On the low-energy spectrum of spontaneously broken Φ^4 theories

0.14 MB·English

by  Maurizio Consoli

On the low-energy spectrum of spontaneously broken Φ4
theories
1
1
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2

n
M. Consoli
a
J

0 INFN - Sezione di Catania, I-95123 Catania, Italy
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h
p
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h Abstract
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1 The low-energy spectrum of a one-component, spontaneously broken Φ4 theory
v
4 is generally believed to have the same simple massive form p2+m2 as in the
h
9
symmetric phase where Φ = 0. However, in lattice simulatqions of the 4D Ising
8
h i
1 limit of the theory, the two-point connected correlator and the connected scalar
.
1
propagator show deviations from a standard massive behaviour that do not exist
0
1 in the symmetric phase. As a support for this observed discrepancy, I present a
1
: variational, analytic calculation of the energy spectrum E (p) in the broken phase.
v 1
i Thisanalytic result, whileprovidingthetrendE (p) p2+m2 atlarge p , gives
X 1 ∼ h | |
r an energy gap E1(0) < mh, even when approaching theqinfinite-cutoff limit Λ
a → ∞
withthatinfinitesimalcouplingλ 1/lnΛsuggested by thestandardinterpretation
∼
of “triviality” within leading-order perturbation theory. I also compare with other
approaches and discuss the more general implications of the result.
1. Introduction
In the case of a one-component, spontaneously broken Φ4 theory, one usually assumes a
form of single-particle energy spectrum, say E (p) = p2 +m2, as in a simple massive
1 h
theory with no qualitative difference from the symmetric phase where Φ = 0.
p
h i
One can objectively test [1] this expectation with lattice simulations, performed in the
4D Ising limit of the theory, and study the exponential decay of the connected two-point
correlator C (p,t) e−E1(p)t and the connected scalar propagator G(p). Differently from
1
∼
the symmetric phase, where the simple massive picture works to very high accuracy, the
results of the low-temperature phase show unexpected deviations. Namely, when the 3-
momentum p 0, the fitted E (p) deviates from (the lattice version of) the standard
1
→
massive form p2 +const. and, when the 4-momentum p (p,p ) 0, the measured
µ 4
≡ →
G(p) deviates from (the lattice version of) the form 1/(p2 +const.) .
p
After the first indications of Ref.[1], Stevenson [2] checked independently the existence
of this discrepancy in the lattice data of other authors. To this end, he started from the
lattice data of Ref.[3] for the time slices of C (p = 0,t) and used the Fourier-transform
1
relation to generate equivalent data for the connected scalar propagator G(p). The result-
ing behaviour of G(p) is in complete agreement with the analogous plots obtained from
Ref.[1] (compare Figs.6c, 7, 8 and 9 of Ref.[2]).
The whole issue was later re-considered in Ref.[4]. According to these authors, at the
present, after taking into account various theoretical uncertainties, the deviations are not
so statistically compelling. In their opinion, the conventional scenario of a simple, weakly
coupled, massive theory, ”unfortunately can only be nailed down by analytic proofs”.
The aimofthisLetter istopresent, inSects.2 and3, apossible analyticproofunder the
form of a variational calculation of the energy spectrum in the broken-symmetry phase.
This analytic result, while indeed providing a behaviour E (p) p2 +m2 at larger
1 ∼ h
p , gives theoretical support for deviations in the p 0 limit. In particular, the energy
p
| | →
gap E (0) is definitely smaller than the m parameter that enters the asymptotic form of
1 h
the spectrum. I emphasize that the estimate, being of variational nature, constrains from
above theratio E1(0) whose value, by enlarging the variational subspace, canonly decrease.
mh
Inaddition, theresult persists whentaking theinfinite cutofflimit Λ withthetypical
→ ∞
trend of the coupling constant λ 1/lnΛ that is expected in the standard interpretation
∼
of “triviality” [5] within leading-order perturbation theory. Finally, in Sect.4, I will also
compare with other approaches and discuss the more general implications of the result.
1
2. Stability analysis of Φ4 theory
The preliminary starting point, for any variational calculation in the broken-symmetry
phase of a one-component Φ4 theory, is the basic Hamiltonian operator (λ > 0)
1 λ
Hˆ = : d3x Π2 +( Φ)2 +Ω2Φ2 + Φ4 : (1)
2 ∇ o 4!
Z (cid:20) (cid:21)
(cid:0) (cid:1)
where ( ω (Ω) = √k2 +Ω2)
k
d3k
Φ(x) = a expik x+a† exp ik x (2)
2ω (Ω )(2π)3 k · k − ·
Z k o (cid:16) (cid:17)
and p
d3k ω (Ω )
Π(x) = i k o a† exp ik x a expik x (3)
(2π)3 2 k − · − k ·
r
Z (cid:16) (cid:17)
In Eq.(1) normal orderingpis defined with respect to a reference state 0 which is the
| i
†
vacuum of the creation and annihilation operators (a 0 = 0 a = 0) with commutation
k| i h | k
relations[a ,a† ] = δ(3)(k k′). The standardstabilityanalysis fortheaboveHamiltonian
k k′ −
is performed in the class of the normalized gaussian ground states Ψ(0) Ψ(0)(Ω,ϕ)
| i ≡ | i
with [6, 7]
Ψ(0) Φ Ψ(0) = ϕ (4)
h | | i
and
Ψ(0) Φ(x)Φ(y) Ψ(0) = ϕ2 +G(x,y) (5)
h | | i
where
d3k
G(x,y) = expik (x y) (6)
2ω (Ω)(2π)3 · −
k
Z
is the equal-time propagator of the shifted fluctuation field
h(x) = Φ(x) ϕ (7)
−
with
d3k
h(x) = b expik x+b† exp ik x (8)
2ω (Ω)(2π)3 k · k − ·
Z k (cid:16) (cid:17)
Thus, the relation with thpe reference vacuum state is 0 Ψ(0)(Ω ,ϕ = 0) at which
o
| i ≡ | i
b a . Equivalently, one could switch to a functional formalism where the gaussian
k k
≡
ground states are described by the class of functionals [8]
1
Ψ(0)[Φ] = (Det G)−1/4exp d3x d3y(Φ(x) ϕ)G−1(x,y)(Φ(y) ϕ) (9)
−4 − −
Z Z
2
In this equivalent approach, the field operator Φ(x) acts on Ψ(0)[Φ] multiplicatively while
the momentum operator acts by functional differentiation
1 δ
Π(x)Ψ(0)[Φ] = Ψ(0)[Φ] (10)
i δΦ(x)
In the following, I shall maintain the standard second-quantized representation (1)-(8) for
its more intuitive character.
As shown in Ref.[8], the states Ψ(0)(Ω,ϕ) can be represented as coherent states built
| i
up with the original a and a† operators. In this sense, they represent forms of condensed
k k
vacua and the old operators are related to the new ”quasiparticle” b and b† operators
k k
(whose vacuum is Ψ(0)(Ω,ϕ) ) by a Bogolubov transformation that includes a shift of the
| i
zero-momentum mode.
The expectation value of the Hamiltonian in the class of the gaussian ground states
gives the gaussian energy density W (ϕ,Ω)
G
Ψ(0) Hˆ Ψ(0) = d3x (W (ϕ,Ω) W (0,Ω )) (11)
G G o
h | | i −
Z
where (I (Ω) = G(x,x), I (Ω) = 1G−1(x,x) )
o 1 8
1 λ 1 λ λ
W (ϕ,Ω) = I (Ω)+ m2 ϕ2 + ϕ4 + m2 + ϕ2 Ω2 + I (Ω) I (Ω) (12)
G 1 2 B 4! 2 B 2 − 4 o o
(cid:18) (cid:19)
and, just for simplicity of notation, the quantity
λ
m2 Ω2 I (Ω ) (13)
B ≡ o − 2 o o
has been introduced. It plays the role of a ‘bare mass’ for the quantum theory but his
origin depends on the normal ordering prescription adopted for the Hamiltonian Eq.(1).
Now, theexistence oftheΦ4 critical point[9]impliesthat, forsufficiently largeandneg-
ative values of m2 , the cutoff theory will exhibit spontaneous symmetry breaking. In this
B
regime, one can explore the conditions for non-trivial minima with ϕ = 0. Minimization
6
of W with respect to ϕ gives
G
∂W (ϕ,Ω) λ λ
G = ϕ m2 + ϕ2 + I (Ω) = 0 (14)
∂ϕ B 6 2 o
(cid:18) (cid:19)
while minimization with respect to Ω yields
λ λ
Ω2(ϕ) = m2 + ϕ2 + I (Ω) (15)
B 2 2 o
3
Finally, the replacement Ω = Ω(ϕ) in W (ϕ,Ω) provides the gaussian effective potential
G
(GEP)
V (ϕ) = W (ϕ,Ω(ϕ)) W (0,Ω ) (16)
G G G o
−
By combining Eqs.(14) and (15), non-trivial extrema ϕ = 0 can only occur at those values
6
ϕ = v where
±
λ
Ω2(v) = v2 m2 (17)
3 ≡ h
The standard identification of m with the energy-gap of the broken phase derives from
h
the following argument. At the absolute minima ϕ = v, the same Hamiltonian in Eq.(1)
±
becomes also normal ordered in the creation and annihilation operators b and b† [8],
p p
namely one finds
Hˆ = E +Hˆ +Hˆ (18)
0 2 int
Here
E = d3x V (v) < 0 (19)
o G
Z
is the gaussian ground-state energy. The quadratic operator
Hˆ = d3p ω (m ) b†b (20)
2 p h p p
Z
describes free-field quanta with energies ω (m ) = p2 +m2 and finally
p h h
p
λv λ
Hˆ = d3x : h3(x)+ h4(x) : (21)
int
3! 4!
Z (cid:18) (cid:19)
takes into account the residual self-interactions that have not been reabsorbed into the
vacuum structure and in the mass parameter m . In the above relation, normal ordering
h
of the b† and b operators is now defined with respect to one of the two equivalent absolute
p p
minima of the GEP for ϕ = v. In this way, by introducing the one-quasiparticle states
±
(see Eq.(6.4) of Ref.[7])
1,p = b† Ψ(0) 2ω (2π)3 (22)
p p
| i | i
q
one finds
1,p (Hˆ E ) 1,p
h | − 0 | i = p2 +m2 (23)
1,p 1,p h
h | i q
and it becomes natural to identify m with the energy-gap of the broken phase. In the
h
following section, I will check this expectation with a variational calculation.
4
3. Variational calculation of the energy gap in the broken phase
The variational procedure is of the same type considered by Di Leo and Darewych [10]
and by Siringo [11] when discussing the bound-state problem in the Higgs sector, namely
Ψ = A(q)b† Ψ(0) + d3k B(k,q)b† b† Ψ(0) (24)
| 1i q| i k+q −k| i
Z
with B(k,q) = B( k q,q).
− −
The two complex functions A(q) and B(p,q) have to be determined in order to solve
theeigenvalueproblemfortheHamiltonianHˆ Eq.(18)inthechosensubspace. Bydenoting
with E = E (q) the corresponding eigenvalue, one gets coupled equations (everywhere
1 1
ω = ω (m ))
p p h
δ Ψ (Hˆ E E ) Ψ
h 1| − o − 1 | 1i = A(q)(ω E )+f(q) = 0 (25)
δA∗(q) q − 1
and
δ Ψ (Hˆ E E ) Ψ
h 1| − o − 1 | 1i = 2B(k,q)[ω +ω E ]+g(k,q) = 0 (26)
δB∗(k,q) k k+q − 1
In Eqs.(25) and (26) f(q) and g(k,q) are defined as
λv B(k,q)
f(q) = d3k (27)
8π3/2√ωq √ωkωk+q
Z
and
λv A(q) λ B(p,q)
g(k,q) = + d3p (28)
8π3/2√ωq√ωkωk+q 32π3√ωkωk+q √ωpωp+q
Z
The two functions f(q) and g(k,q) contain the same integral up to numerical factors.
This allows to eliminate exactly B(k,q) in favour of A(q) as
B(k,q) = A(q) ωq ωq −E1 − 32mωq2h (29)
8v π3/2 ω ω ω +ω E 
k k+q k k+q 1
r −
 
after using the relation (17) m2 = λv2. By replacing in Eq.(25), one obtains
h 3
3m2 λ
A(q)(ω E )+A(q) ω E h J(q) = 0 (30)
q − 1 q − 1 − 2ω 16π2
(cid:18) q (cid:19)
where
1 d3p
J(q) = (31)
4π ω ω [ω +ω E (q)]
p p+q p p+q 1
Z −
5
Therefore, for A(q) = 0, one obtains the final relation for the eigenvalue
6
3m2
E (q) = ω 1 hF(q) (32)
1 q
− 2ω2
(cid:18) q (cid:19)
where
λ J(q)
F(q) = 16π2 (33)
1+ λ J(q)
16π2
Now, the integral in Eq.(31) diverges logarithmically
Λ p2dp 1 Λ
J ln (34)
∼ 2(p2 +m2)3/2 ∼ 2 m
Z0 h h
so that any conclusion on the energy spectrum depends on the possible behaviours of the
coupling constant λ when the ultraviolet cutoff Λ . A straightforward Λ limit
→ ∞ → ∞
for λ =fixed would yield F(q) 1 and a negative E (0). However, a more meaningful
1
→
continuum limit could beobtained, for instance, byinterpreting λas thevalue ofa running
coupling λ(µ) at some scale µ and then requiring λ(µ) 1/ln(Λ/µ) as suggested by the
∼
standard interpretation of “triviality” within leading-order perturbation theory.
For a self-consistent derivation of this trend within our Hamiltonian formalism, let us
return to equation (15) and use relation (13) to replace the bare mass. For simplicity, I
shall first consider the case Ω = 0, i.e.
o
λ
m2 = I (0) (35)
B −2 o
By using the identity of Ref.[7]
Ω2 Λ 1
I (Ω) I (0) = ln + (36)
o − o −8π2 Ω 2
(cid:18) (cid:19)
equation (15) for ϕ = v, where Ω is given in Eq.(17), reduces to the relation
±
λ Λ 1
1 = ln + (37)
8π2 m 2
(cid:18) h (cid:19)
One can give different interpretations to this equation. On the one hand, if Φ4 theory
were just considered a cutoff theory, it might simply express m in terms of the two
h
basic, fixed parameters λ and Λ. On the other hand, in a Renormalization Group (RG)
perspective, it could also be used to determine a suitable flow of the coupling constant
λ = λ(Λ), in the two-parameter (λ,Λ) space, that corresponds to the same value of m .
h
As anticipated, from this latter RG point of view and within leading-order perturbation
theory, the resulting trend λ 1/ln Λ would be similar to the Λ dependence of the
∼ mh −
6
”renormalized” coupling λ , usually identified with the value of a running coupling λ(µ)
R
at a typical finite scale µ m . However, in principle, λ might also be considered a ”bare”
h
∼
coupling λ , and thus identified with a running coupling λ(µ) at an asymptotic ultraviolet
B
scale µ Λ. As discussed in Ref.[12], this latter point of view cannot be ruled out. In fact,
∼
the trend λ 1/lnΛ represents a completely consistent solution that yields ”triviality”
B
∼
(i.e. λ = 0) to any finite order in perturbation theory by avoiding the problems posed by
R
the 1-loop, 3-loop, 5-loop,.. Landau poles and by the 2-loop, 4-loop,... spurious ultraviolet
fixed points at finite coupling that arise in the conventional interpretation. In the more
general context of the ǫ expansion, these two distinct points of view might also reflect
−
the existence of two separate Φ4 theories inhabiting in d = 4+ǫ and d = 4 ǫ space-time
−
dimensions [13].
In any case, regardless of these interpretative aspects, the consistency of the whole
calculation requires to adopt Eq.(37) to fix the (λ,Λ,m ) interdependence. In this way,
h
onecancontroltheultraviolet divergence inJ(q)andobtainafinitevalueforF(q). Notice
however that, independently of the given finite value of F(q), one gets
E (q) q2 +m2 (38)
1 ∼ h
q
at large q and
| |
E (0) < m (39)
1 h
consistently with J(0) and F(0) being positive-definite quantities for any E (0) < 2m .
1 h
The numerical estimate of the energy gap can be obtained from the relation
3 λ J(0)
E (0) = m 1 16π2 (40)
1 h − 2 1+ 16λπ2J(0)!
with
Λ p2dp
J(0) = (41)
(p2 +m2)[2 p2 +m2 E (0)]
Z0 h h − 1
Thus, by defining p = mhsinht and introducinpg ǫ1 E1(0)/mh, one obtains
≡
tmax sinh2t dt
J(0) = (42)
cosht[2cosht ǫ ]
Z0 − 1
or
t 1 π ǫ2 π
J(0) = max + 1 1 arcsin(ǫ /2)+ (43)
1
2 ǫ 2 − − 4 2
1 " r #
(cid:16) (cid:17)
where t = ln(2Λ/m ). In this way, in a double limit t and λ 0, such that
max h max
→ ∞ →
λt is finite, ǫ is definitely smaller than unity. With the trend in Eq.(37), one finds
max 1
7
λ J(0) = 1/4+ ( 1 ) or F(0) = 1/5+ ( 1 ) so that
16π2 O tmax O tmax
E (0) 1
1
= ǫ = 0.7 1+ ( ) (44)
1
m O t
h (cid:18) max (cid:19)
Inthesame approximation, where alsoF(q) F(0)represents anon-leading ( 1 )effect,
− O tmax
the form of the spectrum becomes very simple and one finds
3 m2
E (q) ω h (45)
1 q
∼ − 10 ω
q
I emphasize that the result in Eq.(44) is of variational nature. Therefore, by maintaining
the same relation Eq.(37) for the coupling constant, and by enlarging the variational sub-
spacefortheHamiltonianEq.(18)toincludehigher-ordercomponents b†b†b† , b†b†b†b† ,...,
| i | i
the ratio E1(0) can only decrease.
mh
Exactly the same procedure can be repeated in the more general case where the Ω
o
mass parameter of the symmetric phase is non vanishing. As one can check, by requiring
the broken phase to represent anyway the absolute minimum of the gaussian effective
potential, Eq.(37) can only be modified up to non-leading (λ) terms. As a consequence,
O
Eq.(44) is also modified up to non-leading ( 1 ) terms and the basic result remains
O tmax
unaffected.
Before concluding this section, I have to explain the considerable differences between
the conclusions of the present Letter and those of Ref.[11]. There, the analysis was per-
formed directly in the broken-symmetry phase without considering the overall stability of
the basic Φ4 Hamiltonian (1) in the class of the gaussian ground states. For this reason,
there was no obvious guiding principle to relate λ to the ultraviolet cutoff Λ and to m as
h
in Eq.(37). Thus, differently from the approach followed in the present Letter, one could
try to take the Λ limit at λ = fixed in such a way that λJ(0) and
→ ∞ → ∞
λ J(0)
F(0) = 16π2 1 (46)
1+ λ J(0) →
16π2
Inthisframework, itwasadoptedaparticularmassrenormalizationcondition(seeEqs.(20),
(21), (27) and (30) of Ref.[11])
δm2 = λv2F(0) (47)
−
in order to get, in the broken-symmetry phase, an exactly free massive spectrum up to
termsthatvanishintheΛ limit. Now, itwouldbeveryhardtounderstandthechoice
→ ∞
of such a vacuum-dependent mass counterterm in the context of the basic Hamiltonian
(1). Therefore, it should not come as a surprise that, by changing the renormalization
conditions, thesametypeofvariationalstructurecanleadtodifferent physical conclusions.
8
4. Summary and outlook
In Sects.2 and 3, I have illustrated an analytic, variational calculation of the energy spec-
trum for the broken-symmetry phase of the basic Φ4 Hamiltonian (1). In a continuum
limit where the ultraviolet cutoff Λ and the coupling constant λ 0, such that
→ ∞ →
λlnΛ is finite, the variationally determined spectrum E (p) approaches the free-field form
1
at large p , namely
| |
m2
E (p) p2 +m2 1+ ( h) (48)
1 → h O p2
q (cid:18) (cid:19)
However, in the same continuum limit, the energy-gap E (0) remains definitely smaller
1
than the m parameter that controls the asymptotic shape of the spectrum. With the
h
trend for the coupling coupling constant in Eq.(37), which is self-consistently determined
by the overall minimization of the effective potential, one finds
E (0) 1
1
= 0.7 1+ ( ) (49)
m O lnΛ
h (cid:18) (cid:19)
and the simple leading behaviour Eq.(45). The variational nature of the result implies
that, by enlarging the subspace to include higher-order b†b†b† , b†b†b†b† ,...contributions
| i | i
in the Fock space, the ratio E1(0) can only decrease.
mh
A possible objection might concern the simplest form Eq.(1) adopted for the Hamil-
tonian operator. Would the variational result persist by employing for the contact in-
teraction more sophisticated de-singularized operators as, for instance, the generalized
normal-ordering prescriptions of Ref.[14] ? There is no obvious answer to this question.
By replacing the Hamiltonian operator Hˆ of Eq.(1) with a new operator, say Hˆ′, one
should first repeat the whole stability analysis within the class of the gaussian ground
states and later check the consistency of Hˆ′ with the variational calculation in the b† and
| i
b†b† sectors. In our case, by using Eq.(1) (or equivalently the ‘bare mass’ in Eq.(13)), one
| i
obtains finite results at all stages, once Eq.(37) is used self-consistently to determine the
cutoff dependence of the coupling constant. For this reason, the operator ∆Hˆ = Hˆ′ Hˆ
−
should only introduce non-leading divergent terms in the calculations, at least if the trend
λ 1/lnΛ has to be maintained.
∼
Therefore, one is naturally driven to interpret the peculiar infrared behaviour of the
broken phase as a true physical effect due to the existence of a non-trivial vacuum conden-
sate associated with the typical scale m . When the momentum increases, the differences
h
with the trivial empty vacuum become unimportant and the energy spectrum approaches
a standard massive form with m2 λv2. However, when p 0, the presence of the
h ∼ →
9