0 for all j and for any G and R.
j FIG. 1: Plot of γψ¯ψ,IR,∆p for R = F, Nc = 2, and 1 ≤
f
In Figs. 1 and 2 we plot γIR,∆pf for R=F, Nc =2, 3 p ≤ 4 as a function of Nf ∈ IIRZ. From bottom to top, the
and 1 ≤ p ≤ 4. In Table I we list values of these γIR,∆pf curves(withcolorsonline)refertoγψ¯ψ,IR,∆f (red),γψ¯ψ,IR,∆2f
[37]. These all satisfy the upper bound γIR < 2 from (green), γψ¯ψ,IR,∆3f (blue),and γψ¯ψ,IR,∆4f (black).
conformal invariance [25]. Below, we will often omit the
ψ¯ψ subscript, writing γψ¯ψ,IR ≡ γIR and γψ¯ψ,IR,∆p ≡
f
γIR,∆p.
f Nc =4, Nf =16, γIR,4ℓ =0.269, while γIR,∆4 =0.352.
For this R = F case we first remark on the compar- f
ison of γIR,∆4 with calculations of γIR,nℓ from analyses We next compare our new results with lattice mea-
f surements, restricting to cases where the lattice stud-
ofpowerseriesinα, whichwereperformedton=4loop
ies are consistent with the theories being IR-conformal
level in [11]-[14] using b and c in the MS scheme (with
ℓ ℓ
[18, 30]. For N = 3, we compared our calculations of
studies of scheme dependence in [15]) and extended to c
n = 5 loop level for N = 3 in [16]. We have noted that γIR,∆4 with lattice measurements for Nf = 12 in [10],
c f
β5ℓ does not have a physical αIR,5ℓ for Nf in the lower finding general consistency with the range of lattice re-
part of the interval IIRZ [16]. Although we were able sults,althoughourγIR,∆4f andextrapolationtotheexact
tosurmountthisproblemviaPad´eapproximantsin[16], γIR were higher than some of the lattice values. We also
thesearestillscheme-dependent,whilethe∆f expansion found consistency for the cases Nf = 10 and Nf = 8
has the advantage of being scheme-independent. In gen- [10]. Here,wecomparewithlattice resultsfor γIR inthe
eral, we find that for a given Nc and Nf, the value of case Nc =2, Nf =8. (It is not clear from lattice studies
γIR,∆p that we calculate to highest order,namely p=4, if the SU(2), R = F, Nf = 6 theory has a conformal
is somfewhat larger than γ calculated to its high- IRFPornot[18,30,38].) Followinglatticestudiesofthe
IR,nℓ
est order [10, 13]. For example, for Nc = 3, Nf = 12, SU(2), R=F, Nf =8 theory by severalgroups [18, 39],
γIR,4ℓ = 0.253, γIR,5ℓ ≃ 0.255 (using a value of αIR,5ℓ arecentmeasurementisγIR =0.15±0.02≡0.15(2)[40].
fromaPad´eapproximant[10,16]), whileγIR,∆4f =0.338 OlatutricvearlueseuγltI.R,∆4f = 0.298 is somewhat higher than this
and an extrapolation yields the estimate 0.400(5) for
γIR = limp→∞γIR,∆p [10]. Similarly, for Nc = 2 and We proceedto discussd5 for R=F. InFig. 3 we plot
Nf = 8, γIR,4ℓ = 0.2f04, while γIR,∆4 = 0.298; and for βI′R,∆p for R = F, Nc = 3, and 2 ≤ p ≤ 5. In Table II
f f
4
′
FIG. 2: Plot of γψ¯ψ,IR,∆pf for R = F, Nc = 3, and 1 ≤ FIG.3: PlotofβIR,∆pf forR=F,Nc =3,and2≤p≤4asa
p ≤ 4 as a function of Nf ∈ IIRZ. From bottom to top, the functionofNf ∈IIRZ. Frombottomtotop,thecurves(with
curves(withcolorsonline)refertoγψ¯ψ,IR,∆f (red),γψ¯ψ,IR,∆2f colors online) refer to βI′R,∆2f (red), βI′R,∆3f (green), βI′R,∆4f
(green), γψ¯ψ,IR,∆3f (blue), and γψ¯ψ,IR,∆4f (black). (blue),βψ′¯ψ,IR,∆5 (black).
f
TABLEI:Valuesofthescheme-independentanomalousdimension TABLEII: Scheme-independentvaluesofβI′R,∆p with2≤p≤4
γIR,∆pf with1≤p≤4forR=F andNc=2, 3. fIoIrRRZ.=TFhe,Nnocta=ti2o,n3aea-snfumnecatinosnaso×f1N0f−nin.theresfpectiveintervals
Nc Nf γIR,∆f γIR,∆2f γIR,∆3f γIR,∆4f Nc Nf βI′R,∆2 βI′R,∆3 βI′R,∆4 βI′R,∆5
2 6 0.337 0.520 0.596 0.698 f f f f
2 6 0.499 0.957 0.734 0.6515
2 7 0.270 0.387 0.426 0.467
2 7 0.320 0.554 0.463 0.436
2 8 0.202 0.268 0.285 0.298
2 8 0.180 0.279 0.250 0.243
2 9 0.135 0.164 0.169 0.172
2 9 0.0799 0.109 0.1035 0.103
2 10 0.0674 0.07475 0.07535 0.0755
2 10 0.0200 0.0236 0.0233 0.0233
3 9 0.374 0.587 0.687 0.804
3 9 0.467 0.882 0.7355 0.602
3 10 0.324 0.484 0.549 0.615
3 10 0.351 0.621 0.538 0.473
3 11 0.274 0.389 0.428 0.462
3 11 0.251 0.415 0.3725 0.344
3 12 0.224 0.301 0.323 0.338
3 12 0.168 0.258 0.239 0.228
3 13 0.174 0.221 0.231 0.237
3 13 0.102 0.144 0.137 0.134
3 14 0.125 0.148 0.152 0.153
3 14 0.0519 0.0673 0.0655 0.0649
3 15 0.0748 0.0833 0.0841 0.0843
3 15 0.0187 0.0220 0.0218 0.0217
3 16 0.0249 0.0259 0.0259 0.0259
3 16 2.08e-3 2.20e-3 2.20e-3 2.20e-3
we list values of βI′R,∆p for R = F, Nc = 2, 3 and 2 ≤ case were given for κp with 1 ≤ p ≤ 3 in [8] and for dp
f
p≤5. ForR=F andgeneralNc,d2 andd3 arepositive, with 1≤p≤4 in [9]. Here we find
wc1a2hl,ciluwelaedt4igoaentndyβiId′eRl5d,∆eadr5feβ=n′eg0a.t2i2v=8e..0F.T2o9rh5et5hceaonncdavseβen′tSioUn(a3l=),nN0-l.fo2o8=p2 κ4,adj = 523738·9331943 + 336108ζ3
IR,3ℓ IR,4ℓ 2170 33952
[41], so βI′R,∆5f is slightly smaller than βI′R,4ℓ. A recent + (cid:18)− 310 + 311 ζ3(cid:19)Nc−2
latticemeasurementyieldsβ′ =0.26(2)[42],consistent
with both our β′ and β′IR . = 0.0946976+0.193637Nc−2 (9)
IR,∆5 IR,4ℓ
f
WenextdiscussthecaseR=adj,forwhichN =11/4 and
u
andNℓ =17/16,soIIRZ includesthesingleintegervalue 7141205 5504
Nf = 2 (whence ∆f = Nu−2 = 3/4). Results for this d5,adj = − 23·316 + 312 ζ3
5
adj,alreadydiscussedabove. ForSU(3), wefocusonthe
− (cid:18)30391428 + 463511352ζ3(cid:19)Nc−2 Nf = 2 theory, for which we find βI′R,∆5f = 0.333; and
γIR,∆2 = 0.789, γIR,∆3 = 0.960, and γIR,∆4 = 1.132
= −(0.828739×10−2)−0.357173Nc−2 . [37]. fFor comparison, four n-loop results frofm [13] for
this case are γIR,3ℓ = 0.500 and γIR,4ℓ = 0.470. Lattice
(10) studies of this theory include one that concludes that it
is IR-conformal and gets γ < 0.45 [50] and another
IR
We remark on the SU(2), N = 2, R = adj theory,
f that concludes that it is not IR-conformal and gets an
which has been of interest [43]. Extensive lattice studies effective γ ≃1 [51].
IR
of this theory have been performed and are consistent
with IR conformality [18]. We get βI′R,∆5 = 0.147; and anIdnβsu′mmataray,cwoenfhoarvmeaplrIeRsenfitxededcaplcouinlattioofnsanofaγsψy¯ψm,IpR-
f IR
γIR,∆2 = 0.465, γIR,∆3 = 0.511, and γIR,∆4 = 0.556. totically free gauge theory with fermions, to the high-
f f f
These γIR,∆p values are close to our n-loop calcula- est orders yet achieved. We believe that these results
f
tions in [13] for this theory, namely γIR,3ℓ = 0.543, are of fundamental value for the understanding of con-
γIR,4ℓ =0.500. Latticemeasurementsofthistheoryhave formal field theory, especially because they are scheme-
yielded a wide range of values of γ including, 0.49(13) independent.
IR
[44], 0.22(6) [45], 0.31(6) [46], 0.17(5) [47], 0.20(3) [48], This research was supported in part by the Danish
0.50(26) [49], and 0.15(2) [40] (see references for details National Research Foundation grant DNRF90 to CP3-
of uncertainty estimates). Origins at SDU (T.A.R.) and by the U.S. NSF Grant
Finally, we discuss the case R = S. For SU(2), S = NSF-PHY-16-1620628(R.S.).
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