Infrared analysis of quantum field theories

0.1 MB·English

by  Marco Frasca

Infrared analysis of quantum field theories
Marco Frasca
∗
Via Erasmo Gattamelata, 3
00176 Roma (Italy)
(Dated: February 2, 2008)
Weshowthatforaλφ4 theoryhavingmanycomponents,thesolution withallequalcomponents
in the infrared regime is stable with respect to our expansion given by a recently devised approach
to analyze strongly coupled quantum field theory. The analysis is extended to a pure Yang-Mills
theoryshowinghow, inthiscase, thegivenasymptoticseries exists. Inthisway,manycomponents
theories in the infrared regime can be mapped to a single component scalar field theory obtaining
theirspectrum.
PACSnumbers: 11.15.Me,11.15.-q
7
0
The study ofquantum fieldtheories in the infraredregimeis generallyapproachedby pure numericalmethods. No
0
perturbative approaches have been known until recently for an analytical study of these theories. The difficulty can
2
be traced back to the increasing of the coupling constant with momentum approaching in this case too large values
n
for perturbation theory to be useful at all.
a
In a recent work we have proposed a new approach [1] that relies on the duality principle in perturbation theory
J
[2, 3] permitting to use perturbation theory also when an increasingly large parameter enters into the equations of a
6
field theory. This approach has been successfully applied in general relativity proving that a gradient expansion is a
2
strong coupling expansion [4].

4 This methodhaspermittedtoextractthe spectrumofaλφ4 theorywithasinglecomponentinthe infraredregime
v givingthemassgapinthis case. Ouraiminthis paperisto extendthis approachto multi-componentsquantumfield
8 theories.
4 So, let us consider the Hamiltonian (here and in the following we take ~=c=1)
1
0 1 1 1 1
1 H = dD−1x 2π2+ 2(∂xφ)2+ 2µ20φ2+ 4λφ4 . (1)
6 Z (cid:20) (cid:21)
0 After a normalization to µ is taken, Hamilton equations are
0
/
h ∂tφ = π (2)
t
p- ∂tπ = ∂x2φ−φ−λφ3.
e We take
h
: τ = √λt (3)
v
i 1 1
X π = √λ π(0)+ π(1)+ π(2)+...
λ λ2
r (cid:18) (cid:19)
a 1 1
φ = φ(0)+ φ(1)+ φ(2)+....
λ λ2
obtaining the set of dual perturbation equations
∂τφ(0) = π(0) (4)
∂τφ(1) = π(1)
∂τφ(2) = π(2)
.
.
.
∂τπ(0) = φ(0)3
−
∂τπ(1) = 3φ(0)2φ(1)+∂x2φ(0) φ(0)
− −
∂τπ(2) = 3φ(0)φ(1)2 3φ(0)2φ(2)+∂x2φ(1) φ(1)
− − −
.
.
.
∗[email protected]
2
and one sees that at the leading order a homogeneous equation rules the dynamics of the field. Then, one can prove
numerically that in the limit λ for the classical theory one can still use a Green function method [1] with the
→ ∞
leading order homogeneous equation
G¨+λG3 =δ(t), (5)
having expicitly restated λ. This equation has the exact solution
1 1
2 4 λ 4
G(t)=θ(t) sn t,i . (6)
λ 2
(cid:18) (cid:19) "(cid:18) (cid:19) #
and its time reversedversion, with sn the snoidal elliptic Jacobi function. The quantum field theory is then given by
i δ δ
Z[j]=exp dDy dDy ( 2+1)δD(y y ) Z [j] (7)
1 2 1 2 0
2 δj(y ) −∇ − δj(y )
(cid:20) Z 1 2 (cid:21)
being
i
D D
Z [j]=exp d x d x j(x )∆(x x )j(x ) (8)
0 1 2 1 1 2 2
2 −
(cid:20) Z (cid:21)
with the Feynman propagator
∆(x x )=δD 1(x x )[G(t t )+G(t t )]. (9)
2 1 − 2 1 2 1 1 2
− − − −
Noting the following relation for the snoidal function[5]
2π ∞ ( 1)ne−(n+21)π πu
sn(u,i)= − sin (2n+1) (10)
K(i) 1+e (2n+1)π 2K(i)
n=0 − (cid:20) (cid:21)
X
being K(i) the constant
π
2 dθ
K(i)= 1.3111028777, (11)
Z0 1+sin2θ ≈
the Fourier transform of the Feynman propagatorpis
∞ Bn
∆(ω)= (12)
ω2 ω2 +iǫ
n=0 − n
X
being
π2 ( 1)n+1e−(n+21)π
Bn =(2n+1)K2(i) −1+e (2n+1)π , (13)
−
and the mass spectrum of the theory given by
1
1 π λ 4
ωn = n+ (14)
2 K(i) 2
(cid:18) (cid:19) (cid:18) (cid:19)
proper to a harmonic oscillator. In this case, the mass gap in the limit λ is
→∞
1
π λ 4
δS = (15)
2K(i) 2
(cid:18) (cid:19)
correspondingto the choicen=0,thatis producedby the self-interactionofthe scalarfield. As showninRef.[1], the
next correctionin the dual perturbation series is proportionalto the missing term in the propagator,i.e. k2+1 as it
should be.
We now go one step further with respect to [1] proving that the perturbation series given in eqs.(5) is meaningful
by solving them. This series should prove to give increasingly smaller terms with increasing λ. The existence of a
3
classical solution in the strong coupling limit is essential to prove that, already at this stage, no instabilities set in
andalsotolookatthe analyticalformofthis solution. Inthis waywecanbe surethatalsoaquantumfieldtheoryin
the same limit can be built. Our aim is to prove existence and we do not care about explicit Lorentz invariance but
we note that all this computations can be carried out with explicit Lorentz invariance by introducing a fifth variable
and working with an Euclidean theory[17].
Then, we can write immediately the solution at the leading order by imposing Lorentz invariance as
A
φ(0) =Asn (√λt k x),i (16)
√2 − ·
(cid:18) (cid:19)
thatisaJacobisnoidalwavewithawavevectorkandanormalizationfactordependentontheenergyasalsohappens
for ordinary plane waves. Normalization on a box will yield a quantization of the wave vector. The next order will
give the equation to be solved
∂2φ(1)+3φ(0)2φ(1) = k2φ(0)3 φ(0). (17)
τ
− −
What we have to prove is that this term is bounded so that the strong coupling expansion is meaningful in the
classical model. For our aims we will content with a WKB solution to the above equation. This is consinstent with
the requirement of a perturbation going to infinity (formally λ )[2]. This gives
→∞
2 1 τ k2φ(0)(τ,x)3+φ(0)(τ,x)
φ(1) tan √3 dτ φ(0)(τ ,x) . (18)
′ ′
≈−r3A2 (cid:18) Z0 (cid:19)cn √A2(τ −k·x),i dn √A2(τ −k·x),i
(cid:16) (cid:17) (cid:16) (cid:17)
We can see that the solution is finite except for those sets of points where WKB approximation is expected to fail,
that is, caustics. Then, the next correction can be written down straightforwardly as
2 1 τ ∂2φ(1)(τ,x) φ(1)(τ,x) 3φ(0)(τ,x)φ(1)(τ,x)2
φ(2) tan √3 dτ′φ(0)(τ′,x) x − − . (19)
≈r3A2 (cid:18) Z0 (cid:19) cn √A2(τ −k·x),i dn √A2(τ −k·x),i
(cid:16) (cid:17) (cid:16) (cid:17)
thatbehavessimilarlytothefirstcorrection. Thenextorderscanbewrittendowninasimilarmanner. Theconclusion
isthatwehavebuiltastrongcouplingsolutionoftheclassicalequationofascalarfieldwithasinglecomponent. The
series exists, apartfor caustics, and is meaningful. Then, whenever we will be able to map a field theory to this case,
a meaningful quantum field theory in the infrared is obtained and the spectrum in this regime can be immediately
derived.
Inordertoexploitthisapproach,letusconsideraU(1)-invariantscalarfieldtheory. ThistheoryhastheHamiltonian
1 1 1 1 1 1
H = dD−1x 2π12+ 2π22+ 2(∂xφ1)2+ 2(∂xφ2)2+ 2µ20(φ21+φ22)+ 4λ(φ21+φ22)2 . (20)
Z (cid:20) (cid:21)
This gives the following Hamilton equations (after normalization to µ )
0
∂tφ1 = π1 (21)
∂tπ1 = ∂x2φ1−φ1−λφ1(φ21+φ22)
∂tφ2 = π2
∂tπ2 = ∂x2φ2−φ2−λφ2(φ21+φ22).
Applying the above procedure we get the perturbation equations
∂2φ(0) = φ(0)(φ(0)2+φ(0)2) (22)
τ 1 − 1 1 2
∂2φ(1) = (3φ(0)2+φ(0)2)φ(1) 2φ(0)φ(0)φ(1)+∂2φ(0) φ(0)
τ 1 − 1 2 1 − 1 2 2 x 1 − 1
∂2φ(2) = 3φ(0)2φ(2) 3φ(0)φ(1)2 2φ(1)φ(0)φ(1) 2φ(0)φ(0)φ(2) φ(0)φ(1)2 φ(2)φ(0)2+∂2φ(1) φ(1)
τ 1 − 1 1 − 1 1 − 1 2 2 − 1 2 2 − 1 2 − 1 2 x 1 − 1
.
.
.
∂2φ(0) = φ(0)(φ(0)2+φ(0)2)
τ 2 − 2 1 2
∂2φ(1) = (3φ(0)2+φ(0)2)φ(1) 2φ(0)φ(0)φ(1)+∂2φ(0) φ(0)
τ 2 − 1 2 2 − 1 2 1 x 2 − 2
∂2φ(2) = 3φ(0)2φ(2) 3φ(0)φ(1)2 2φ(1)φ(0)φ(1) 2φ(0)φ(0)φ(2) φ(0)φ(1)2 φ(2)φ(0)2+∂2φ(1) φ(1)
τ 2 − 2 2 − 2 2 − 2 1 1 − 1 2 1 − 2 1 − 2 1 x 2 − 2
.
.
.
4
So, a fully mapping of this perturbation series with the one of a single scalar field can be obtained by looking for a
solution with all the components being equal. This in turn means that the spectrum in the infrared of this theory is
obtained by taking into eq.(14) the substitution λ 2λ. This is a general rule as for a theory with N components
→
one has to take the substitution λ Nλ into the spectrum giving the result we were looking for. This result holds
→
because both the classicalandthe quantumfield theoryfor a single componentscalarfield do existas we haveshown
and we can mapa N-componentscalarfield theory onit. We note as the presence of a naturalenergy scale µ makes
0
thespectrumofthetheorynotdependentonanyarbitraryparameter. ThingswillbedifferentforaYang-Millstheory
as we will see below.
We now apply the above procedure to a Yang-Mills theory for the gauge group SU(N). We note that in this case
the only constant entering into the equation, the charge, can be removed from the equations. Anyhow, we will keep
it as a placeholder for the series terms. Our computation can be just carried out when the gauge is fixed. So, the
Hamilton equations in the gauge Aa =0 can be written down as [6, 7]
0
a a
∂tAk = F0k (23)
a a abc b c
∂tF0k = ∂lFlk+gf AlFlk
abc a a a abc b c
being g the coupling constant, f the structure constants of the gauge group, Flk =∂lAk ∂kAl +gf AlAk and
the constraint∂kF0ak+gfabcAbkF0ck =0 does hold. So, let us introduce the following equatio−ns, as done for the scalar
field,
τ = gt (24)
Fa = gFa(0)+Fa(1)+ 1Fa(2)+...
0k 0k 0k g 0k
Fa = Fa(0)+ 1Fa(1)+ 1 Fa(2)+...
lk lk g lk g2 lk
a a(0) 1 a(1) 1 a(2)
A = A + A + A +....
k k g k g2 k
Then, one has the perturbation equations
∂τAak(0) = F0ak(0) (25)
∂τAak(1) = F0ak(1)
.
.
.
∂τF0ak(0) = fabcfcdeAbl(0)Adl(0)Aek(0)
∂τF0ak(1) = fabcfcdeAbl(1)Adl(0)Aek(0)+fabcfcdeAbl(0)Adl(1)Aek(0)+fabcfcdeAbl(0)Adl(0)Aek(1)
+fabc∂l Abl(0)Ack(0) +fabcAbl(0) ∂lAck(0) ∂kAcl(0)
−
. (cid:16) (cid:17) (cid:16) (cid:17)
.
.
and we can recognize at the leading order the homogeneuos Yang-Mills equations. These equations display a rich
dynamics as e.g. Hamiltonian chaos [8, 9, 10]. It is commonly accepted that when equal components are considered
asa leadingordersolution,this solutionis unstable[18]. Here wehaveprovedthatthis isnottrue with respectto our
series. Indeed, we are able to map Yang-Mills theory on a single scalar field theory by taking all equal components
and the series exists both as a classicaland as a quantum field theory as we have already seen. But the mapping can
be just completed by the introduction of an arbitrary energy scale Λ that can be only obtained through experiment
or by a lattice computation. Here we derive its value through the mass of the lightest unflavoredresonance. In order
to obtain the spectrum we easily realize that the right substitution is given by λ g2NΛ4. This gives the spectrum
→
1
1 π g2N 4
ωn = n+ Λ. (26)
2 K(i) 2
(cid:18) (cid:19) (cid:18) (cid:19)
We see here the proper ’t Hooft scaling Ng2 as also shown in lattice Yang-Mills quantum field theory computations
[11]. This gives all the spectrum of a 0++ glueball. We just note that the ground state n=0 gives the mass scale of
the propagator [12].
5
So, we give the glueball spectrum to be compared with [13]. We get from eq.(26) for the scalar glueball and its
excited states
mG
=1.198140235(2n+1) (27)
√σ
where we have put the tension σ =Λ2 g2N/2.
So, let us compare the ground state n = 0 with the lightest unflavored resonance f (600). As recently shown [14]
0
oMneeVs,hionuladgrheaevmeeinnttwhiitshc[a1s2e],Mthσat=ca4n4p1b+−e186usMedeVtowciotmhpsuamteethmeamgnaistsudoef tohrede0r+a+sggliuveebnaliln. [15]. This gives √σ ≈ 368
Indeed, other excited states are given by n=1, n=2 and n=3 as
mG
(n=1) = 3.594420705 (28)
√σ
mG
(n=2) = 5.990701175
√σ
mG
(n=3) = 8.386981645
√σ
in very good agreement with the values in ref.[13] 3.55(7) and 5.69(10) for SU(3) and 3.307(53) and 6.07(17) for the
extrapolated values for N to infinity. Finally, the 0++ mass is given for n=1 by mG =1323 MeV.
A note about the gluon propagatorin the infrared is needed. Indeed, we can see that this is finite and does not go
to zero (12). Rather, it is in nice agreement with recent analysis [16].
In conclusion we have proved a general approach to analyze multi-component quantum field theories in infrared
regime obtaining their spectrum with respect to an effective single component scalar quantum field theory. When
applied to a pure Yang-Mills theory the agreement with lattice computations turns out to be very satisfactory. It
wouldbereallyinterestingtoseeasolutionofascalarquantumfieldtheoryonalatticetostudyitsinfraredproperties
in the light of our results.
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[17] I haveto thank Hagen Kleinert for pointing meout this.
[18] I was not able to trace this claim in the literature.