Non-renormalization for the Liouville wave function

0.17 MB·English

by  J. Alexandre

Non-renormalization for the Liouville wave
function
8
0
J. Alexandre∗, Anna Kostouki†and N. E. Mavromatos‡
0
2 Department of Physics, King’s College London

n WC2R 2LS, UK
a
J February 2, 2008

6
1

]
h Abstract
t
- Usinganexactfunctionalmethod,withintheframework ofthegradi-
p
ent expansion for the Liouville effective action, we show that the kinetic
e
h term for the Liouville field is not renormalized.
[

1 1 Introduction
v
7
It is knownthatthe effective potentialfor the two-dimensionalLiouville theory
5
5 remains an exponential, with renormalizedcoupling constant and mass param-
2 eter [1]. This property respects the symmetry of the classical action, under
. which a translation in the Liouville field is equivalent to a change in the mass
1
0 parameter.
8 We study here the wave function renormalization Z of the Liouville field,
0 using an exact functional method, which leads to a self-consistent equation
:
v for the effective action (the proper graphs generator functional), in the spirit
i of a Schwinger-Dyson equation, and which is therefore not based on a loop
X
expansion. The idea is to look atthe evolutionof the quantum theory with the
r
a amplitude of the central charge deficit Q2 of the Liouville theory [2], since it
wasshownin[3]thatitispossibletoobtainexactflowsforthequantumtheory
with Q2. As we emphasize below, these flows are regularized by a fixed world
sheetcutoff, unlike the Wilsonianapproach. Using this method, it wasalready
found in [3], in the approximation where Z does not depend on the Liouville
field, that Z does not get quantum corrections and keeps its classical value.
We extend here this study to the more general situation where Z could be a
polynomialoftheLiouvillefield. Thisisthenextstepinordertohaveacomplete
picture,consistentwiththe gradientexpansion. As we shalldemonstratebelow
∗[email protected]
†[email protected]
‡[email protected]
1
the result is similar to that of [3]: the wave function renormalization remains
trivial, and the kinetic term for the Liouville field does not get dressed by
quantum fluctuations.
We note that the functional approach used here, which serves as an alter-
native to Wilsonian renormalization, has proven to give new insights into the
quantumstructure ofa theory,andledtonon-trivialresultsinavarietyofcon-
texts so far, including scalar field theory [4], Quantum Electro-Dynamics [5],
Wess-Zumino [6] and Kaluza-Klein [7] models, time-dependent bosonic strings
[8].
Thestructureofourarticleisthefollowing: Insection2,weexplaininsome
detailthefunctionalmethod,alreadyusedin[3],andderivetheexactequations
fortheevolutionofthepotentialandthewavefunctionrenormalizationwiththe
central charge deficit, Q2. The details of the derivations are given in Appendix
A.We emphasizethe specific rˆoleplayedby the two-dimensional fieldtheoryin
ensuringthewavefunctionnon-renormalization,andwegivethesolutionforthe
corresponding effective potential. In Section 3 we demonstrate the consistency
of our results with the Wilsonian approach, where we explain that this trivial
solution for Z is consistent with an exact renormalization equation. Finally, in
AppendixBweshowtheequivalencebetweentheWilsonianandtheone-particle
irreducible effective potentials.
2 Evolution equations
The bare action for the Liouville field, on a flat world sheet we assume in this
work, reads:
Q2
S = d2ξ ∂ φ∂aφ+µ2eφ , (1)
a
2
Z (cid:26) (cid:27)
wheretheamplitudeofthekinetictermiscontrolledbythecentralchargedeficit
Q2. Upon quantization of this theory, as we explain below [3], Q2 controls the
amplitude of quantum fluctuations:
for Q2 >> 1, the quadratic kinetic term dominates the bare Lagrangian
•
and therefore the quantum theory is almost classical;
when Q2 decreases,quantum fluctuations gradually appear in the system
•
and the full quantum theory is obtained when Q2 finite constant.
→
Ourmotivationistofindtheevolutionofthepropergraphsgeneratorfunctional
with Q2, and therefore obtain information on the quantum theory.
2.1 Path integral quantization
In order to define the corresponding quantum theory, one first defines the par-
tition function (assuming a Euclidean world sheet, as required for convergence
of the respective path integral)
Z[j]= [φ]exp S jφ =exp W[j] , (2)
D − − {− }
Z (cid:26) Z (cid:27)
2
where j is the source and W is the connected graphs generatorfunctional. The
classical field is defined as
δW
φ = , (3)
c
δj
and the proper graphs generator functional Γ, describing the quantum theory,
is obtained as the Legendre transform of W:
Γ[φ ]=W[j] d2ξ jφ , (4)
c c
−
Z
where the sourcej is to be understood asa functionalof φ , found by inverting
c
thedefinition(3). Oneobtainsthenafamilyofquantumtheories,parametrized
byQ2;itwasshownin[3]thattheeffectiveactionΓsatisfiesthefollowingexact
evolution equation with Q2 (we omit the subscript for the classical field)
c
1 1 ∂ ∂ δ2Γ −1
Γ˙ = d2ξ ∂ φ∂aφ+ Tr , (5)
2 a 2 ∂ξ ∂ζa δφ δφ
Z ( a (cid:18) ξ ζ(cid:19) )
wherethedotrepresentsaderivativewithrespecttoQ2. Theevolutionequation
(5) is exact and does not rely on any loop expansion: it is a self-consistent
equation, in the spirit of a differential Schwinger-Dyson equation. We stress
here that the trace appearing in eq.(5) is regularized with a fixed world sheet
cut off Λ, and the running parameter is Q2, unlike the Wilsonian approach
where, for a fixed Q2, one would study the evolution of Γ with a running world
sheet cut off.
In the framework of the gradient expansion, which we adopt in this work, we
consider the projection on a specific subspace of functionals in the functional
space where Γ lives, for which we assume the following form of the effective
action
Z (φ)
Γ= d2ξ Q ∂ φ∂aφ+V (φ) . (6)
a Q
2
Z (cid:26) (cid:27)
As we show in Appendix A, the evolution equations with Q2 for the potential
V (φ) and the wave function renormalizationZ (φ) are
Q Q
Λ2 V′′ ZΛ2
V˙ = ln 1+
8πZ − 8πZ2 V′′
(cid:18) (cid:19)
1 Z′ 2 ZΛ2 47
Z˙ = 1+ 5ln 1+
8πZ Z ! (cid:20) (cid:18) V′′ (cid:19)− 6 (cid:21)
′ ′′′
7 Z V
+ , (7)
24πZ Z V′′
! !
where Z =Z (φ) and V =V (φ), and a prime denotes derivative with respect
Q Q
to φ.
As can be seen from the evolution equations (7), a solution where Z does
′
not depend on the Liouville field (i.e. Z = 0) is consistent, for which case we
also obtain Z˙ =1, and therefore no renormalizationof the wave function.
3
One could seek for other solutions, different from Z = Q2, but we will
give below several arguments in favour of the uniqueness of the φ-independent
solution:
As discussed in section3, the solutionZ =Q2 is consistent with an exact
•
renormalization equation for the potential, using a sharp cut off.
Also in the Wilsonian context, the Liouville theory has been studied us-
•
ingtheaverageactionformalism[9],basedonasmoothcutoffprocedure,
therebyallowingthestudyoftheevolutionofthewavefunctionrenormal-
ization. Inthiswork,thewavefunctionrenormalizationZ (φ), wherek is
k
the running cut off, does depend on the Liouville field, as a consequence
of the initial condition of the flows, which is chosen so as to satisfy the
respective Weyl-Ward identities. The authors argue, though, that the IR
limitk 0oftheaverageaction,whichcorrespondstotheeffectiveaction
→
we consider here, is consistent with this non-renormalizationproperty.
WecanimagineintegratingtheequationforZ numerically,startingfrom
Q
•
the initial condition Z (φ) Q2 for Q2 >> 1, since the theory is then
Q
≃
almost classical. The step Q Q dQ corresponds to Q2 Q2+dx,
→ − →
with dx= 2QdQ+dQ2, and we have then
−
ZQ−dQ =ZQ+dx(Z˙Q)=ZQ+dx
′
because Z =0 for the initial condition. Therefore
ZQ−dQ =ZQ 2QdQ+dQ2 =(Q dQ)2,
− −
In this way we arrive, step by step, to the result that Z = Q2 for any
value of Q.
We show in the next subsection that, for a field-independent Z, this non-
•
renormalizationpropertyispossibleindimensiond=2only,whichgivesa
strongindicationthatthe solutionZ =Q2 isthe relevantoneinthe more
general case studied here. This is also consistent with the world-sheet
conformal-invariance restoring properties of the Liouville theory [2];
Finally, in the case of a curved world sheet, the bare action contains an
additional term, linear in the Liouville field, and reads
Q2
S = d2ξ√γ γ ∂aφ∂bφ+Q2R(2)φ+µ2eφ , (8)
ab
2
Z (cid:26) (cid:27)
where γ is the world sheet metric, with determinant γ and curvature scalar
ab
R(2). The gradient expansion for the effective action would then have to take
into account this linear term in φ, but because of the second functional deriva-
tive, which appears in the evolution equation (5), this additional linear term
does not play a rˆole in the generation of quantum fluctuations. It is in this
sense that working, from the beginning with flat world sheets, suffices for our
purposes.
4
2.2 Specificity of two dimensions (d = 2)
Inthissubsectionwegothroughthe samestepsasthosedescribedinAppendix
A, for a wave function renormalizationindependent of φ, but in any dimension
d. We showthen thatthe renormalizationofthe wavefunctionrenormalization
vanishes only for the case d=2.
We assume that the effective action has the form
Z
Γ= ddξ Q∂ φ∂aφ+V (φ) , (9)
a Q
2
Z (cid:26) (cid:27)
such that its second functional derivative in configuration space reads:
δ2Γ = Z∂ ∂a+V′′ δ(2)(ξ ζ). (10)
a
δφ δφ − −
ξ ζ
(cid:16) (cid:17)
For the configuration φ = φ + 2ρcos(kξ), where φ ,ρ,k are constants, the
0 0
second functional derivative in Fourier space is:
δ2Γ = Zp2+V′′ +ρ2V(4) (2π)2δ(2)(p+q)
δφ δφ
p q
(cid:16) (cid:17)
+ρV(3)(2π)2 δ(2)(p+q+k)+δ(2)(p+q k)
−
ρ2 (cid:16) (cid:17)
+ V(4) δ(2)(p+q+2k)+δ(2)(p+q 2k)
2 −
+higher o(cid:16)rders in ρ. (cid:17) (11)
The inverse of this second functional derivative is calculated using
(A+B)−1 =A−1 A−1BA−1+A−1BA−1BA−1+ (12)
− ···
where A stands for the diagonal contribution and B for the off-diagonal one,
proportional to ρ. The relevant term for the evolution of Z is
Tr p2A−1BA−1BA−1
= ρ(cid:8)2 V(3) 2 ddp (cid:9)p2 1 + 1
A (2π)df2(p) f(p+k) f(p k)
(cid:16) (cid:17) Z (cid:18) − (cid:19)
2
= 2 ρ2 V(3) I(k), (13)
A
(cid:16) (cid:17)
where is the two-dimensional volume, f(p)=Zp2+V′′ and
A
ddp p2
I(k)= (14)
(2π)df2(p)f(p+k)
Z
The evolution of the wavefunction renormalization, Z, is proportional to the
quadratic-order-in-k part of I(k), and we have
ddp 4Z2p2(k p)2 Zk2p2
I(k) = I(0)+ · + (k4) (15)
(2π)d (Zp2+V′′)5 − (Zp2+V′′)4 O
Z (cid:26) (cid:27)
5
ddp 2p4 p2
= I(0)+k2Z−d/2 + (k4)
(2π)d (p2+V′′)5 − (p2+V′′)4 O
Z (cid:26) (cid:27)
πd/2 Z−d/2 Γ(3 d/2) Γ(5 d/2)
= I(0)+k2 2 − 2 −
(2π)d[V′′]3−d/2 Γ(3) − Γ(5)
(cid:26)
Γ(4 d/2) dΓ(3 d/2)
2d − − + (k4)
− Γ(5) −2 Γ(4) O
(cid:27)
Using the property Γ(n+1) = nΓ(n), together with Γ(1) = 1, the expansion
(15) can be written
πd/2 Z−d/2 d d
I(k)=I(0)+k2 Γ(3 d/2) 1 + (k4), (16)
(2π)d[V′′]3−d/2 − 24 2 − O
(cid:18) (cid:19)
whichshowsthatthe termofquadraticorderink vanishes for d=2only. This
is a strong indication that the solution Z˙ = 1 found previously is the relevant
one.
2.3 Solution for the potential
Fromnowon,weconsiderZ =Q2. Theevolutionequation(7)forthepotential
becomes then
V′′ Q2Λ2
V˙ = ln 1+ , (17)
−8πQ4 V′′
(cid:18) (cid:19)
where the quadratic divergence was disregarded, as it is field-independent.
The equation (17) has been studied in [3] for the specific regimes Q2 0
→
and Q2 . We give here some details of the derivation for finite values of
→ ∞
Q2. We therefore assume that
Q2Λ2
>>1. (18)
V′′
With this condition in mind, eq.(17) is then satisfiedby a potential of the form
V (φ)=Λ2v exp(ε φ), (19)
Q Q Q
where v and ε are dimensionless functions of Q (for the condition (18) to
Q Q
be satisfied we need v <<1). Indeed, plugging this ansatz into the evolution
Q
equation (17) gives, in the limit (18),
vε2 Q2
v˙ = ln
−8πQ4 vε2
(cid:18) (cid:19)
ε3
ε˙ = . (20)
8πQ4
The latter evolution equation for ε can be integrated exactly. The appropriate
boundary condition is ε 1 when Q2 , since the system is then classical.
→ →∞
6
The integration over Q2 leads to
4πQ2
ε = . (21)
Q s1+4πQ2
We remind the reader that the solution (21) is exact in the framework of the
gradient expansion (6), and is not based on a loop expansion. The evolution
equation for v is not solvable exactly, and we thus leave the study of the
Q
potential amplitude for the next section, where this is achieved by means of a
Wilsonian exact renormalizationapproach.
Before closing this section, we note that, for the specific conformal charge
deficit Q2 = 8, corresponding to c = 1 conformal field theories, there are two
cosmologicalconstantoperators,dressingtheidentity(µ2+µ2φ)exp(√2φ)[10],
1 2
where µ ,µ are constants. Our solution above cannot include the operator
1 2
proportional to µ2 [3], since we consider a continous set of values for Q2 and
2
this operator exists only for a discrete isolated value. To incorporate this case,
one shouldstudy the flowwith respectto another parameterin the bare theory
with fixed Q2 =8, such as α′ or µ .
i
3 Consistency with the Wilsonian picture
We now exhibit the Wilsonian properties of the solution (19), using the exact
renormalization method of [11]. We consider an initial two-dimensional bare
theory, with running cut-off Λ. The effective theory defined at the scale Λ
−
δΛ is derived by integrating the ultraviolet degrees of freedom from Λ to Λ
−
δΛ. The idea of exact renormalization methods is to perform this integration
infinitesimally, i.e. take the limit δΛ/Λ 0, which leads to an exact evolution
→
equation for S . The procedure was detailed in [11], and here we reproduce
Λ
only the main steps for clarity and completeness. Note that we consider here a
sharpcut-off,whichispossibleonlyifweconsiderthe evolutionofthepotential
part of the Wilsonian action, as explained now.
We consider a Euclidean two-dimensional spacetime, and we assume that,
for each value of the energy scale Λ, the Euclidean action S has the form
Λ
Z (φ)
S = d2ξ Λ ∂ φ∂aφ+V (φ) . (22)
Λ a Λ
2
Z (cid:26) (cid:27)
The integration of the ultraviolet degrees of freedom is implemented in the fol-
lowing way. We write the dynamical fields φ = φ +ψ, where the φ is the
IR IR
infraredfield with non-vanishing Fourier components for p Λ δΛ, andψ is
| |≤ −
the degree of freedom to be integrated out, with non-vanishing Fourier compo-
nents for Λ δΛ < p Λ only. An infinitesimal step of the renormalization
− | | ≤
group transformation reads:
exp( SΛ−δΛ[φIR]+SΛ[φIR]) (23)
−
= exp(S [φ ]) [ψ]exp( S [φ +ψ])
Λ IR Λ IR
D −
Z
7
δS [φ ] 1 δ2S [φ ]
Λ IR Λ IR
= [ψ]exp ψ(p) ψ(p)ψ(q) ,
D − δψ(p) − 2 δψ(p)δψ(q)
Z (cid:18) ZΛ ZΛZΛ (cid:19)
+higher orders in δΛ,
where represents the integration over Fourier modes for Λ δΛ < p Λ.
Λ − | | ≤
Higher-order terms in the expansion of the action are indeed of higher order in
R
δΛ, since each integral involves a new factor of δΛ. The only relevant terms
are of first and second order in δΛ [11], which are at most quadratic in the
dynamical variable ψ, and therefore lead to a Gaussian integral. We then have
SΛ[φIR] SΛ−δΛ[φIR]
−
δΛ
Tr δS [φ ] δ2S [φ ] −1 δS [φ ]
Λ Λ IR Λ IR Λ IR
=
δΛ δψ(p) δψ(p)δψ(q) δψ(q)
( (cid:18) (cid:19) )
Tr δ2S [φ ]
Λ Λ IR
ln + (δΛ), (24)
−2δΛ δψ(p)δψ(q) O
(cid:26) (cid:18) (cid:19)(cid:27)
where the trace Tr is to be taken in the shell of thickness δΛ, and is therefore
Λ
proportional to δΛ.
We are interested in the evolution equationfor the potential only, for which
it is sufficient to consider a constant infrared configuration φ = φ , and this
IR 0
is the reason why a sharp cut-off can be used: the singular terms that could
arisefromtheθ function,characterizingthesharpcut-off,arenotpresent,since
the derivatives of the infrared field vanish. In this situation, the first term on
the right-hand side of eq.(24), which is a tree-level term, does not contribute:
δS /δψ(p) is proportional to δ2(p), and thus has no overlap with the domain
Λ
of integration p =Λ. We are therefore left with the second term, which arises
| |
from quantum fluctuations, and the limit δΛ 0 gives, with the ansatz (22),
→
Λ Z (φ )Λ2+V′′(φ )
∂ V (φ ) ∂ V (0)= ln Λ 0 Λ 0 , (25)
Λ Λ 0 − Λ Λ −4π ZΛ(0)Λ2+VΛ′′(0) !
Eq.(25) provides a resummation of all the loop orders, since it consists in a
self-consistentequation. Asaresult,theevolutionequation(25)isexactwithin
the framework of the ansatz (22), and is independent of a loop expansion.
Inordertomakethe connectionwiththe solution(19),we nowconsiderthe
following ansatz
Z (φ ) = Q2 (26)
Λ 0
V (φ ) = Λ2v exp(εφ ),
Λ 0 Λ 0
where ε is the constant (21) and v depends on the running cut off only. One
Λ
should keep in mind here that Q2 is now constant, whereas the cut off Λ is
running. WhenpluggedintheWegner-Houghtonequation(25),theansatz(26)
leads to
Λ Q2+ε2vexp(εφ )
2Λv+Λ2∂ v (exp(εφ ) 1)= ln 0 . (27)
Λ 0 − −4π Q2+ε2v
(cid:18) (cid:19)
(cid:0) (cid:1)
8
One can see that this equation is consistent in the limit v <<1 only, which we
are interested in: keeping the first orders in v, the φ -dependence cancels out
0
and the remaining equation is
Λ
2Λv+Λ2∂ v = ε2v, (28)
Λ −4πQ2
which is easily integrated to
µ 2+ε2/(4πQ2)
v = . (29)
Λ
Λ
(cid:16) (cid:17)
This solution indeed satisfies v << 1, since we are interested in the regime of
largecutoff,inthespiritofthecondition(18). Takingintoaccountthesolution
(21), the potential is finally
µ 1/(1+4πQ2) 4πQ2
V (φ)=µ2 exp φ , (30)
Λ Λ s1+4πQ2!
(cid:16) (cid:17)
We stress again that this solution is not the result of a loop expansion. An
importantremarkisinorderhere: itwaspossibletofindthesolution(30)ofthe
Wilsonianexactrenormalizationgroupequation,becauseZ doesnotdependon
φ . Indeed, it is the only possibility for the φ -dependence to cancel in eq.(27),
0 0
at the first order in v. This shows the consistency of the choice Z = Q2 made
in the previous section.
Note that the solution (30) does not need to satisfy the evolution equation
(17), since the Wilsonian potential defined in this section is not obtained by
meansoftheLegendretransformasthepotentialdefinedintheprevioussection.
The equivalence between these potentials is obtained in the limit where the
runningcutoffgoesto0(see Appendix B),butinthis case,theexpression(30)
is not valid since it was derived in the limit of large cut off.
Finally, the limit of the Wilsonian potential (30) when Q2 , for fixed
→ ∞
cut off (in the spirit of section 2), gives the expected bare Liouville potential
shown in eq.(1).
4 Discussion
In this work we have analysed Liouville field theory on the world-sheet from
the perspective of a novel functional method, suggested in [3]. In particular,
we have demonstrated that the function Z(φ) appearing in the Liouville ki-
netic term, is not renormalized, that is it preserves its classical form Z = Q2
in the full quantum theory. As we have discussed, this is a specific feature of
the two-dimensional field theory, and is not the case in general, e.g. in four
dimensions [4]. In fact, this feature is also essential for maintaining the confor-
malpropertiesofthe Liouville field, in particularits rˆolein restoringconformal
symmetry [2].
9
It shouldbe stressedthat in the presentworkwe haveassumeda functional
dependenceoftheeffectivetheorybasedonthegradientexpansion,namelythat
Z(φ)isonlyapolynomial functionoftheLiouvillefieldφandnot itsworld-sheet
derivatives ∂φ. This assumption is dictated by the above-mentioned argument
ofconformalcovarianceoftheLiouvilleaction,whichwewishtomaintaininthe
full quantum theory [2]. It is remarked that in case one included such higher
derivative terms, the allowed structures in the effective action should involve
terms of the form
∂ ∂b n
Z (φ)∂ φ b ∂aφ (31)
n a µ2
(cid:18) (cid:19)
where Z (φ) are dimensionless polynomialsof φ andn is aninteger. It remains
n
tobeseenwhetherinsuchcasestheabove-mentionednonrenormalizationresult
is valid. However, we expect such terms not to be present, as their presence
wouldappearinconflictwiththestandardconformalpropertiesoftheLiouville
field. In this sense we think that the analysis in the present article is complete.
As a final remark we note that for a curved world sheet, with a curvature
scalar R(2), replacing µ2 in (31), one could in principle have structures of the
form
∂ ∂b n
Y (φ)∂ φ b ∂aφ (32)
n a R(2)
(cid:18) (cid:19)
where Y (φ) aredimensionless polynomials ofφ. However,suchstructures can-
n
not appear for n = 0, as they should vanish in the limit of flat world sheet
6
R(2) 0, and since Y does not depend on the curvature scalar, it cannot
n
→
vanish in this limit in order to leave the terms (32) finite.
Before closing we note that the above analysis can be extended to incorpo-
rate Liouville-dressed non-critical stringy σ-models, involving the coupling of
Xµ fields with the Liouville mode φ. In such a case there are more compli-
cated potential terms, since for each non conformal vertex operator V(X) of
thenon-criticalstring,thereisaconformal-symmetryrestoringfactoreαφ,with
α the appropriate Liouville dimension [2], multiplying V(X), d2σeαφV(X).
Nevertheless, the application of the exact method for the Liouville sector and
R
the associate Q2 flows applies to this case, with similar results, as far as the
Liouville wavefunction non renormalizationis concerned.
Acknowledgements
J.A.wouldliketothankJanosPolonyiforusefuldiscussionsrelatedtotheflow
equations, and the explanation given in Appendix B. We wish to thank the or-
ganisersofthe 1st Annual School Of EU Network “UniverseNet, The Origin Of
The Universe: Seeking Links Between Fundamental Physics And Cosmology,
Mytilene (Island of Lesvos, Greece), September 24-29 2007, for the hospital-
ity and for giving the opportunity to A.K. to present preliminary results of
this work. The work of of A.K. and N.E.M is partially supported by the Eu-
ropean Union through the FP6 Marie Curie Research and Training Network
UniverseNet (MRTN-CT-2006-035863).
10