Chiral phase structure of QCD with many flavors

0.2 MB·English

by  Holger Gies

HD-THEP-05-16
DESY-05-126
Chiral phase structure of QCD with many flavors
Holger Gies1 and Joerg Jaeckel2
1Institut fu¨r theoretische Physik, Philosophenweg 16, 69120 Heidelberg
2Deutsches Elektronen Synchrotron, Notkestrasse 85, 22607 Hamburg
We investigate QCD with a large number of massless flavors with the aid of renormalization
group flow equations. We determine the critical number of flavors separating the phases with and
withoutchiralsymmetrybreakinginSU(Nc)gaugetheorywithmanyfermion flavors. Ouranalysis
includes all possible fermionic interaction channels in the pointlike four-fermion limit. Constraints
from gauge invariance are resolved explicitly and regulator-scheme dependencies are studied. Our
findings confirm the existence of an Nf window where the system is asymptotically free in the
ultraviolet, butremainsmassless andchirally invarianton all scales, approaching aconformal fixed
pointintheinfrared. Ourpredictionforthecriticalnumberofflavorsofthezero-temperaturechiral
phase transition in SU(3) is Ncr = 10.0±0.29(fermion)+1.55(gluon), with the errors arising from
6 f −0.63
approximations in thefermionic and gluonic sectors, respectively.
0
0
PACSnumbers: 11.10.Hi,11.15.Tk,11.30.Rd
2

n
a I. INTRODUCTION interactions which in turn depends on the amount of
J
fermionic screening,we expect the conformalscenarioto

5 hold for large N sufficiently close to Na.f.. The broken
2 In order to investigate the dynamics of strongly in- f f
phase is supposed to occur for smaller N with a critical
teracting gauge systems, it has often been a successful f
Ncr separating the two phases. In fact, evidence for this
2 strategyto deformthe desiredsysteminto a more acces- f
phasestructurehasbeencollectedfromvariousmethods,
v sible one. For instance, the approximation of a contin-
1 uumgaugetheoryby afinite lattice gaugetheorycanbe including (improved) ladder truncated Dyson-Schwinger
7 equations [2, 3, 4], lattice QCD [5], anomaly-induced ef-
viewed as such a deformation. In the same spirit, the
1 fective potentials [6], and instanton models [7], with es-
addition of more symmetries as in supersymmetric ver-
7 timates for the critical number of flavors ranging from
0 sions of gauge theories represents such a more tractable Ncr ≃5 [8] to Ncr ≃12 for SU(3) gauge theory.
5 deformation. In the present work, we consider the de- f f
0 formation of SU(N ) gauge theories by many massless An analysis of the quark-scattering amplitude using
c
/ fermion flavors. Since light fermions have the perturba- the functional methods [2, 3] particularly reveals that
h
p tivetendencytoscreenlong-rangeforces,alargefermion the nature of the phase transition, though continuous,is
- flavor number Nf has the potential to move the system not conventionally second order.1 This is most promi-
p
towardsweakergaugeinteractionforwhichanalytictools nently visibleinthe factthatthere appearto be nolight
e
are more powerful. scalar states in terms of which an effective theory could
h
v: timTahteelryeloebvsaenrcveabolfeffeorrmNion>icNsca.rfe.e:n=in1g1Nat lwarhgeereNafsyismupl-- pbheacsoensttrrauncstietidonon[2t,he3,sy9m].mAetnryicasitdteem(Nptf &atNccforn)sotrfutchte-
i f f 2 c
X totic freedom is lost. But even for smaller N, fermionic ingaGinzburg-Landau-typeofeffectivepotentialforthe
f
r screening is first signaled by the second β function co- chiral order parameter hence produces discontinuities in
a efficient of the gauge coupling, reversing its sign for the potential parameters. On the other hand, the or-
N > 34N3c . This sign change induces a second zero der parameter in the form of the chiral condensate, i.e.,
f 13N2c−3 the minimum of this would-be potential, changescontin-
ofthe perturbativeβ function,implyinganinfrared(IR)
uously across the phase transition.
attractive fixed point of the gauge coupling α∗ > 0. As
In this work, we analyze the phase structure of many-
wasfirstarguedin[1],thefixed-pointvalueα∗issmallfor
flavor QCD with the aid of the functional renormal-
N . Na.f., supporting the possibility of a perturbative
f f ization group (RG). Beyond its conceptual simplicity,
analysis. Ifthis fixedpointpersistsonallscales,thesys-
theapproachoperatesincontinuousspacetime,supports
temapproachesaconformallyinvariantlimitinthe deep
an explicit implementation of chiral symmetry, and in-
IR, keeping massless quark and gluon excitations in the
cludesaresummationofbeyond-ladderdiagramsalready
spectrum. Thisscenariocancollapseowingtononpertur-
in simple truncations. Moreover, we monitor gauge in-
bative phenomena such as the spontaneous break-down
variance with the aid of Ward-Takahashi identities and
ofchiralsymmetry. Thelatterrendersthefermionsmas-
study regulator-scheme dependencies. Since the critical
sive, implying their decoupling at low scales. Fermionic
screeningpropertiesarethusswitchedoffinthedeepIR,
and the systemis characterizedby stronglycoupled glue
and the Goldstone bosons of chiral symmetry breaking
1 Owing to the different conformal properties of the system on
(χSB).
eachside,thetransitionisoftenreferredtoasaconformalphase
Since χSB is triggered by the strength of the gauge transition[2].
2
number of flavors is a universal quantity, the scheme- ghost sector is tacitly assumed). The gauge-field kinetic
dependencies are a direct measure of the approximation term is accompanied by a k-dependent wave-function
uncertainties of our nonperturbative truncation. Apart renormalization Z , the fermionic one by Z . Similarly,
A ψ
from a quantitative prediction for Ncfr, our results re- Z1,Z+,Z−,Zσ andZVA arethevertexrenormalizations,
veal further facets of the nature of the phase transi- whereas g, λ denote the bare couplings. Renormalized
tion. Acrossthephasetransition,weparticularlyobserve dimensionless couplings are defined by
discontinuities in the effective fermionic self-interactions
which are generated by the RG flow. gZ Z k2λ
1 i i
g = , λ = . (3)
latFeodrinoutreramnasloyfsias,flwowe eemqupalotyiotnhfeorfutnhcetieoffneacltiRvGeafvoerrmague- ZA1/2Zψ i Zψ2
action Γ [10],
k We workin the Landaugauge,α=0, whichis knownto
1 k beafixedpointoftherenormalizationgroup[12]andhas
∂ Γ = STr∂ R (Γ(2)+R )−1, t=ln , (1)
t k 2 t k k k Λ the additional advantage that the fermionic wave func-
UV
tion is not renormalized in our truncation, such that we
serving as an alternative definition of quantum field the-
can choose Z =1.
ψ
ory. Γ is a free-energy functional that interpolates be-
k The four-fermioninteractionscanbe classifiedaccord-
tween the bare action Γ =S and the full quantum
k=ΛUV ing to their color and flavor structure. Color and flavor
effective action Γ = Γ . Here, R denotes a regulator
k=0 k singlets are
functionthatspecifiesthe detailsofthe momentum-shell
integrations, the variation of which will provide us with (V–A) = (ψγ ψ)2+(ψγ γ ψ)2, (4)
µ µ 5
anerrorestimateofourapproximations. WedevoteSect.
II to a discussion of our approximation scheme, which (V+A) = (ψγµψ)2−(ψγµγ5ψ)2, (5)
represents a truncation of the full theory to the relevant
operators for the present problem. Results and conclu- wherecolor(i,j,...)andflavor(a,b,...)indicesarecon-
sions are presented in Sects. III and IV. tracted pairwise, e.g., (ψψ) ≡ (ψaψa). The remaining
i i
operators have non-trivial color or flavor structure
II. FLOW EQUATIONS (S–P) = (ψaψb)2−(ψaγ ψb)2 (6)
5
≡ (ψaψb)2−(ψaγ ψb)2,
i i i 5 i
Ourprimaryaimisareliabledeterminationofthecrit-
(V–A)adj = (ψγ Tzψ)2+(ψγ γ Tzψ)2, (7)
ical flavor number Ncr. For this, the approach of the µ µ 5
f
phase transition from the symmetric side is technically
advantageous, since the relevant degrees of freedom are where(ψaψb)2 ≡ψaψbψbψa,etc.,and(Tz)ij denotesthe
expected to remain the same during the flow from the generators of the gauge group in the fundamental repre-
ultraviolet(UV) tothe infrared(IR):quarksandgluons. sentation. Thesetoffermionicself-interactionsoccurring
Therefore, we solve the flow equation (1) in a truncated in Eq. (2) forms a complete basis. Any other pointlike
subspace of all possible action functionals. four-fermion interaction invariant under SU(Nc) gauge
In addition to standard gauge and fermion sectors, symmetry and SU(Nf)L × SU(Nf)R flavor symmetry is
weincludeallpossiblepointlikefour-fermioninteractions reducible by means of Fierz transformations.
that are compatible with SU(N ) gaugesymmetry and a It is importantto stress that the effective action(2) is
c
chiralSU(N ) ×SU(N) flavorsymmetryforN fermion in the QCD universality class [13], as long as the four-
f L f R f
species2, fermion interactions are small in the UV and thus RG
irrelevant. Only nonperturbatively large (Z λ )’s would
Z (∂ Aµ)2 i i
Γ = ψ(iZ ∂/+Z gA/)ψ+ AFµνFz + µ bring us into another universality class with χSB of
k Z ψ 1 4 z µν 2α Nambu–Jona-Lasinio(NJL) type, whichwill notbe con-
1 sidered in this work.
+ Z−λ−(V–A)+Z+λ+(V+A)+Zσλσ(S–P)
2 Plugging the truncated effective action (2) into the
h
+Z λ [2(V–A)adj+(1/N )(V–A)] . (2) flow equation, we obtain the following β functions for
VA VA c
i the dimensionless couplings λi (see [11]):
Here A = AzTz, F = Fz Tz denote the nonabelian
µ µν µν
gauge potential and field strength (the inclusion of a ∂tλ−=2λ−−4v4l1(F,1B),4(cid:20)N3 g2λ−−3g2λVA(cid:21) (8)
c
1 12+9N2
− v l(FB),4 cg4
8 4 1,2 (cid:20) N2 (cid:21)
2 Wenotethatonlythefour-fermioninteractionsaremanifestlyin- c
voafrtiahnetcuonudpelrinlgosc.alGgaauuggeetirnavnasrfioarnmcaetioofnsthfeorraelmlpaionsisnibgletecrhmoisceiss −8v4l1(F),4n−NfNc(λ2−+λ2+)+λ2−
governed by (modified) Ward-Takahashi identities as discussed
forthepresentsystem in[11]. −2(Nc+Nf)λ−λVA+Nfλ+λσ+2λ2VA ,
o
3
∂ λ = 2λ −4v l(FB),4 − 3 g2λ (9) η˘g2g2.ηA, ηψ and η1 are the gluon, quark and quark-
t + + 4 1,1 (cid:20) N +(cid:21) gluon-vertexanomalousdimensions,respectively,defined
c
1 12+3N2 by ηi = −∂tlnZi; ηψ, in fact, vanishes in the present
−8v4l1(F,2B),4(cid:20)− N2 cg4(cid:21) truncation using the Landau gauge. The ˘ symbol de-
c notes those standard contributions which are not gener-
−8v4l1(F),4 −3λ2+−2NcNfλ−λ+ ated by the modified Ward-Takahashi identities; for in-
n stance, the difference between η and η˘ is exactly given
1 1
−2λ+(λ−+(Nc+Nf)λVA)+Nfλ−λσ by the term ∼∂ λ in Eq. (13).
t i
+λVAλσ + 41λσ2 , Instead of computing η˘g2 in the RG scheme, we use
o the four-loop result obtained in the MS scheme [18, 19]
∂ λ =2λ −4v l(FB),4 6C (N )g2λ −6g2λ (10) as an estimate. For a qualitative discussion, let us list
t σ σ 4 1,1 2 c σ +
1 2(cid:2)4−9N2 (cid:3) the first two coefficients (all four known coefficients are
− v l(FB),4 − c g4 used in the numerics),
4 4 1,2 N
h c i
−8v4l1(F),4 2Ncλ2σ−2λ−λσ−2NfλσλVA−6λ+λσ , β = 11N − 2N, β = 34N3c +3Nf−13N2cNf, (15)
n o 0 3 c 3 f 1 3N
3 c
∂tλVA=2λVA−4v4l1(F,1B),4(cid:20)Ncg2λVA−3g2λ−(cid:21) (11) with β reversing its sign for N > 34N3c . Since we
1 24−3N2 1 f 13N2c−3
− v l(FB),4 − cg4 do not expect that the RG- and MS-scheme β functions
8 4 1,2 (cid:20) Nc (cid:21) exactlycoincide,thisfour-loopapproximationintroduces
−8v4l1(F),4 −(Nc+Nf)λ2VA+4λ−λVA− 14Nfλ2σ . anerrorthatisestimatedbelow.3 Itshouldbestressedat
n o this point that some crucial properties of the β function
are not scheme dependent; in particular, the existence
Here C (N ) = (N2−1)/(2N ) is a Casimir operator of
2 c c c of zeros, i.e., fixed points, is a universal result which is
thegaugegroup,andv =1/(32π2). Forbetterreadabil-
4 relevant for the present problem.
ity, we have written all gauge-coupling-dependent terms
One further technical point needs to be mentioned:
insquarebrackets,whereasfermionicself-interactionsare
groupedinside braces. The thresholdfunctions l charac- whereastherunninggaugecouplingisdominatedbyη˘g2,
the threshold functions l of the fermionic sector require
terizetheregulatordependence[14]. Forthe“optimized”
the separate knowledge of η . As a rough but sufficient
linear regulator [15], these read A
estimate, we simply employ the one-looprelationfor the
l(F),4 = 1, l(FB),4 =1− ηA, l(FB),4 = 3 − ηA. (12) Landau gauge, ηA1-loop = 2123η˘g12-loop. Since the anoma-
1 2 1,1 6 1,2 2 6 lous dimensions turn out to remain numerically small of
O(0.1) and the threshold functions only depend on ηA,
6
Next, let us turn to the running of the gauge coupling. this additional higher-order approximation has little ef-
Eventhoughanon-perturbativeestimate canalreadybe fect on the result.4
obtained within the current simple truncation [16], a re-
liable estimate to higher loops requires the inclusion of
many further vertex operators in the truncation 2. In- III. RESULTS
stead, we construct the β function according to the fol-
lowinglineofargument: First,wenotethatgaugeinvari-
The onset of χSB is detected by the four-fermion
ance in the RG scheme canbe monitoredwith the aidof
part of our model (2). Chiral symmetry remains pre-
regulator-dependent Ward-Takahashi identities [16, 17].
served in the pointlike limit, as long as the fermion self-
For the present system, these identities constrain the
interactions remain finite. In turn, χSB is signaled by a
four-fermioncontributionstotherunninggaugecoupling
diverging λ . This seeming Landau-pole behavior points
to be of the form [11]: i
g2
∂tg2 =η˘g2g2−4v4l1(F),4 1−2v4l1(F),4 ciλi ∂tXciλi, 3 Note that the additional terms ∼∂tλi in Eq. (13) already con-
cσ =1+Nf, c+ =0, c− =−2, cVA =P−2Nf, (13) tributetotheresultingβfunctionattwo-looporder. Byapprox-
imating η˘g2 by the MS result, we introduce a potential double
where countingofdiagramsatthisandhigherorder. However,the∂tλi
contributions vanish at the fixed point. In addition, we would
liketostressthatweareusingamass-dependent regularization
η˘g2 =ηA−2η˘1+2ηψ (14) scheme, and universalityofthe secondcoefficient inthe β func-
=−4v g2 β +β (2v g2)+β (2v g2)2+... tion does not hold and further contributions at this order are
4 0 1 4 2 4 expected.
(cid:0) (cid:1) 4 Asacheck,wehaveusedthepureone-loopresultη1-loop,which
is the standard gauge-coupling anomalous dimension, A
isnon-vanishingatthefixedpoint. Thecorrespondingresultslie
trivially related to the standard β function by β˘2 = wellwithintheestimatederror(shadedregioninFig.3).
g
4
2
∂ λ α= 0 α
t i cr
1.75
1.5
α& 0
1.25
1
λ
i
α> αcr 0.75
0.5
0.25
FIG. 1: Sketch of a typical β function for the fermionic self- 0 5 10 15 20 25
interactions λi: at zero gauge coupling, α = 0 (solid curve),
the Gaußian fixed point λi = 0 is IR attractive (the second Nf
fixed point at λi > 0 corresponds to the IR repulsive criti- FIG. 2: Critical coupling for the four-fermion system. From
cal coupling of NJL type). For small α & 0 (dashed curve), toptobottom,thenumberofcolorsincreasesfromNc=2to
the fixed-point positions are shifted on the order of α2. For Nc =8(thicklinecorrespondstoNc =3). Thered/dark-grey
gaugecouplingslargerthanthecriticalcouplingα>αcr(dot- (green/light-grey) line shows the fixed-point gauge coupling
dashedcurve),nofixedpointsremainandtheself-interactions for Nc = 3 at four (two) loop. At the crossing, the critical
quickly grow large, signaling χSB. We emphasize that both numberof flavors can be read off.
fixed-points values remain finite until the fixed points even-
tually vanish at αcr.
ciently weak gauge coupling; their values are modulated
by the logarithmic increase of the gauge coupling. If the
to the formation of chiral condensates, since the λi’s gauge coupling grows larger than the critical value, the
are inversely proportional to the mass parameter of a fermionic quasi-fixed points vanish (the parabola drops
Ginzburg-Landau potential for the order parameter in a belowtheλ axisinFig.1),andthesystemrunsintothe
i
(partially) bosonized language, λi ∼1/m2i. χSB regime. In this way, the IR physics is solely deter-
Therefore, let us first consider the λi flow separately. mined by the gauge coupling, as expected. Whether or
For vanishing gauge coupling, the flow is solved by van- notthesystemendsupintheχSBregimefinallydepends
ishing λi’s, which is the Gaußian fixed point. This on the maximal value of the gauge coupling. For a large
fixed point is IR attractive, implying that these self- number of flavors, N . Na.f., this value is given by the
f f
interactions are RG irrelevant for sufficiently small bare IR fixed-point value α∗, induced by the higher β func-
couplings, as they should be. Since the λi flows are tion coefficients. Lowering Nf increases α∗, eventually
quadratic in the λi’s corresponding to a parabola, there exceeding αcr required for χSB.
are 15 further non-Gaußian fixed points (2(#λi=4) = 16 We determine the resulting critical number of flavors
intotal). Thesearerelatedtothesetofcriticalcouplings Ncr by comparing the fixed point of the gauge coupling,
f
λi,cr beyondwhichthe systemis in the universalityclass α∗ ≡ α∗(Nf), to αcr. The task of solving the complete
of NJL type, see Fig. 1. coupledsetoffixed-pointequations∂ g2 =0, ∂ λ=0be-
t t
Forfixedsmallgaugecoupling,theGaußianfixedpoint comes simplified by the following observation: Eq. (13)
isshiftedabittofixedpointvaluesoftheorderλ∗,i ∼g4, reveals that the only contributions to ∂tg2 involving λ
effectively describing scattering of masslessquarks. This areproportional∂ λ. By definition this termvanishes at
t
fixed point is still IR attractive, see Fig 1; hence there anyfixed point inthe subsystem offour-fermioninterac-
is no χSB as long as the gauge coupling remains small tions. Therefore,thegaugecouplingatanyfixedpointof
enough. the complete systemis determinedby η˘g2 =0,whichde-
If the gauge coupling becomes larger than a critical pendsonlyong. InFig.2,thefixed-pointgaugecoupling
value αcr = gc2r/4π, all fixed points of the λi flows are α∗(Nf) is plotted for Nc = 3 with a two- (green/light-
removed,suchthattheλi’squicklyrunintoadivergence, grey) and a four-loop (red/dark-grey)result for η˘g2.
signaling χSB [20]. The critical number of flavorsNcr can now be read off
f
In Fig. 2, we study the four-fermion system at fixed from the intersection between this line and the critical
gauge coupling. For various numbers of colors Nc, we coupling αcr of the four-fermion subsystem for Nc = 3
plot the critical gauge coupling αcr = gc2r/4π at which (thick solid line). In the same way, we determine the
the fermion system becomes critical and exhibits χSB. critical number of flavors as a function of N , as plotted
c
For an increasing number of colors, αcr decreases. The in Fig. 3. 5
dependence on the number of flavors is rather weak.
Now, let us switch on the running of the gauge cou-
pling. Theabove-mentionedfixedpointsinthefermionic
system become quasi-fixed points which persist for suffi- 5 AnequivalentwaytodetermineNcfr(Nc)istodirectlysolvethe
5
The regulator dependence of the flow equation offers
a possibility for estimating the error of our quantitative
N 4400
results which is introduced by the truncation. Exact re- f
sults for physical quantities like Ncr are universal with-
f
outanyregulatordependence. However,thisuniversality 3300
does not hold for the truncated flow, such that the reg-
ulator dependence of would-be universal quantities di-
2200
rectly translates into an error of the truncation. Let us
start with the fermionic subsystem, for which the point-
like approximation of vertices and the use of “classical” 1100
propagators represents the most severe approximation.
If these truncatednontrivialmomentum dependencies of 00
verticesandpropagatorswereimportantforthedetermi- 22 33 44 55 66 77 88
nationoftheuniversalN ,wewouldexpectastrongreg-
f
ulator dependence within the point-like approximation. N
c
This is because the flow equation is localized in momen- FIG. 3: Critical number of flavors for SU(Nc) gauge theory.
tumspaceanddifferentregulatorsprobetheverticesand Theresultbasedonthefour-loopMSbetafunctionisdenoted
propagatorsat different momentum shells; consequently, byblack circles whichlie almost on top ofstars, representing
erroneously truncated momentum dependencies of the thethree-loop result; blackboxescorrespond tothetwo-loop
vertices generically imply strong variations of would-be beta function. The (inner) green/dark-grey shaded region
around the four-loop result displays our error estimate for
universalquantitiesfordifferentregulators. Inthissense,
the fermionic sector. The (outer) turquois/light-grey shaded
thestudyofregulatordependenciesisanimportantifnot
region shows the approximate gluonic error, estimated by a
crucialtestofourpoint-likeapproximationinthefermion
variation of the higher-loop coefficients. The red/grey line
sector for the determination of N . In the present case,
f showsthenumberofflavorsabovewhichasymptoticfreedom
the regulator dependence occurs in the threshold func-
is lost.
tions l. Our pointlike truncation is reminiscent of a
derivative expansion, for which a flow-optimization pro-
cedure has led to the construction of a linear regulator
here. Of course, we should stress that the point-like ap-
[15] used in Eq. (12). Following the stability arguments
proximation is expected to break down for many other
of [15], we consider the results from this regulator as
questions, in particular, those related to the properties
our “optimized” predictions. The opposite limit of non-
of the χSB phase.
optimized regulators for pointlike interactions is marked
bythesharpcutoff[21]withl(F),4 =l(FB),4 =l(FB),4 =1, The largest uncertainty arises from the gauge sector.
1 1,1 1,2 Asaerrorestimate,onemightusethedifferencebetween
neglecting the anomalous dimension. Therefore, a con-
servativeerrorestimateisgivenbythedifferencebetween thevariousloopordersintheβ-functioncontributionη˘g2.
As depicted in Fig. 3, the difference between two and
the sharp cutoff and the optimized regulator.6 This re-
three loops in the MS scheme is of the order ∆Ncr ∼ 1.
sults in the green/dark-grey shaded region in Fig. 3. f
The difference between the three- and the four-loop re-
The resulting error is actually very small; in particular,
sult is already much smaller. This suggests that, for our
we observe strong nontrivial cancellations between the
purposes, the gauge sector is reliably approximated by
contributions of different thresholdfunctions to the final
the four-loop β function. However, as discussed above,
result. Since this exactly matches with the expectation
the(known)MSresultisnotexpectedtocoincideexactly
foruniversalquantities,itprovidesevidencefortherelia-
with the (unknown) result to be obtained with the lin-
bility ofthe pointliketruncationforthe presentpurpose.
ear regulator used so far. For a quantitative estimate of
As a further evidence for the consistency of the point-
the resultingerror,wecanperformacomparisonattwo-
like approximationinthe symmetric phase,wenote that
loop level, where the mass-dependent character of the
thegluonaswellasthequarkanomalousdimensionvan-
flow-equation regularization already introduces scheme
ish at the IR fixed point in our approximation,implying
dependencies. On a quantitative level, this introduces
that the use of ”classical” propagators is self-consistent
differences on the 10% level [22].7 For the 3- and 4-loop
coefficients we assume larger uncertainties on the 30%
and 50% level, respectively. This results in an error de-
picted by the turquois/light-grey shaded region in Fig.
flow equations (8)-(13) with λi(t = 0) = 0 and a small value
g2(t=0)asinitialconditions. OnecanthenfindNcr bylooking 3. Quadraticallyaddingbothuncertainties,theresulting
f
fortheinfimumofallNfforwhichthefour-fermioncouplingsdo error is dominated by the gluonic sector.
not diverge at a finite t, (or alternatively, for the supremum of
thosewhichdodiverge).
6 The sharp cutoff can be considered as the limit of a sequence
of powerlike regulators, the results of which all lie within the
estimatederror. 7 For an exact computation of the universal two-loop coefficient
withinthefunctional RGapproach,see[23].
6
It has been argued [24] that a regularization scheme wehaveestimatedfromtheregulatordependence ofuni-
exists in which the two-loop β function for the gauge versal quantities.
couplingisexact. Westressthatthisdoesnotimplythat Our restriction to a truncation in terms of quark and
thetwo-loopcurveinFig.3isexact,sincethisparticular gluonfieldswithpointlikeinteractiondoesnotfacilitatea
unknown regularization scheme may involve a regulator studyoftheχSBregime;forthis,theinclusionofbosonic
withstrongquantitativeinfluenceonthefermionicsector composite fields along the lines of [13, 20, 26, 27] repre-
(possiblyrequiringoperatorsbeyondthepointlikelimit). sents a powerful tool.
Hence, it is well possible that the resulting curve of that Nevertheless, important aspects of the unusual nature
special scheme is again close to the estimated four-loop of the phase transition can already be read off from
result in our scheme. the fermionic sector: as long as the system is in the
It is worth pointing out that the total estimated er- symmetric phase, Ncr < N < Na.f., the fermionic self-
f f f
ror is significantly smaller than the distance to the line interactions approach fixed-point values in the IR which
Na.f.(N )whereasymptoticfreedomislost. Thisstrongly are finite numbers. Even arbitrarily close to the phase
f c
confirms the existence of a phase of many-flavor QCD transition, these numbers remain finite and are elements
whichisasymptoticallyfree,buthasnoχSBandnostrict ofacompactregioninparameterspace,λi|k→0 ∈Ωλ. By
confinement in the IR. contrast,theself-interactionsdivergeinthebrokenphase
where all fixed points have vanished. As a consequence,
aninfinite regioninparameterspaceoutside Ω remains
λ
IV. SUMMARY AND CONCLUSIONS inaccessible. Varying N continuously across the phase
f
transitioncomingfromthesymmetricside,thefermionic
We have used flow equations to determine the criti- self-interactionsjumpdiscontinuouslyatNf =Ncfr. Now,
calnumber offlavorsNcfr separatingthe phaseswith and theλi’sareinverselyproportionaltothemassparameter
without χSB. Our truncation includes all pointlike four- of a Ginzburg-Landau potential for the order parameter
fermion interactions allowed by gauge and chiral sym- in a (partially) bosonized language, λi ∼ 1/m2i; hence,
metry which are generated by ladder as well as non- our observation in the fermionic sector agrees with the
ladder processes; thereby, our result represents a sig- observations of [2, 3, 9] that the phase transition is not
nificant improvement over conventional approximation conventionally second order, even though the chiral or-
techniques. Gauge symmetry is monitored by modi- derparameterisknowntochangecontinuouslyacrossthe
fiedWard-Takahashiidentities,resultinginanadditional chiral phase transition [13]. Our work thus reveals fur-
four-fermion contribution to the flow of the gauge cou- theraspectsofthenatureofthistypeofzero-temperature
pling. We have demonstrated that these terms have no phase transitions which may find further application in
effect on Ncfr, owing to the structure of the modified related systems such as QED3 or models of electroweak
Ward-Takahashi identity (by contrast, these terms are symmetry breaking.
ofquantitativeimportance,forinstance,fortheLandau-
pole problem of QED [25]).
Our findings confirm the existence of a sizable pa- Acknowledgments
rameter region Ncr < N < Na.f. where chiral sym-
f f f
metry remains unbroken, in agreement with earlier re- J.J. would like to thank the ITP in Heidelberg for its
sults [2, 3, 4, 5, 6]. Our result for the critical num- hospitalityandA.Ringwaldforusefulcomments. H.G.is
ber of flavors, for instance, in SU(3) gauge theory, is grateful to J.M. Pawlowski for discussions and acknowl-
Ncr = 10.0±0.29(fermion)+1.55(gluon). The errors re- edges support by the Deutsche Forschungsgemeinschaft
f −0.63
sult from the fermionic and gluonic sectors of our trun- (DFG) under contract Gi 328/1-3 (Emmy-Noether pro-
cation, respectively, the quantitative influence of which gram).
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