Fluctuation pressure of a fluid membrane between walls through six loops

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by  Boris Kastening

Fluctuation pressure of a fluid membrane between walls through six loops
Boris Kastening
Institut fu¨r Materialwissenschaft, Technische Universit¨at Darmstadt, Petersenstraße 23, D-64287 Darmstadt, Germany
and
Institut fu¨r Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, D-14195 Berlin, Germany
6
email: [email protected]
0
(Dated: August 2005)
0
2 The fluctuation pressure that an infinitely extended fluid membrane exerts on two enclosing

n parallel hard walls is computed. Variational perturbation theory is used to extract the hard-wall
a limit from a perturbative expansion through six loops obtained with a smooth wall potential. Our
J result α = 0.0821 0.0005 for the constant conventionally parametrizing the pressure lies above
±
3 earlier Monte Carlo results.
1
PACSnumbers: 05.40.-a,46.70.Hg,87.16.Dg,05.10.-a

]
h
c I. INTRODUCTION limit A . By scaling analysis, the fluctuation pres-
e sure of t→he∞membrane has the form [1]
m
Membranesarefrequentstructuresinchemicalandbi-
t- ological systems. Their dynamic behavior at finite tem- p=α(kBT)2, (2)
a perature is of great interest, since their dominant repul- κ(d/2)3
t
s sive force is given by thermal out-of-plane fluctuations
and we are interested in the numerical value of α. Esti-
.
at [1, 2]. If the temperature is sufficiently high, the details mates of α range from crude theoretical estimates α
ofthe potentialthatinhibits theirmutualpenetrationor ≈
m 0.0242byHelfrich[1]andα 0.0625byJankeandKlein-
that causes them to be confined to a certain geometrical ≈
ert[3] (this reference also containsan early Monte Carlo
- region are unimportant. Then the membranes’ thermal
d result α=0.060 0.003) through Monte Carlo results
n fluctuationsmaybe describedbyatwo-dimensionalfield ±
o theory with a hard-wall potential that describes their α=0.079 0.002 (3)
c mutual interactions and the boundary conditions of the ±
[ space accessible to them. by Janke, Kleinert, and Meinhart [4] and

2 In an important class of membranes, their constituent
α=0.0798 0.0003 (4)
v molecules are able to move freely within them. The ±
4 thermalfluctuationsofthese“fluid”membranesarecon-
by Gompper and Kroll [5], and a theoretical estimate
1 trolledbytheirbendingrigidityκ. Thecurvatureenergy
α 0.0771 by Kleinert [6] based on the analogy with
6
of such a membrane is, in the harmonic approximation, ≈
8 a quantum mechanical particle in a box to a theoretical
described by
0 estimateα 0.0797byBachmann,Kleinert,andPelster
≈
5 κ [7] using variational perturbation theory. Recently, we
0 E = d2x[∂2ϕ(x)]2, (1) haveextendedthefour-loopcalculationin[7]tofiveloops
2
t/ ZA [8] and found a value α 0.0820, outside the error bars
a ≈
wherethe subscriptreferstoaplanewithanareaAthat of the Monte Carlo results. We were, however, unable
m
servestoparametrizethemembranes’surface,andwhere to quote an error bar for our own result. In this work,
- ϕ(x) describes the location of the membrane orthogonal we extend our computation through six loops. Together
d
n to the point x on this plane. For the harmonic approx- with improved resummation methods, this allows us to
o imation to be valid, the membrane must not fluctuate confirm the disagreement with the Monte Carlo results
c too wildly and thus the temperature must also not be and put stringent error bars around our result.
:
v too high. It is difficult to describe the membranes’ fluc- Ourworkisstructuredasfollows. InSec.II,webriefly
i tuations outside the range of validity of the harmonic remind the reader how the hard walls may be modeled
X
approximation, since then, e.g., overhangs with respect usingananalyticpotentialandhowaperturbativeseries
r toanygivenplaneandstericself-interactionsofthemem- forαmaybederived. InSec.III,thecentralresultsofthe
a
brane are possible. technically similarquantummechanics (QM)problemof
There have been various theoretical approaches to a particle in a box are listed since they are instrumen-
computethepressureofasinglemembranebetweenwalls tal for extracting α for the membrane problem from its
[1, 3–8] or of a stack of membranes [1, 3–5, 9]. Here we perturbativeexpansioninSec.V. InSec.IV,wedirectly
consider a fluid membrane between two rigid walls and resumthe perturbative seriesfor α. InSec. V, we adjust
ask what pressure its classical statistical bending fluctu- the potential modeling the boundary conditions for the
ations exert on the walls. The plane parametrizing the membrane problem so that the perturbative series for α
membrane is taken to be midway between the enclosing of the QM problem is obtained. Appropriate resumma-
walls, which are a distance d apart, and we consider the tion schemes let us then infer the distance of the walls
2
describedbytheresultingpotential,andthisinformation with the expansion parameter
is trivially translatedinto a value of αfor the membrane
1
problem. In Sec. VI, we summarize and briefly discuss
g = . (9)
our results. m2d2
The perturbative coefficients a are functions of the ǫ .
l 2k
Combining (2), (6), and (8), we obtain a finite-g version
II. MODELING OF THE HARD WALLS
α(g)ofαsuchthatanexpansionofα(g)throughLloops
has the form
Consider a tensionless membrane between two large
flat parallel walls of area A separated by a distance d, 1 L
α(g) a gl, (10)
whosecurvatureenergyisgivenby(1). Thed-dependent ≈ 4g2 l
part fd of the free energy density of the system at tem- Xl=0
perature T is given by the functional integral
of which we need to extract the limit
exp(cid:18)−kABfTd(cid:19)= x Z−+d/d2/2dϕ(x)exp(cid:18)−kBET(cid:19). (5) α=gl→im∞α(g). (11)
Y
In Secs. IV and V, we consider several resummation
The pressure is then obtained as schemesforextractingthevalueofαfromalimitednum-
ber of coefficients a .
∂f l
d
p= (6)
− ∂d
III. QM PARTICLE IN A BOX
and has the form (2) [1, 3]. Our goal is to find the nu-
merical value of the constant α.
Followinganideaintroducedin[6]andutilizedalsoin A one-dimensional problem similar to the two-dimen-
[7, 8], we implement the restriction d/2 < ϕ < d/2 by sional case above is finding the ground state energy of a
adding a potential term m4d2 d2xV−(ϕ/d) to E, where QM particle in a one-dimensional box [6, 10] (which, in
V is an even function that is analytic inside a circle turn, is equivalent to finding the classicalpartition func-
with radius 1/2 and has sufficRiently strong singularities tion of a string with tension between one-dimensional
at 1/2. We then expand the potential V in a Taylor walls [6, 10, 11]). Introduction of a potential to model
seri±es in ϕ and drop the restriction on ϕ. At the end of thehardwallsleadstoaquantityα(g)parametrizingthe
the calculation, we let m 0 to recover the hard-wall groundstateenergyofaparticlemovinginthispotential
limit. → (see [6, 8, 10] and the appendix for details; our notation
follows [8]). This quantity has a loop expansion of the
Since the functional form of p in terms of κ, d, and T
form (10) and due to the trivial topologies of the Feyn-
is known and since we differentiate f only with respect
d
man diagrams through two loops, the coefficients a , a ,
to d, we set k T = κ = 1 in the sequel. The energy 0 1
B
and a are identical to those of the membrane case.
functional may then be written as 2
For the particular potential
1 1
E = d2x [∂2ϕ(x)]2+ m4ϕ(x)2 1
2 2 V (z)= , (12)
Z (cid:26) c 2π2cos2(πz)
∞
+m4ǫ d2+m4 ǫ d2(1−k)ϕ(x)2k ,(7)
0 2k the exact ground state energy is known (see, e.g., [10])
Xk=2 (cid:27) and translates into
wherethe ǫ aretheexpansioncoefficientsofthepoten-
2k π2 16 1 4 π4g2
tial V. α(g)= + + 1+ , (13)
128 π4g2 2 π2g 64
The above procedure defines a finite-m version f (m) r !
d
of the free energy density f of (5), such that f =
d d
giving the limit
limm→0fd(m). fd(m) may be expanded in a perturba-
tive series in terms of vacuum diagrams—i.e., Feynman
π2
diagrams without external legs [6–8]. The technical de- α= =0.07710628438... (14)
tailsofthisprocedurearedescribedin[8],anddeviations 128
from the treatment in [8] are delegated to the appendix.
for g . The QM result (13) will be utilized by the
Theresultisthatanexpansionoff (m)throughLloops → ∞
d resummationschemesofSec.Vtoextractαforthemem-
has the form
brane problem.
In the sequel, we will always contrast the membrane
L
1
f (m) a gl−2, (8) results with those for the QM problem for the same re-
d ≈ d2 l summation scheme.
l=0
X
3
TABLE I: Expansion coefficients for the poten- TABLE III: α from VPT as applied in Sec. IV for both the
tial Vc and for a potential Vmb that gives the QM and the membrane problem. αmb for L = 2,3,4 and
QM coefficients al also for the membrane problem. L = 5 were already obtained in [7] and [8], respectively.
Vc Vmb L αqm αmb
ǫ0 1/(2π2)= 0.0506606 same 2 0.0385531 0.0385531
ǫ2 1/2= 0.5 same 3 0.0719411 0.0737974
ǫ4 π2/3= 3.28987 same 4 0.0758821 0.0794726
ǫ6 17π4/90= 18.3995 18.0284 5 0.0767518 0.0813538
ǫ8 31π6/315= 94.6129 89.5702 6 0.0769910 0.0820175
ǫ10 691π8/14175= 462.545 419.568
ǫ12 10922π10/467775= 2186.57 1890.91
0.086
TABLE II: Perturbative expansion coefficients for both
0.084
the QM and the membrane problem for the potential Vc.
0.082
QM membrane
a0 1/2π2 = 0.0506606 same 0.08
a1 1/8= 0.1250000 same
a2 π2/64= 0.1542126 same 0.078
a3 π4/1024= 0.0951261 0.105998 0.076
a4 0 0.026569
a5 π8/262144= 0.0361959 0.034229 0.074
− − −
a6 0 −0.083246(13) 0.072
1 2 3 4 5 6
IV. α FROM DIRECT RESUMMATION OF α(g) FIG. 1: αqm (lower lines) and αmb (upper lines) as
a function of the number of loops L. The horizon-
tal line is the exact QM result. The dotted-dashed,
Knowing only a few low-order coefficients a , we are short-dashed, solid, long-dashed, and dotted lines repre-
l
looking for the g limit of the series (10). This sent data from Tables III, V, VII, VIII, and IX, respec-
limit corresponds p→hy∞sically to removing the regulator tively. Further explanations are given in the main text.
that suppresses fluctuations in the infrared. In the con- At the right, the Monte Carlo results (3) and (4) (boxes)
as well as our final result (27) (diamond) are displayed.
textofcriticalphenomena,suchserieshavebeensuccess-
fully resummedusing Kleinert’svariationalperturbation
theory (VPT; see [12–14] and Chaps. 5 and 19 of the
textbooks [15] and [16], respectively; improving pertur- towards its high-order behavior. However, for the VPT
bationtheorybyavariationalprinciplegoesbackatleast resummation scheme to work well and give trustworthy
to[17]). Accuratecriticalexponents [13,14,16]andam- results,itisimportantthatthetruncatedseriestobere-
plitude ratios [18] have been obtained using VPT. summed resembles already the behavior at high orders.
Inthissection,wepresenttheresultsofapplyingVPT Consequently,thedependenceonthevariationalparame-
directlytotheseries(10)asdescribedin[7]andinSec.IV terinVPT,whenapplieddirectlytotheseries(10),does
of [8]. We refer the reader to [8] for the details and just not develop increasingly flatter plateaus through the or-
mention that we only present the results of the q = 1 ders considered. Such plateaus are, however,an internal
versionofVPT,sincetheresultsforself-consistentdeter- consistency check of the method, and we therefore de-
mination of q from the series (10) remain too imprecise velop other resummation variants for obtaining α in the
even at the six-loop level. sectionsbelow. Nevertheless,we provideinTable III the
For the potential (12), which also plays an important extension to six loops of Eq. (24) in [8] for the potential
role in the resummation variants considered in the sec- Vc and for comparison also list the corresponding QM
tions below, the ǫ are listed in Table I, and the corre- results, taken from Table I in [8]. The results are also
2k
sponding perturbative coefficients a are listed for both plotted in Fig. 1 (dotted-dashed lines), and in spite of
l
the QM and the membrane problem in Table II. The the above critical remarks they agree perfectly well with
membrane’s coefficients start deviating fromthe QMco- the results of the more refined resummation variants to
efficients at the three-loop level. At the beginning, the be discussed below.
deviation from the particular feature of the QM series In [19], an attempt was made to extractα from the a
l
that even loop orders beyond two loops have zero coeffi- throughsixloops usingso-calledfactorandrootapprox-
cients is small. This givesthe membrane’sseriesa struc- imants. However, the achieved accuracy was not high
ture that can be expected to be in a transitional phase enough for any decision about a discrepancy with the
4
TABLE IV: Expansion coefficients of (16) TABLE V: Results for α for both the QM and
for the quantities 1/√Vc and 1/√Vmb. the membrane problem using the simple resum-
mation scheme from the beginning of Sec. V.
QM membrane
v0 √2π= 4.44288 same L αqm αmb
v2 π3/√2= 21.9247 same 1 0.0625000000 same
v4 π−5/12√2=−18.0324 same 2 0.0792468245 same
v6 π7/360√2= 5.93242 10.3394 3 0.0770188844 0.0845718
− − 4 0.0771087134 0.0817113
v8 π9/20160√2= 1.04555 −18.7293 5 0.0771062388 0.0816335
vv1102 π−13π/121/319851040480000√√22== −00..01018456752787 −252..426409070 6 0.0771062850 0.0818696(2)
TABLE VI: Expansion coefficients of (18) for
Monte Carlo results (3) and (4). both the QM and the membrane problem.
QM membrane
u1 2√2/π= 0.900316 same
V. α FROM ZERO OF POTENTIAL
u2 10/3= 3.33333 same
− −
u3 128√2π/45= 12.6375 13.1791
Instead of resumming (10) directly, we apply here the u4 104π2/21= 48.878 54.6843
− − −
strategy of Sec. VI of [8]. That is, as a first step we fix u5 6904√2π3/1575= 192.214 235.065
the ǫ2k in (7) order by order such that the expansion of u6 −81784π4/10395=−766.379 −1037.10
α(g) for the membrane problem V is identical to that
mb
oftheQMcasewithapotentialV . Theresultingǫ are
c 2k
listed in Table I. The second step is then to ask where
the resulting potential V (z) has the singularities z least in QM, the singularities of this quantity nearest to
mb 0
closesttotheoriginontherealaxis. Thescalingrelat±ion the origin are simple poles. The resulting series
f 1/d2 when m2 = 0 allows us then to recover α for
∞
∝
the membrane case through F(z) V(z) V(0)= f z2l (17)
2l
≡ −
α =4z2α , (15) p p Xl=1
mb 0 qm
may then be inverted to
with α from (14). Since the nearest singularities of
qm
∞
V are of quadratic type, we may assume that the re-
c z2 = u Fl. (18)
sulting membrane potential V has approximately such l
mb
a behavior. We may therefore assume that 1/√Vmb has Xl=1
approximatelinearbehavioratitsfirstzero—i.e.,at z .
0 Thefirstfewcoefficientsu forboththe QMpotentialV
± l c
The simplest investigation of z0 is to truncate the ex- and the resulting membrane potential Vmb are listed in
pansion of 1/√V at L loops, Table VI. We are interested in finding
L z2 = lim z2(F). (19)
1/ V(z) v z2l, (16) 0 F→∞
2l
≈
p Xl=0 Motivated by the successes of such an ansatz in crit-
ical phenomena, we assume that the function F can be
and subsequently numerically determine the first zero of
expanded around its first singularities z as
theright-handsideof(16). Thev2l fortheQMandmem- ± 0
branecasesarelistedinTableIV. Theresultingvaluesof ∞
αmaybe found inTableV. While the correctQMvalue F = u¯ (z2 z2)−q/2+k, (20)
k − 0
(14)isapproachedexponentiallyfast,theconvergencein
k=0
X
the membrane case is also remarkable. The results are
plotted as short-dashed lines in Fig. 1. where q =2 for QM. Inversion of (20) gives
We interpret the fact that the last two differences
∞
among the membrane values are comparable as a signal z2 = u′ F−2m/q, (21)
m
that the maximum achievable accuracy with the current
m=0
method has been reached. A more refined approach is X
then needed to take into account a likely more compli- with z2 = u′. We may either set q = 2 as in QM in the
0 0
cated analytic structure of the potential V . Let us hope that the deviation for the membrane case is small,
mb
therefore employ VPT to improve the naive resumma- ordetermine q self-consistently. We usebothapproaches
tion above. Consider the quantity V(z) V(0). At below.
−
p p
5
TABLE VII: Results for α when the al are fixed to TABLE VIII: Results for q and α when the al
be those of the QM problem and q = 2 is assumed. are fixed to be those of the QM problem and
q is determined from its own resummed series.
L αqm αmb
2 0.0750000 0.0750000 L qqm αqm qmb αmb
3 0.0766754 0.0791616 3 2.09487 0.0786643 2.26290 0.0856888
4 0.0769828 0.0806435 4 2.05356 0.0777648 2.21951 0.0846057
5 0.0770794 0.0812768 5 2.02049 0.0772965 2.11817 0.0829441
6 0.0770973 0.0815743 6 2.00822 0.0771659 2.08666(1) 0.0825299(2)
Now apply VPT [15, 16]. In a truncated expansion
TABLE IX: Results for q and α when the
L al are fixed to be those of the QM problem
z2 u Fl, (22) and q is determined from optimized plateaus.
l
≈
Xl=1 L qqm αqm qmb αmb
3 2.09730 0.0787132 2.28225 0.0861317
we replace
4 2.04990 0.0777101 2.21532 0.0845332
2/q 2/q −lq/2 5 2.02405 0.0773337 2.13061 0.0830987
F F 6 2.00948 0.0771767 2.09303(1) 0.0825984(2)
Fl (tF)l +t 1
→ ((cid:18)Fˆ(cid:19) " −(cid:18)Fˆ(cid:19) #)
2/q −lq/2
Fˆ
= (tFˆ)l 1+t 1 , (23)
  F! −  first dlnz2/dlnF in VPT [13, 16], since it has the same
 
q as z2(F) and since
 
reexpand the resulting expression in tthrough tL, set
t = 1, and then optimize the resulting expression in Fˆ,
dlnz2
whereoptimizingreferstofindingappropriatestationary lim =0 (26)
or turning points according to the principle of minimal F→∞ dlnF
sensitivity [20]. That is, we replace
bythe assumptionofasingularityofthe potential. That
L−l lq/2 Fˆ 2/q k is, we resum the expansion of dlnz2/dlnF as detailed
Fl Fˆl − 1 (24) above and tune q such that optimization with respect to
→ k=0(cid:18) k (cid:19) F! −  Fˆ leads to (26). Through two loops, the expansion of
X
  dlnz2/dlnF is q-independent, and we start with L=3.
and optimize the resulting expression in Fˆ. In the limit It turns out that through the order we are working, we
F of interest to us, this amounts to must use turning points for even L and maxima for odd
→∞ Lwhendeterminingq. Forsubsequentlydeterminingz2,
L L−l lq/2 the situation is reverse—namely,as above for q =2. 0
z02 ≈optFˆ"l=1ulFˆlk=0(cid:18)−k (cid:19)(−1)k#, (25) The results for q and α through six loops are listed in
X X Table VIII. The results for α are plotted as long-dashed
which is the L-loop approximation to z2—i.e., using the lines in Fig. 1. Note how q approaches 2 rapidly for the
0
expansion coefficients through u . It turns out that, QM problem and that also for the membrane problem a
L
through the order we are working, there is exactly one value around 2 appears to be approached.
extremum for even L and exactly one turning point and
An alternative to using (26) for the determination of
no extremum for odd L. This makes the choice of the
q is to tune q such that the plateaus at which the result
optimization unique at each order. The value of α is in dependsleastonvariationsofFˆ areoptimized[21]. This
each case obtained through (15).
strategy has been successfully applied in [21, 22] in the
Theresultsforq =2aresummarizedinTableVII and
context of critical phenomena. In practice, this means
plottedassolidlinesinFig.1. ThecorrectQMvalue(14) finding Fˆ and q such that first and second derivatives of
isapproachedexponentiallyfast. Theconvergenceinthe the right-hand side of (25) with respect to Fˆ vanish (for
membranecaseisalsoremarkable. Thoughthevaluesare
turning points) or such that first and third derivatives
slightly lower than those reported in Tables III and V,
with respect to Fˆ vanish (for extrema).
they clearly point towards a value of α above the results
(3) and (4). The results for q and α through six loops are listed
If we refrain from making assumptions about q for in Table IX and are very similar to those of Table VIII.
z2(F), we can determine it self-consistently by treating They are plotted as dotted lines in Fig. 1.
6
VI. SUMMARY AND DISCUSSION if present, should be very small. The best qmb value can
thus be expected to lie at most slightly below 2, leading
In summary, we have computed the constant α toonlyasmalldecreaseofthevaluesofTableVIIandthe
parametrizing the pressure law (2) of an infinitely ex- upper solid curve in Fig. 1 and to a slightly lower mean
tended fluid membrane between two parallel hard walls. value and larger error bar than given in (27). It is re-
The hard wall was replaced by a smooth potential, al- assuring that the results of both the direct resummation
lowing for a perturbative loop expansion for α. Several of α(g) employed in Sec. IV and the naive resummation
resummationschemeswereusedto extractthe hard-wall from the beginning of Sec. V lie very close to the mean
limit from expansion coefficients through six loops with value of (27).
results listed in Tables III, V, VII, VIII, and IX and All things considered, we are rather confident about
plotted in Fig. 1. our result (27), but further studies of the system are re-
The values from Table VII on the one hand and Ta- quired to settle the question of the correct value of α.
bles VIII and IX on the other hand approacheach other
with increasing numbers of loops from below and above,
respectively. Aconservativeprocedureforcombiningour APPENDIX: FEYNMAN DIAGRAMS
resultsforαistoaveragethe lowestandhighestsix-loop
values for α obtained above (i.e., the values for α from
In [8], we have described at length how recursionrela-
TablesVIIandIX,respectively)andtaketheirdifference
tions along the lines of [23, 24] can be used to construct
to be the full error bar. This provides our final result
the vacuum diagrams needed for the computation of the
perturbative coefficients a in (10) through a given loop
α=0.0821 0.0005, (27) l
± order,andthereis noneedto repeatthe derivationhere.
displayedin Fig.1. It lies abovethe Monte Carloresults GiventheFeynmandiagrams,avertexwith2klinesrep-
(3) and(4), alsodisplayedin Fig.1. The simplestexpla- resents a factor
nation we have to offer for this discrepancy is that their
error bars, in particular that of (4), may have been cho- m4d2(1−k)ǫ , (28)
2k
−
sen too optimistic. Also, finite-size or other systematic
effects may not have been taken into account properly. andtheperturbativecoefficientsa areobtainedfromthe
l
On the other hand, it is possible that the treatment sum of l-loop diagrams as
in our work is inflicted by systematic errors. Experience
with VPT tells that the internal consistency checks, es- a = c g I , (29)
l l-n l-n l-n
pecially the development of increasingly flatter plateaus −
n
X
inthe optimizationprocedurewith higherorders,arere-
liableindicatorsofVPTtowork. Thesecheckshavesuc- wherecl-n isacombinatorialfactor,gl-n isamonomialin
cessfully been implemented for the procedures of Sec. V. the ǫ2k, and Il-n is the corresponding momentum space
Nevertheless,itcannotberuledoutthatthemethodused integral. The integration measure is dDk/(2π)D with
here is not flexible enough to adequately take into ac- D =1 for QM and D =2 for the membrane. The mem-
R
count the unknown true analytical structure of α(g). brane propagator carrying a momentum k is given by
Another concern is the value of q . While the treat- 1/(k4 +m4), while the QM propagator carrying a mo-
mb
ments leading to the α values listed in Tables VIII and mentum k is given by 1/(k2+m2).
IXandthusleadingtotheupperdottedandlong-dashed Fromthe listof diagramsthroughfive loopsin [8]it is
curvesinFig.1 determine q self-consistently,the QM- obviousthat a major reasonfor the rapidincreaseof the
mb
inspired value q = 2 was somewhat arbitrarily chosen number of Feynman diagrams with the number of loops
mb
toobtainthevaluesforαinTableVII. Whatifthevalue isaproliferationofthemomentum-independentone-loop
of q describing α(g) best differs from 2? A slightly propagatorinsertionsoftheform . Asimplemeasure
mb
larger value (but below the ones from the self-consistent to reduce the number of diagrams jto be considered is a
determinations of q ) leads to slightly increased values one-loop resummation where we abqsorb the above inser-
mb
ofαandthereforealsotoalargermeanvalueandsmaller tion into the parameter m in the propagator and at the
error bar in (27). On the other hand, a slightly smaller end reexpand the resulting modified perturbative series
valueleadstoslightlydecreasedvaluesofαandtherefore inpowersofg aswascarriedoutin[25]andtreatedona
alsoto asmallermeanvalueandlargererrorbarin(27). more formal level in [24]. Note that many other resum-
In both cases, the q = 2 curve may still approach the mations in the form of momentum-independent propa-
mb
correct result. For an optimal value of q below 2 this gator and vertex corrections are possible, leading to a
mb
mayhappenbydecreasingαvaluesathigherorders(that further reduction of the number of Feynman diagrams.
suchabehaviorispossibleinprincipleiseasilytestedby However, at the current level of computing vacuum dia-
setting q to a value above2.085,whichcausesthe cor- gramsthroughsixloops,thisisunnecessary. Inanycase,
mb
respondingsix-loopvalueforαtodropbelowthatatfive the diagrams most difficult to evaluate are always those
loops). The smooth behavior and slow flattening of the with the full loop topology and remain after any such
q = 2 curve lets us believe, though, that such a drop, resummation.
mb
7
To implement the one-loop resummation in the mem-
TABLE X:Numbersof diagrams for low loop orders.
brane case, we must compute diagrams with a modified
propagator H such that numberof loops l 0 1 2 3 4 5 6 7
diagrams 1 1 1 3 7 24 83 376
G−1 =H−1 12L(4) H , (30) diags. after one-loop resum. 1 1 1 2 3 11 29 125
12 12 − 1234 34
diags. with l-loop topology 1 1 0 1 1 5 8 37
where the notation of [8] has been employed. Writing
G−1 = k4 + m4 and H−1 = k4 + M4, this condition
translates into
k4+m4 =k4+M4 3m4d−2ǫ M−2 (31) 1. Discardallvacuumdiagramswithoneormoreone-
− 2 4 looppropagatorinsertions ,withthe exception
or of the two-loopdiagram,whijchchanges sign(here,
M2 3 M2 3 the combinatorics do not woqrk out; loosely speak-
m2 − m2 − 2ǫ4g =0, (32) ing,itisundefinedwhichpartofthediagramisthe
(cid:18) (cid:19) (cid:18) (cid:19) insertion).
where we have used (9) and the one-loop result
2. Compute all remaining diagrams with the replace-
d2k 1 1 ment m M in the propagators.
= . (33) →
(2π)2k4+m4 8m2
Z 3. ReplaceM2 Z(g)m2 forthemembranecaseand
Define Z(g) by M Z(g)m→for the QM case and reexpand the
→
M2 =Z(g)m2. (34) perturbative series in powers of g.
Although (32) can be solved analytically for Z(g), it is InTableX,wegivetheoriginalnumbersofdiagramsat
more useful for our purposes to write somelowlooporders,thenumbersleftafterourone-loop
resummation,andthe numbers ofdiagramswith the full
∞
respective loop topology. The latter is necessarily the
Z(g)=1+ c gk (35)
k same both before and after the one-loop resummation.
k=1
X In Table XI, we list all diagrams through six loops left
and extract after the one-loop resummation. Also given are their
c = 3ǫ (36) combinatorialfactorscl-n,theircouplingconstantfactors
1 4 4 g ,andthevaluesI ofthecorrespondingintegralsfor
l-n l-n
and the recursion relation M =1forboththeQMandthemembraneproblem(the
k−1 k−2k−i multiplyingpowerofM canimmediatelybeinferredfrom
3 1
ck = cick−i ck−i−jcicj, k >1. (37) thenumberofloopsandpropagatorsofagivendiagram).
−2 −2 The techniques for evaluating the integrals are ex-
i=1 i=1j=1
X XX
plained in [8]. With the exception ofI , the membrane
6-5
Throughsixloopsinthevacuumdiagrams,weneedZ(g)
integralshavebeenevaluatedtotheprecisiongiveneither
through g5 and obtain
inmomentumspaceorinbothmomentumandconfigura-
Z(g) = 1+ 34ǫ4g− 3227ǫ24g2+ 2176ǫ34g3− 28054085ǫ44g4 toibotnaisnpeadcei.nFcoornIfi6g-5u,rathtieonindspicaactee.dSpinrecceistihoenscloiguhldtloynlloywbeer
+ 729ǫ5g5+ (g6). (38)
64 4 O precisionofI6-5 introducesthemaincomputationalerror
The same resummation (30) can be implemented for intothe determinationofαatthe six-looplevel,wehave
the QM case. Writing G−1 = k2+m2 and H−1 = k2+ indicated the ensuing numericalerrorin the other tables
M2, the condition (30) translates now into of this work, where applicable.
k2+m2 =k2+M2 6m2d−2ǫ4M−1 (39T)ABLEXI:Diagramsl-n(nthl-loopdiagram)throughsixloops,theircombinato-
− rialfactorscl-n,couplingconstantfactorsgl-n,andvaluesIl-nofthecorresponding
or integralsforM =1. D=1andD=2correspondtotheQMandmembraneprob-
lems,respectively.
3
M M 3
ǫ4g =0, (40) l−n diagram cl-n gl-n IlD-n=1 IlD-n=2
m − m − 2
(cid:18) (cid:19) (cid:18) (cid:19) 0-1 1 −ǫ0 1 1
wherewehaveusedg =4/md2[8]andtheone-loopresult q
1 1
1-1 1 −1 −
+∞ dk 1 1 (cid:15)(cid:12) 2 4
= . (41)
Z−∞ 2πk2+m2 2m 2-1 (cid:14)(cid:13) −3 −ǫ4 1 1
(cid:15)(cid:15)(cid:12)(cid:12) 4 64
ThistimewedefineZ(g) M/m. ThenZ(g)isthesame q
as in the membrane case≡considered above. (cid:14)(cid:14)(cid:13)(cid:13)
The resulting modifications in the Feynman rules are 3-1 (cid:23)(cid:20) 12 ǫ24 312 4.04576×10−4
(cid:7) (cid:4)
for both the membrane and the QM case: q q
(cid:6) (cid:5)
(cid:22)(cid:21)
8
l−n diagram cl-n gl-n IlD-n=1 IlD-n=2 l−n diagram cl-n gl-n IlD-n=1 IlD-n=2
1 1
3-2 q 15 −ǫ6 8 512 6-1 #q q q 1245416 −ǫ54 13130572 3.74650×10−8
4-1 (cid:23)qTT(cid:20)(cid:20)q(cid:20) 288 −ǫ34 5132 1.63237×10−5 6-2 (cid:15)(cid:15)"(cid:12)(cid:15)q (cid:15)(cid:12)q (cid:15)!(cid:12)q (cid:12)248832 −ǫ54 29479112 3.12644×10−8
(cid:22)q (cid:21) q q q
4-2 (cid:23)(cid:20) 360 ǫ4ǫ6 1 5.05719×10−5 (cid:14)(cid:14)(cid:13)(cid:14)(cid:13)(cid:14)q (cid:13)
(cid:7) (cid:15)(cid:4) (cid:12) 64 (cid:14) (cid:13)
q q
(cid:6) (cid:5) (cid:15)(cid:12)
4-3 (cid:22)(cid:21)(cid:14)(cid:13) 105 −ǫ8 116 40196 6-3 (cid:7)@q@(cid:14)q (cid:0)(cid:0)(cid:13)q(cid:4)q 165888 −ǫ54 1173966748 4.52023×10−8
q
(cid:22)q (cid:21)
5-1 (cid:23)q q(cid:20) 2592 ǫ44 40596 7.55133×10−7 6-4 (cid:23)qq(cid:0)@(cid:0)@q qq(cid:20)497664 −ǫ54 1172966948 2.85447×10−8
(cid:22)(cid:21)
5-2 (cid:15)(cid:22)q(cid:15)q (cid:12)qq(cid:21)(cid:12) 2304 ǫ44 1212988 1.04187×10−6 6-5 #q q q 3315776 −ǫ54 245576 2.37861(5)×10−8
(cid:14)(cid:13)
q q
(cid:15)(cid:12) "!
(cid:14)(cid:14)q (cid:13)q (cid:13) 6-6 (cid:23)(cid:7)q(cid:4)(cid:23)(cid:20)(cid:7)q(cid:4)(cid:20)27648 −ǫ54 6525536 6.39380×10−8
q
5-3 (cid:15)(cid:15)(cid:12)(cid:15)(cid:15)(cid:12)q (cid:12)10368 ǫ44 61744 6.71540×10−7 (cid:22)(cid:6)q(cid:0)(cid:5)(cid:22)(cid:21)@(cid:6)q(cid:5)(cid:21)
(cid:14)q (cid:13)(cid:14)q (cid:14)q (cid:13) 6-7 #@(cid:0) q @ 414720 ǫ34ǫ6 14764556 6.27924×10−8
(cid:14) (cid:13) q @ q
5-4 (cid:31)@@(cid:0)(cid:0)(cid:28)5760 −ǫ24ǫ6 30772 1.50770×10−6 "q !
q (cid:28)(cid:31)q
(cid:30)q (cid:29) 6-8 #q 207360 ǫ34ǫ6 122588 5.50218×10−8
q
(cid:15)(cid:12)
(cid:20)T q q
5-5 (cid:23)(cid:20)(cid:14)qT(cid:13)(cid:20) 12960 −ǫ24ǫ6 10324 2.04047×10−6 "!
(cid:22)q q(cid:21) 6-9 (cid:15)(cid:14)(cid:15)q (cid:13)(cid:12)q (cid:12) 138240 ǫ34ǫ6 2415976 1.30234×10−7
(cid:15)(cid:12)
5-6 (cid:23)(cid:7)(cid:6)q q (cid:20)(cid:5)(cid:4)q 4320 −ǫ24ǫ6 10524 3.95093×10−6 6-10 (cid:23)(cid:0)@(cid:0)@(cid:14)(cid:14)q@(cid:0)q@(cid:0)(cid:20)(cid:15)(cid:13)q (cid:13)(cid:12)155520 ǫ34ǫ6 81592 9.43917×10−8
(cid:22)(cid:21) q q
5-7 '(cid:23)(cid:7)(cid:6)q (cid:20)(cid:4)(cid:5)$q 360 ǫ26 1912 3.76084×10−6 6-11 (cid:15)(cid:22)(cid:12)(cid:15)(cid:15)q (cid:15)(cid:12)(cid:21)(cid:14)(cid:15)(cid:13)(cid:12)q (cid:12)622080 ǫ34ǫ6 122788 8.39425×10−8
q q
(cid:22) (cid:21)
(cid:14)(cid:14)(cid:13)(cid:14)(cid:13)(cid:14)q (cid:13)
&%
5-8 (cid:15)(cid:12)(cid:23)(cid:20)(cid:15)(cid:12)2700 ǫ26 1218 6.32149×10−6 (cid:14) (cid:13)
(cid:7) (cid:4)
(cid:14)(cid:13)(cid:22)(cid:6)q (cid:21)(cid:5)(cid:14)q (cid:13) 6-12 (cid:23)qTT(cid:20)q(cid:20)q(cid:20)155520 ǫ34ǫ6 10124 1.70039×10−7
5-9 (cid:15)(cid:12) 2025 ǫ26 614 655136 (cid:22)q (cid:21)
(cid:14)q (cid:13)q 6-13 (cid:23)(cid:7) (cid:23)(cid:20)(cid:4)(cid:7)q(cid:4)(cid:20)34560 ǫ34ǫ6 81592 1.02301×10−7
q q
(cid:6) (cid:5)
5-10 (cid:23)(cid:20) 5040 ǫ4ǫ8 1 6.32149×10−6 (cid:22)(cid:22)(cid:21)(cid:6)q(cid:5)(cid:21)
(cid:7)(cid:6)q (cid:5)(cid:4)q 128 6-14 (cid:23)(cid:7)q(cid:4)(cid:20)(cid:15)(cid:12) 51840 ǫ34ǫ6 40596 2.46933×10−7
(cid:22)(cid:21) q q
(cid:14)(cid:13)
1 1 (cid:22)(cid:6)q(cid:5)(cid:21)
5-11 945 −ǫ10 @
q 32 32768 6-15 (cid:31)q @(cid:0)(cid:28)(cid:15)(cid:12)161280 −ǫ24ǫ8 61744 1.88463×10−7
(cid:0)
(cid:30)(cid:31)q
(cid:14)(cid:13)
(cid:30)q (cid:29)
9
l−n diagram cl-n gl-n IlD-n=1 IlD-n=2
6-24 (cid:15)(cid:15)(cid:12)(cid:12) 24300 −ǫ4ǫ26 2516 10481576
(cid:20)T q q q
6-16 (cid:23)(cid:20) qT(cid:20)181440 −ǫ24ǫ8 20348 2.55058×10−7 (cid:14)(cid:14)(cid:13)(cid:13)
(cid:22)q q(cid:21) 6-25 (cid:23)(cid:20) 75600 ǫ4ǫ10 2516 7.90187×10−7
6-17 (cid:23)(cid:20)(cid:23)(cid:20)20160 −ǫ24ǫ8 10124 1.63681×10−7 (cid:7)(cid:6)q (cid:5)(cid:4)q
(cid:7)q (cid:7)(cid:4)q (cid:4)q (cid:22)(cid:21)
(cid:6) (cid:5)(cid:6) (cid:5)
(cid:22)(cid:21)(cid:22)(cid:21) 6-26 '(cid:23) (cid:20)$20160 ǫ6ǫ8 1 4.70105×10−7
(cid:15)(cid:12) 384
(cid:7) (cid:4)
6-18 (cid:23)(cid:7) q (cid:20)(cid:4) 40320 −ǫ24ǫ8 20548 4.93867×10−7 (cid:22)(cid:6)q (cid:14)(cid:21)(cid:5)q (cid:13)
(cid:6)q (cid:5)q &%
(cid:22)(cid:21) 6-27 (cid:23)(cid:20) 75600 ǫ6ǫ8 1 7.90187×10−7
6-19 '(cid:23)(cid:7)q (cid:7)(cid:4)q (cid:4)(cid:20)$q 64800 −ǫ4ǫ26 11152 1.31735×10−7 (cid:14)(cid:15)(cid:22)(cid:7)(cid:13)(cid:12)(cid:6)q (cid:21)(cid:5)(cid:4)q 256
(cid:6) (cid:5)(cid:6) (cid:5)
1 1
(cid:22) (cid:21) 6-28 37800 ǫ6ǫ8
&% (cid:15)(cid:12) 128 524288
q q
6-20 (cid:31)@ (cid:0)(cid:15)(cid:28)(cid:12)172800 −ǫ4ǫ26 61744 1.88463×10−7 (cid:14)(cid:13)
@(cid:0)
q (cid:31)(cid:28)q
1 1
(cid:14)(cid:13) 6-29 10395 −ǫ12
(cid:30)q (cid:29) 64 262144
q
6-21 (cid:15)(cid:23)qT(cid:12)T(cid:20)(cid:20)(cid:15)q(cid:20)(cid:12)194400 −ǫ4ǫ26 20348 2.55058×10−7
(cid:14)(cid:13)(cid:14)(cid:13)
(cid:22)q (cid:21)
6-22 (cid:23)(cid:20)(cid:15)(cid:12) 32400 −ǫ4ǫ26 5112 3.95093×10−7
(cid:7) (cid:4)
q q q
(cid:6) (cid:5)
(cid:14)(cid:13)
(cid:22)(cid:21)
6-23 (cid:23)q (cid:20)(cid:15)(cid:12) 129600 −ǫ4ǫ26 20548 4.93867×10−7
(cid:7) (cid:4)
q q
(cid:6) (cid:5)
(cid:14)(cid:13)
(cid:22)(cid:21)
[1] W. Helfrich, Z. Naturforsch. 33A, 305 (1978); [11] P.-G. de Gennes, J. Chem. Phys. 48, 2257 (1968);
W.Helfrich and R.M. Servuss,NuovoCimento D3, 137 R.Lipowsky,Z.Phys.BCondens.Matter97,193(1995);
(1984). C.HiergeistandR.Lipowsky,PhysicaA244,164(1997).
[2] I.BivasandA.G.Petrov,J.Theor.Biol. 88,459(1981); [12] H. Kleinert, Phys. Lett. A 207, 133 (1995) [quant-
D.SornetteandN.Ostrowsky,J.Phys.(France)45,265 ph/9507005].
(1984). [13] H. Kleinert, Phys. Rev. D 57, 2264 (1998); 58, 107702
[3] W.JankeandH.Kleinert,Phys.Lett.A117,353(1986). (1998) [cond-mat/9803268].
[4] W. Janke, H. Kleinert, and M. Meinhart, Phys. Lett. B [14] H. Kleinert, Phys. Rev. D 60, 085001 (1999) [hep-
217, 525 (1989). th/9812197]; Phys. Lett. A 277, 205 (2000) [cond-
[5] G. Gompper and D.M. Kroll, Europhys. Lett. 9, 59 mat/9906107].
(1989). [15] H. Kleinert, Path Integrals in Quantum Mechanics, Sta-
[6] H. Kleinert, Phys. Lett. A 257, 269 (1999) [cond- tistics, Polymer Physics, and Financial Markets, 3rd ed.
mat/9811308]. (World Scientific, Singapore, 1995), Chap. 5.
[7] M. Bachmann, H. Kleinert, and A. Pelster, Phys. Lett. [16] H. Kleinert and V. Schulte-Frohlinde, Critical Proper-
A 261, 127 (1999) [cond-mat/9905397]. ties of φ4-Theories, 1st ed. (World Scientific, Singapore,
[8] B. Kastening, Phys. Rev. E 66, 061102 (2002) [cond- 2001), Chap. 19.
mat/0205073]. [17] V.I. Yukalov,Moscow Univ.Phys. Bull. 31, 10 (1976).
[9] W. Janke and H. Kleinert, Phys. Rev. Lett. 58, 144 [18] H. Kleinert and B. Van den Bossche, Phys. Rev. E 63,
(1987); 056113 (2001) [cond-mat/0011329].
F. David,J. Phys. (France) 51, C7115 (1990); [19] V.I. Yukalov and S. Gluzman, Int. J. Mod. Phys. B 18,
R.R. Netz and R. Lipowsky, Europhys. Lett. 29, 345 3027 (2004) [cond-mat/0412046].
(1995); [20] P.M. Stevenson,Phys. Rev.D 23, 2916 (1981).
M.Bachmann,H.Kleinert,andA.Pelster,Phys.Rev.E [21] B.HamprechtandH.Kleinert,Phys.Rev.D68,065001
63, 51709 (2001) [cond-mat/0011281]. (2003) [hep-th/0302116].
[10] H. Kleinert, A. Chervyakov, and B. Hamprecht, Phys. [22] B. Kastening, Phys. Rev. A 70, 043621 (2004) [cond-
Lett.A 260, 182 (1999) [cond-mat/9906241]. mat/0406035].
10
[23] H.Kleinert,A.Pelster,B.Kastening,andM.Bachmann, [25] B. Kastening, Phys. Rev. D 54, 3965 (1996) [hep-
Phys.Rev.E 62, 1537 (2000) [hep-th/9907168]. ph/9604311]; 57, 3567 (1998) [hep-ph/9710346].
[24] B. Kastening, Phys. Rev. E 61, 3501 (2000) [hep-
th/9908172].