Study of the conformal hyperscaling relation through the Schwinger-Dyson equation

0.28 MB·English

by  Yasumichi Aoki

KEK Preprint 2011-22
Study of the conformal hyperscaling relation through the Schwinger-Dyson equation
Yasumichi Aoki,1 Tatsumi Aoyama,1 Masafumi Kurachi,1 Toshihide Maskawa,1
Kei-ichi Nagai,1 Hiroshi Ohki,1 Akihiro Shibata,2 Koichi Yamawaki,1 and Takeshi Yamazaki1
(LatKMI Collaboration)
1Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464-8602, Japan
2Computing Research Center, High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801, Japan
Westudycorrectionstotheconformalhyperscalingrelationintheconformalwindowofthelarge
N QCDbyusingtheladderSchwinger-Dyson(SD)equationasaconcretedynamicalmodel. From
f
theanalyticalexpressionofthesolutionoftheladderSDequation,weidentifytheformoftheleading
masscorrectiontothehyperscalingrelation. Wefindthattheanomalousdimension,whenidentified
2
throughthehyperscalingrelationneglectingthesecorrections,yieldsavaluesubstantiallylowerthan
1
∗
the one at the fixed point γ for large mass region. We further study finite-volume effects on the
0 m
2 hyperscaling relation, based on the ladder SD equation in a finite space-time with the periodic
boundary condition. We find that the finite-volume corrections on the hyperscaling relation are
n
negligible compared with the mass correction. The anomalous dimension, when identified through
a
the finite-size hyperscaling relation neglecting the mass corrections as is often done in the lattice
J
analyses, yields almost the same value as that in the case of the infinite space-time neglecting the
9 mass correction, i.e., a substantially lower value than γ∗ for large mass. We also apply the finite-
m
1
volume SD equation to the chiral-symmetry-breaking phase and find that when the theory is close

to the critical point such that the dynamically generated mass is much smaller than the explicit
]
t breaking mass, the finite-size hyperscaling relation is still operative. We also suggest a concrete
a form of the modification of the finite-size hyperscaling relation by including the mass correction,
l
- which may beuseful to analyze thelattice data.
p
e
h
I. INTRODUCTION
[

1
Technicolormodel[1,2]hasbeenconsideredasaninterestingpossibilityforthedynamicaloriginoftheelectroweak
v
symmetry breaking. However, it has fatal phenomenological difficulties (especially with the strong suppression of
7
5 flavor changing neutral current processes). The problems can be solved by the walking technicolor [3, 4] having
1 approximatescaleinvariancewith largemass anomalousdimension, γm ≃1,whichwas proposedbasedonthe ladder
4 Schwinger-Dyson (SD) equation. Modern technicolor models often utilize asymptotically free gauge theories with an
. approximate infrared fixed point (IRFP) to achieve the walking behavior.
1
0 TheSU(N)gaugetheorywithalargenumberofmasslessfermionsisoneofthetheoriesthatareexpectedtopossess
2 such a property [5]. In the case of SU(3) gauge theory for example, the two-loop running coupling has an IRFP in
1 the range of 9 ≤ N ≤ 16 (< NAF), where N is the number of massless fermion with fundamental representation,
f f f
:
v and NfAF is the value of Nf above which a theory loses its asymptotic freedom nature [6, 7]. Within this range of
i N , the larger the number of N becomes, the smaller does the value of the running coupling at the IRFP. Because
X f f
of this, it is expected that there is a critical value of flavor, Ncr, below which the theory is in the confining hadronic
r f
phase with broken chiral symmetry, while above which it is in the deconfined phase with unbroken chiral symmetry.
a
An analysis based on the SD equation with the improved ladder approximation estimates that the value of Ncr lies
f
between11and12[8]. Therefore,for12≤N ≤16 (oftencalled“conformalwindow”),the theorypossessesanexact
f
IRFP, while for 9 ≤ N ≤ 11, the chiral symmetry is spontaneously broken, i.e., the IRFP disappears and the scale
f
invariance is only approximate. In Ref. [9, 10], this chiral phase transition at Ncr was further identified with the
f
“conformal phase transition” which was characterized by the essential singularity scaling (Miransky scaling).
Considering the intrinsically non-perturbative nature of the problem, the lattice gauge theory should play an
importantroleforthestudyofthephasestructureofsuchtheories. InadditiontopioneeringworkssuchasRefs.[11–
14], there is growing interest in this subject in recent years [15]. A straightforward way of investigating the infrared
behaviorof a giventheory is to calculate the running coupling constantof the theory. Thoughit requiressimulations
in a wide rangeof parameterspace since extensive rangeof the energyscale has to be coveredto trace the running of
the coupling by step-scaling procedure, there are many groups that devote their efforts to such a direction.
Alternatively, infrared conformality of the theory can also be investigated by deforming the theory with the in-
troduction of a small fermion bare mass, m , as a probe, and study the relation between some low-energy physical
0
quantities (such as the meson masses and the decay constants) and m . In Ref. [16, 17], it is shown that the scaling
0
relation between a low-energy quantity and m can be expressed in terms of the mass anomalous dimension at the
0
2
IRFP,γ∗.1 Inthecaseofthemass(M)ofamesonwithcertainspinandquantumnumbers,forexample,thescaling
m
relation (“hyperscaling relation”) is expressed as
1/(1+γ∗)
M ∼m m . (1)
0
When one considers a theory in a finite space-time, the scaling relation is modified to the “finite-size hyperscaling
relation” as follows:
M =L−1f(x), (2)
where, L is the size of space and time, and f is some function of scaling variable x which is defined as
x≡Lˆmˆ1/(1+γm∗). (3)
0
Here,weintroduceddimensionlessquantities,Lˆ ≡LΛandmˆ ≡m /Λ,wherewetakeΛastheUVscaleatwhichthe
0 0
infraredconformalityterminates. Severalgroups[18–21]triedtojudge whether candidatetheoriespossesanIRFP or
not by measuring the low-energy quantities on the lattice for various combination of input values of Lˆ and mˆ , then
0
∗
checking whether Eq. (2) is satisfied for a certain value of γ .
m
However,acoupleofquestionsarisehereregardinguseof(finite-size)hyperscalingrelationforthestudyofinfrared
conformality: Oneofthemisrelatedtothe factthatthebarefermionmass,m ,whichis introducedasaprobe,itself
0
necessarily breaks the infrared conformality of the original theory. How small m has to be so that the hyperscaling
0
relation is approximately satisfied? What is the form of correction if it is not small enough? When the anomalous
∗
dimension is measured for mass not so small, can it be regarded as γ at IR fixed point at face value? Another
m
question is, when the theory in question does not have an IRFP (namely, in the phase where the chiral symmetry is
spontaneously broken) in the first place, how and how much is the hyperscaling relation violated?
TheladderSDequation,whichis thebirthplaceofthe walkingtechnicolor,isactuallyaconcretedynamicalmodel
to study such questions. In the framework of the ladder SD equation, we know whether a given theory is infrared
conformal or not, and also the value of the anomalous dimension as well. Therefore, we can quantitatively study
the (finite-size) hyperscaling relation and its violation by using the ladder SD equation in a self-consistent manner.
Numerical calculations can be easily done in a wide range of parameter space, and to a certain extent, even an
analytical understanding can be obtained by investigating the solution of the ladder SD equation.
In this paper, we study the (finite-size) hyperscaling relationand its violation,basedon the ladder SD equationby
taking the example of SU(3) gauge theory with various number of fundamental fermions (which is often called the
large N QCD).
f
In the next section, from the analytical expression of the solution of the ladder SD equation, we identify the form
of the leading correction to the hyperscaling relation. We find that the anomalous dimension γ for off the IRFP
m
∗
(with finite mass scale) is substantially smaller than γ , the value at IR fixed point (with vanishing mass) for the
m
larger mass. Our result may shed some light on the value of the anomalous dimension often reported by the lattice
simulations done with relatively large masses.
In Section 3, for the purpose of studying the finite-size hyperscaling relation, we formulate the SD equation in
a finite space-time with the periodic boundary condition. By numerically solving it for various values of the input
parameters (mˆ ,Lˆ), finite-size, as well as mass deformation effects on the finite-size hyperscaling relation is studied
0
in the conformal window. The result suggests that the correction due to the finite-size effect on the finite-size
hyperscaling relation is negligible compared with that coming from large mass corrections. Then we find that the
anomalousdimension, when identified throughthe finite-size hyperscalingrelationneglecting the mass corrections,is
almost the same as that obtained through the hyperscaling in the infinite space-time neglecting the mass corrections
∗
and hence is substantially smaller than γ . We also use the ladder SD equation in a finite space-time to study the
m
chiral-symmetry-breakingphase, and show how the effect of spontaneous breaking of the chiralsymmetry affects the
(finite-size) hyperscaling relation. In the case of N = 11 which is close to the criticality so that the dynamically
f
generated mass is small compared with the explicit mass, the finite-size hyperscaling relation is still operative. We
further suggest a concrete form of the modification of the finite-size hyperscaling relation taking account of the mass
correction, which may be useful for the analysis of the lattice data.
Finally, Section 5 concludes the paper.
1 Thisγm∗ isidentifiedwiththeanomalousdimensionγm relevanttothewalkingtechnicolor,whichisγm measuredatultraviolet(UV)
limit(instead of IR limit),or near the scaleof (pseudo-) UVfixed point, usuallyidentified withthe ETC scale. See discussions below
Eq.(28).
3
α
( µ 2 )
α
0.8
*
0.6
0.4
0.2
0
−60 −40 −20 0 20 40 60
µ2
log Λ2
FIG. 1: Two-loop runningcoupling (solid curve) compared with the approximateform in Eq. (8) (dashed line) in the case of
SU(3) gauge theory with 12 massless fundamental fermions.
N 12 13 14 15 16
f
α∗ 0.75 0.47 0.28 0.14 0.042
γ∗ 0.80 0.36 0.20 0.095 0.027
m
TABLE I: Values of α∗ and also listed are the corresponding γm∗ for SU(3) gauge theory with fundamental fermions in the
conformal window to begiven byEq.(28).
II. CORRECTION TO THE HYPERSCALING RELATION
We start from the study of the hyperscaling relation in the infinite space-time, namely the one in Eq. (1). In this
section, from the analytical expression of the solution of the ladder SD equation, we identify the form of the leading
correction to the hyperscaling relation. Here, we take the example of the large N QCD in the conformal window to
f
study infrared conformal theories.
Thetwo-looprunningcouplingofthelargeN QCDisshowninFig.1. Inthefigure,thetwo-looprunningcoupling
f
(andits approximatedform)inthe caseofSU(3)gaugetheorywith 12masslessfundamentalfermionis plottedasan
example. The solid curve represents the two-loop running coupling which are obtained from the following RGE for
α(µ2) = g¯2(µ2) :
4π
(cid:16) (cid:17)
d
µ α(µ2)=β(α(µ2))=−bα2(µ2)−cα3(µ2), (4)
dµ
where
1 1 N2−1
b= (11N −2N ), c= 34N2−10N N −3 c N . (5)
6π c f 24π2 c c f N f
(cid:18) c (cid:19)
α∗ is the value of the running coupling at the IRFP which is determined as
b
α∗ =− . (6)
c
Valuesofα∗inthecaseofSU(3)gaugetheorieswithvariousnumberoffundamentalfermion(intheconformalwindow)
are shownin Table I.Λ whichappears inFig. 1 is a renormalizationgroupinvariantscale,Λ=µ exp − α(µ) dα ,
β(α)
the two-loop analogue of ΛQCD of the ordinary QCD, which is taken as [8] (cid:16) R (cid:17)
Λ ≡ µ exp − 1 log α∗−α(µ2) − 1 , α(Λ2)≃0.78α∗, (7)
b α∗ α(µ2) b α(µ2)
(cid:20) (cid:18) (cid:19) (cid:21)
4
in such a way that the scale Λ plays the role of the “UV cutoff ” where the infrared conformality we are interested
in terminates, i.e., α(µ2) ∼ const(≃ α∗) for µ2 < Λ2, while α(µ2) ∼ 1/log(µ2/Λ2) for µ2 > Λ2 as in the usual
asymptotically free theory. Actually in the walking technicolor, Λ is taken to be of order of (or even larger than)
the UV scale Λ (“ETC scale”) where the technicolor theory no longer makes sense as it stands and is actually
ETC
converted into a more fundamental theory such as the Extended Technicolor (ETC).
In this paper we use the following form of the running coupling as an approximation of the two-loop running
coupling (dashed line in Fig. 1) :
g¯2(µ2)
α(µ2)= =α∗ θ(Λ2−µ2). (8)
4π
In this approximation the coupling takes the constant value α∗ (the value at the IR fixed point) below the scale Λ
and entirely vanishes in the energy regionabovethis scale. Therefore,the physicalpicture of the largeN QCD with
f
this approximation is the same as that in constant coupling gauge theory with UV cutoff Λ which was extensively
studied long time ago[22–24]. We note that it is possible, at least numerically, to solve the SD equationwithout this
approximation for the two-loop running coupling. However, we adopt this simplification so that we can analytically
study the solution of the SD equation to a certain extent.
A. SD equation
Let us first write down the SD equation for SU(N ) gauge theory with fundamental fermions:
c
d4k 1 (p−k) (p−k)
iS−1(p) = /p−m + C g¯2((p−k)2) g − µ ν γµ iS (k) γν. (9)
F 0 i(2π)4 2 (p−k)2 µν (p−k)2 F
Z (cid:18) (cid:19)
Here, iS−1 ≡A(p2)p/−B(p2) is the full fermionpropagator, C = Nc2−1 is the quadraticCasimir,and g¯((p−q)2)is
F 2 2Nc
the runningcouplingconstant. Inthe aboveexpression,we tookthe Landaugauge,andadoptedthe improvedladder
approximation, in which the full gauge boson propagator is replaced by the bare one and the full vertex function is
replaced by a simple γµ-type vertex with the running coupling constant associated with it. From this equation, we
obtain the following two independent equations:
d4k C g¯2((p−k)2) (p·k) {p·(p−k)}{k·(p−k)}
A(p2) = 1+ 2 A(k2) +2 , (10)
(2π)4 k2A(k2)2+B(k2)2 p2(p−k)2 p2(p−k)4
Z (cid:20) (cid:21)
d4k 3C g¯2((p−k)2) B(k2)
B(p2) = m + 2 . (11)
0 (2π)4 k2A(k2)2+B(k2)2 (p−k)2
Z
ThesearecoupledequationsforA(p2)andB(p2)writteninEuclideanmomentumspace. (Notethatwehavedropped
the subscript “E” for Euclidean momentum variables.)
To further simplify the SD equation, we adopt a simplified form for the argument of the running coupling, g¯2((p−
k)2): we takeitto be a functionofonly p2 andk2 insteadof(p−k)2. Withthis simplification,itbecomes possibleto
carry out the angular integration (in the momentum space), then the SD equation becomes an equation for a single
variable x≡p2. Also, in this case, A(x)=1 is obtained from Eq. (10). Therefore,the SD equation becomes a single
E
integral equation for the mass function Σ(x)(≡B(x)/A(x) =B(x)).
For the analytical study we adopt a practically simple ansatz for the running coupling: [25]
g¯2((p−k)2)⇒g¯2(max{p2,k2}). (12)
Then the ladder SD equation with Eq.(8) reads:
3C Λ2 1 Σ(y)
2
Σ(x)=m0+α∗ 4π dy max{x,y} y+Σ2(y), (13)
Z0
which can readily be converted into an equivalent (nonlinear) differential equation with boundary conditions [23]:
′′ 3C2 Σ(x)
(xΣ(x)) +α∗ = 0, (14)
4π x+Σ(x)2
limx2Σ(x)′ = 0, (15)
x→0
′
(xΣ(x)) = m . (16)
x=Λ2 0
(cid:12)
(cid:12)
5
We mayfurther simplify Eq.(14) byreplacingΣ(x) inthe denominatorofthe secondtermin the LHSby a constant,
m , which is customarily defined by :
P
m ≡Σ(x=m2). (17)
P P
Then the SD equation reads: [26]
′′ 3C2 Σ(x)
(xΣ(x)) +α∗ = 0. (18)
4π x+m2
P
We should note that it is known that the solution obtained from this linearized equation well approximates that
obtained(bynumericalcalculation)fromtheequationwithoutlinearization. Laterweshallshowthatparticularlyfor
theanomalousdimensionthereisaremarkableagreementbetweentheanalyticalresultobtainedfromtheasymptotic
solution of the linearized equation Eq.(18) and the numerical one from the full nonlinear integral SD equation under
a slightly different ansatz for the argument of the running coupling (to be mentioned later).
m defined in Eq. (17) is often called the “pole mass”, though of course m is not the real pole mass (remember
P P
that x is the Euclidean momentum square). Still, m is useful quantity for the investigation of the hyperscaling
P
relation since it is known, from the study with the Bethe-Salpeter equation [27], that m is proportional to meson
P
masses. Therefore, in this paper, we use m which is obtained from the solution of the SD equation as low-energy
P
physical quantity which appears in the hyperscaling relation.
B. Leading correction to the hyperscaling relation
BeforeweproceedtotheinvestigationofthesolutionoftheSDequationtoderivetherelationbetweenm andm ,
P 0
we note that it is known [22] that there is a critical value of α∗, αcr(6=0), such that spontaneous symmetry breaking
solution (Σ(x)6=0) which satisfies Eqs. (18), (15) and (16) for the chiral limit m =m (Λ)≡0 does not exists for
0 0
α∗ ≤αcr, (19)
where [23]
π π
α = = (N =3). (20)
cr c
3C 4
2
Namely, a nontrivial solution (Σ(x) 6= 0) for α < α is the explicit breaking solution which exists only for m =
cr 0
m (Λ)6=0. Then the pole mass m is nothing but a renormalized mass (“current mass”) m :
0 P R
m =m =Z−1m , (21)
P R m 0
where m = m (µ = m ), and Z = Z µ | is the mass renormalization constant. This means that, in the
R R P m m Λ µ=mP
chiral limit m = 0, there is no mass gap (dynamically generated mass) m = 0, and therefore the IRFP of the
R D
(cid:0) (cid:1)
theory is exact for α∗ <αcr. This is exactly the regionin which one expects that the hyperscaling relationshould be
satisfied. Therefore,in the restofthis section, we concentrateonstudying the SD equationin the regionofα∗ <αcr,
or equivalently (through the relation in Eq. (6)) NAF ≥ N ≥ Ncr (conformal window), where NAF = 16.5 and
f f f f
Ncr ≃11.9 for N =3.
f c
A solutionof Eq.(18)which satisfiesboundary conditionEq.(15) canbe expressedin terms ofthe hypergeometric
function as [26]
1+ω 1−ω x
Σ(x) = ξm F , ,2,− , (22)
P 2 2 m2
(cid:18) P(cid:19)
where
α∗
ω ≡ 1− . (23)
α
r cr
ξ is a numerical coefficient which is determined from the definition of m in Eq. (17):
P
1+ω 1−ω
ξ−1 =F , ,2,−1 . (24)
2 2
(cid:18) (cid:19)
6
In the limit of x≫m2, the solution can be expanded as
P
ω−1
Γ(ω) x 2
Σ(x) ≃ ξm + (ω ↔−ω) . (25)
P " Γ(ω+21)Γ(ω+23)(cid:18)m2P(cid:19) #
ByinsertingtheaboveexpressionofΣ(x)intotheremainingboundaryconditioninEq.(16),weobtainthefollowing
relation between m and m :
P 0
ω−1
Γ(ω) Λ2 2
m = ξm + (ω ↔−ω) . (26)
0 P " Γ(ω+21)2 (cid:18)m2P(cid:19) #
From this we have the mass renormalizationconstant Z in Eq.(21) as
m
ω−1
m m Γ(ω) Λ2 2
0 0
Z ≡ = = ξ + (ω ↔−ω) , (27)
m mR mP " Γ(ω+21)2 (cid:18)m2P(cid:19) #
where we note again m =m in the conformal window.
P R
∗
Then we can obtain the mass anomalous dimension at IRFP, γ , as
m
∗ ∂logZm α∗
γ = lim = 1−ω = 1− 1− , (28)
m mP/Λ→0∂log(mP/Λ) (cid:18) r αcr (cid:19)
∗
where the limit is taken as m → 0 with Λ fixed. In Table I, we show values of γ , which can be calculated from
P m
the above expression combined with Eqs. (5), (6) and (20), in the case of SU(3) gauge theory with various numbers
of fundamental fermion in the conformal window.
∗
It shouldbe noted thatthis γ atIRFP is actually the same as the anomalousdimension in the UV limit (Λ→∞
m
withmP fixed)γm(UV) ≡limΛ/mP→∞ ∂l∂oglo(mgZPm/Λ) =1−ω[28]. γm(UV) isthequantityrelevanttothewalkingtechnicolor
(UV)
withγm =1(thevalueat(pseudo-)UVfixedpointinthebrokenphaseα∗ >αcr inthechirallimitmR =0: mP =
m ) [3]: The technifermion condensate hψ¯ψi| at the UV scale Λ=Λ (>103TeV)≫µ(=O(m )=O(TeV)) is
D Λ ETC P
enhanced by γm(UV) =1 as hψ¯ψi|Λ =Zm−1hψ¯ψi|µ with Zm−1 =(Λ/µ)γm(UV) =(Λ/µ)1 ≫1.
∗ ∗
By using this γ , we can rewrite the expression in Eq. (26) in terms of γ :
m m
m Γ(1−γ∗) m 1+γ∗ Γ(−1+γ∗) m 3−γ∗
0 = ξ m P m + m P m . (29)
Λ " Γ(2−2γ∗)2 (cid:16) Λ (cid:17) Γ(γ2∗)2 (cid:16) Λ (cid:17) #
This is the expression which should be compared with the hyperscaling relation in Eq. (1). It is obvious that if we
drop the second term in the RHS of Eq.(29), it reduces to the hyperscaling relation[16]. Therefore,the secondterm
should be identified as the leading correction to the hyperscaling relation.
To see the significance of the correction term, in Fig. 2, we plot ratios of the second term to the first term in the
∗ ∗
RHS of Eq. (29) as functions of m /Λ for various values of γ in the range of 0 ≤ γ ≤ 1.0, which corresponds to
P m m
NAF ≥N ≥Ncr. When m is much smaller than Λ, (except in the case of γ∗ =1.0) the effect of the second term
f f f P m
∗
is very small since in the range of 0 < γ < 1.0, the power of (m /Λ) in the second term is always greater than
m P
that in the first term. This is reasonable considering the fact that small m (or equivalently, small m ) means small
P 0
∗
mass deformation. Another limit in which Eq. (29) approximates well the hyperscaling is γ →0. In this limit, the
m
coefficientofthesecondtermgoesto0whilethatofthefirsttermgoesto1. Also,thepowersuppressionofthesecond
term becomes strongin this limit as well. However,we should remember that phenomenologicallymotivated theories
∗ ∗
have large γ . In the limit of γ → 1, the power of (m /Λ), as well as coefficients of the two terms asymptote to
m m P
the same values. Therefore, we have to take the second term seriously when we study the anomalous dimension of
the candidate theories for viable walking technicolor models with γ ≃1 through the hyperscaling relation from the
m
numerical data on the lattice.
For checking the reliability of our ansatz Eq.(12) and the linearization of the differential equation, as well as the
asymptotic expansionEq.(25), we show inFig. 3 the log-scaleplot ofm −m inEq.(29) for N =12 in comparison
0 P f
with thatobtainedby directly solvingnumerically the full nonlinearSD equationwith morenatural(angle-averaged)
ansatz:
g¯2((p−k)2)⇒g¯2(p2+k2). (30)
7
0
−0.2
γ* = 0
m
= 0.2
−0.4
= 0.4
= 0.6
−0.6 = 0.8
= 0.9
= 1.0
−0.8
−1
0.00001 0.0001 0.001 0.01 0.1 0.5
m Λ
P
∗
FIG. 2: Ratios of thesecond term to thefirst term in theRHSof Eq. (29) as functions of m /Λ for various values of γ .
P m
1
0.1 numerical
analytic
0.01
Λ 0.001
0
0.0001
m
−5
10
−6
10
−7
10
−8
10
0.0001 0.001 0.01 0.1 1
m Λ
P
FIG. 3: Analytical asymptotic solution of the linearized SD equation vs. numerical solution of the full nonlinear one (for
N = 12). Analytic result (blue dotted line) is the plot of Eq. (29) which is from the asymptotic expansion of the linearized
f
SD equation Eq.(18) with the ansatz Eq.(12), while the numerical one (denoted by red plus) is that of the solution of the full
nonlinear integral SD equation Eq.(31) with ansatz Eq.(30).
The SD equation in this case with Eq.(8) reads:
3C Λ2−x 1 Σ(y)
2
Σ(x)=m0+α∗ 4π dy max{x,y} y+Σ2(y), (31)
Z0
which differs from Eq.(13) by Λ2−x in the UV end of the integral. The slope in Fig.(3) corresponds to γ +1. The
m
agreement is remarkable, irrespectively of the different ansatz and the additional approximations.
8
γ
eff
m
0.8
0.7 Nf = 12
Nf = 13
0.6 Nf = 14
0.5 Nf = 15
Nf = 16
0.4
0.3
0.2
0.1
0
0.00001 0.0001 0.001 0.01 0.1 0.5
m Λ
P
FIG. 4: Effective mass anomalous dimension as a function of m /Λ for SU(3) gauge theories with 12, 13, 14, 15 and 16
P
fundamental fermions.
C. Effective anomalous dimension
In the literature, the hyperscaling relation is often used as a tool to judge whether a theory is infrared conformal
or not. However, the importance of the corrections to the hyperscaling relation due to the mass deformation are
often underestimated, or even completely neglected. This is not surprising because, in practical situations, it is very
difficult to notice that data (for example, hadron mass, M , obtained from lattice simulations with various values of
H
input m a)needto be fittedby afunction withcorrectionterm. Eveninthe situationthatthe correctiontermis not
0
verysmallcomparedtotheleadingterm,datacouldbeeasilyfittedbyasimplefunctionofaformm ∼M1+γ unless
0 H
data aretakenin a wide rangeofm (or,equivalently, M ). Especiallywhenthe dataareassociatedwith, say,a few
0 H
percent of error bars, it is very possible that one succeeds in fitting the data with a function of a form m ∼ M1+γ.
0 H
However,thebest-fitvalueofγ obtainedbythisfittingmustbenumericallydifferentfromtheactualmassanomalous
dimensionatthe IRFP.Tomakethe difference clear,we introduceeffective mass anomalous dimension, γeff, whichis
m
defined asthe value ofγ one obtains asa best-fit value when one forcesto do fitting by using a fit-function whichhas
a form of hyperscaling relation. Since the significance of the correction term is different for different values of M ,
H
the value of γeff should change depending on the range of M one uses for fitting to obtain it.
m H
In the framework of the SD equation with the improved ladder approximation, we can identify γeff as γ =
m m
γ (µ/Λ)| obtained from Eq.(27) with Eq.(28) (or equivalently Eq. (29)):
m µ=mP
∂logZ
γeff =γ = m (32)
m m ∂log(m /Λ)
P
∂ Γ(1−γ∗) mP γm∗ Γ(−1+γ∗) mP 2−γm∗
= log ξ + (33)
∂log(mP/Λ) " Γ(2−2γ∗)2 (cid:16) Λ (cid:17) Γ(γ2∗)2 (cid:16) Λ (cid:17) #!
Itisobviousthatwereitnotforthe secondterm, γeff wouldcoincide withγ∗. Noteagainthatthe significanceofthe
m m
correction term is different for different values of m . Therefore, the effective mass anomalous dimension becomes a
P
functionofm . InFig.4,γeff forSU(3)gaugetheorieswith12,13,14,15and16fundamentalfermionsareplottedas
P m
a function of m . For the purpose of making it easier to see the deviationof the effective mass anomalousdimension
P
from the value at the IRFP, we also plot γeff/γ∗ in Fig. 5. From these figures, as we expected, we see that the
m m
deviationbetweenγeff andγ∗ becomes moresignificantin largerm /Λregion. We canalsosee thatthe deviationof
m m P
theeffectivemassanomalousdimensionislargerforsmallerN (or,inotherwords,forN closertoNcr ≃11.9). This
f f f
∗
is also expected from the discussion below Eq. (29) because smaller N means larger γ , with which the correction
f m
termtothehyperscalingrelationbecomesimportant. Aswementionedearlier,phenomenologicallyinterestingtheory
is the one with large mass anomalous dimension. Therefore, it is important to keep this effect of correction term to
the hyperscaling relation in mind when one study such theories.
9
γeff γ*
m m
1.1
1
0.9 Nf = 16
Nf = 15
0.8 Nf = 14
Nf = 13
0.7 Nf = 12
0.6
0.5
0.00001 0.0001 0.001 0.01 0.1 0.5
m Λ
P
FIG. 5: Effective mass anomalous dimension which is normalized by the value of it at the IRFP as a function of m /Λ for
P
SU(3) gauge theories with 12, 13, 14, 15 and 16 fundamental fermions.
III. EFFECT OF THE CORRECTIONS ON THE FINITE-SIZE HYPERSCALING
So farwe haveconcentratedonthe study ofthe hyperscalingrelationinthe infinite space-time. However,since the
lattice simulations are done in a finite space-time, the hyperscaling relation in a form of Eq. (2) are used more often.
Therefore, it is important to study the effect of correction due to mass deformation on the finite-size hyperscaling
relation. For the purpose of studying the finite-size hyperscaling relation, we formulate the SD equation in a finite
space-time with the periodic boundary condition. By numerically solving it for various values of input parameters
(mˆ ,Lˆ), mass deformation effects on the finite-size hyperscaling relation is studied in large N QCD.
0 f
A. SD equation in a finite space-time
To formulate the SD equation in a finite space-time, we start from the SD equation in the infinite space-time in
Eqs. (10) and (11). To put these equations in a finite space-time, all one needs to do is to replace the continuum
momentum variables by the discrete ones:
2πn
p →p˜ = i, (n ∈N), (34)
i i i
L
where, p is the i-th component of the momentum variable. We adopted the periodic boundary condition for all
i
directions, though it is easy to implement the anti-periodic boundary condition. We also assumed that the size of
all space-time directions are the same. It is also straightforwardto introduce different sizes for spacial and temporal
directions. However, we took the same length for every direction just for simplicity. n ’s are integers which label
i
discrete momentum variables. With this replacement of the momentum variables, the SD equation in Eqs. (10) and
10
(11) turn into the following form2:
1 C g¯2((p˜−k˜)2) (p˜·k˜) p˜·(p˜−k˜) k˜·(p˜−k˜)
A˜(p˜) = 1+ 2 A˜(k˜) +2 , (35)
L4 k˜2A˜(k˜)2+B˜(k˜)2 p˜2(p˜−k˜)2 n p˜2(p˜o−nk˜)4 o
mX0,1,2,3
1 3C g¯2((p˜−k˜)2) B˜(k˜) 
B˜(p˜) = m + 2 , (36)
0 L4 k˜2A˜(k˜)2+B˜(k˜)2 (p˜−k˜)2
mX0,1,2,3
where
n m
0 0
2π n 2π m
p˜=  1 , k˜ =  1 . (37)
L n L m
2 2
   
 n   m 
 3   3 
   
Here, n and m are integers, though we should note that the SD equation is not defined at p˜= 0 since one of the
i i
two independent equations is derived by requiring coefficients of /p in LHS and RHS are the same in Eq. (9). In the
above expressions, A(p2) and B(p2) were replaced by A˜(p˜) and B˜(p˜). This is because they are no longer functions of
momentum-squaredsince the rotationalsymmetry is broken(except someresidualdiscreterotationalsymmetry)due
to the hyper-cubic shape of the finite-size space-time. However, at the practical level, the effect of such rotational-
symmetry violation is negligible as far as L is taken large enough compared to the scale of relevant physics. This is
true in the case of the current study. We first numerically solve, by using iteration method, Eqs. (35) and (36) as
coupled equations for A˜(p˜) and B˜(p˜). Then, to obtain the value of m (see Eq. (17)), we plot Σ˜(p˜)≡B˜(p˜)/A˜(p˜) as
P
a function of p˜2. There are always multiple p˜’s which give the same value of p˜2, and those do not necessarily give a
degenerate value of Σ˜ unless those are related by the residual discrete rotational symmetry. However, we confirmed
thatsuchdifferencesarenegligibleintheregionwherem isdetermined. Weshouldalsonotethat,whenweestimate
P
the value of m from Eq. (17), we used a function which is obtained by interpolating Σ˜(p˜) in momentum space.
P
However, we never did extrapolation to the scale below 2π/L since there is no reliable information below that scale.
Therefore, we obtain data only when the value of m2 is greater (2π/L)2.
P
B. Finite-size hyperscaling and its corrections
In this subsection, by numerically solving the finite-volume SD equation formulated in the previous subsection,
we generate data of m for various sets of input parameters (LΛ,m /Λ). We take SU(3) gauge theory with 12
P 0
fundamental fermion as an example here. Then, by using those generated data, we do the analysis based on the
finite-size hyperscaling in Eq. (2). This is a kind of “simulation” of the practical situation we often encounter when
we study a theory by using data obtained from lattice simulations. An interesting point about doing hyperscaling
analysis using data generatedby the SD equation is that we know that the SU(3) gauge theory with 12 fundamental
fermions, in the framework of the SD equation, is the infrared conformal theory, and we also know the value of the
∗
mass anomalous dimension at the IRFP, which is estimated as γ ≃ 0.80 in this case. (See Table I.) Therefore, we
m
clearly see how the finite-size hyperscaling is violated due to the effect of the mass deformation.
InFig.6,we plotthe valuesofm /Λ(horizontalaxis)for variousvaluesofm /Λ(verticalaxis)andLΛ(indicated
P 0
by different symbols). When we solved the finite volume SD equation, we adopted an angle averaged form (as in
Eq. (30)) for the argument of the running coupling. As we explained in the previous section, this is the procedure
which is needed to make the angular integration (in momentum space) possible in the case of infinite-volume SD
equation, and it is actually not needed for finite-volume SD equation since we numerically solve them by iteration
without doing angular integration. However, for the purpose of putting finite- and infinite-volume SD equations on
2 During the summations of mi, one encounters singularities at mi = ni. Also, in the limit of B˜ → 0, the contribution from
(m0,m1,m2,m3) = (0,0,0,0) diverges. However, these can be identified as unphysical artifacts considering the fact that these are
integrable singularities in the case of infinite space-time. Therefore, in the numerical calculation of the SD equation, we simply drop
thesingularpointsfromsummations. (Thelattersingularitycanalsobeavoidedbyadoptingtheanti-periodicboundarycondition. We
didthenumericalcalculationswiththeanti-periodicboundaryconditioninthetemporaldirection,andcomparedthesolutionwiththe
one obtained from the periodic boundary condition with the prescription explained above. We found that the difference between two
arenegligible.)