Van der Waals Forces and Photon-Less Effective Field Theory

Abstract

In the ultra-cold regime Van der Waals forces between neutral atoms can be represented by short range effective interactions.
We show that universal low energy scaling features of the underlying vdW long range force stemming from two photon exchange
impose restrictions on an Effective Field Theory without explicit photons. The role of naively redundant operators, relevant
to the definition of three body forces, is also analyzed.

Full text

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arXiv:1012.2284v1 [cond-mat.other] 10 Dec 2010
Few-Body Systems manuscript No.
(will be inserted by the editor)
Van der Waals forces and Photon-less Effective Field Theories
E. Ruiz Arriola
Presented at 21th European Conference On Few-Body Problems In Physics: EFB21
29 Aug - 3 Sep 2010, Salamanca (Spain)
Abstract In the ultra-cold regime Van der Waals forces between neutral atoms can be rep-
resented by short range effective interactions. We show that universal low energy scaling
features of the underlying vdW long range force stemming from two photon exchange im-
pose restrictions on an Effective Field Theory without explicit photons. The role of naively
redundant operators, relevant to the definition of three body forces, is also analyzed.
Keywords Van der Waals forces · Effective Field Theory · Ultracold collisions
PACS 34.10.+x · 34.50.Cx · 33.15.-e · 03.75.Nt
1 Introduction
From a fundamental QED point of view the underlying mechanism responsible for van
der Waals (vdW) forces corresponds to two photon-exchange (see e.g. [1] and references
therein). As compared to the short range and exponentially suppressed chemical bonding
forces on sizes about a few Bohr radii, vdW forces are long range. For interatomic separa-
tions aB≪ r ≪ ¯ hc/∆E, two photons are exchanged in a short time ∼ 2r/c while transitions
with excitation energy ∆E take a much larger time ∼ 2¯ h/∆E, yielding the potential
V(r) = −C6
where Cnare the dispersion coefficients which are accurately known for many diatomic
systems (see e.g. a compilation in [2]). The vdW length R = (MC6/¯ h2)
size of the forces. For such potentials, low energy scattering with kR ≪ 1 is dominated by
S-waves which phase-shift, δ0(k) , fulfills the effective range expansion (ERE) [3]
kcotδ0(k) = −1
where α0is the scattering length, and r0is the effective range. Note that for this potential
the long-range character stars at O(k4) due to the logarithmic piece.
r6−C8
r8−C10
r10−...
(1)
1
4 characterizes the
α0+1
2r0k2+v2k4log(k2R2)+...
(2)
E. Ruiz Arriola
Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain.
E-mail: earriola@ugr.es

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2 Low energy Scaling of vdW forces
Remarkably, the effective range was computed analytically [4,5] when Cn≥8= 0 yielding
r0
R=16Γ (5/4)2
3π
33π
−4
R
α0+4Γ (3/4)2
R2
α2
0
= 1.395−1.333R
α0
+0.6373R2
α2
0
(3)
The scaling of the effective range r0in the vdW length and the quadratic 1/α0behaviour is
just a particular case of a more general result [6] (see also [7] in these proceedings). In fact,
the dominance of the leading long distance C6term tacitly assumed in Refs. [4,5] was to be
expected a priori by suitably re-writing higher order C8, C10contributions in vdW units
R2MV(r) = −(R/r)6?1+g1(R/r)2+g2(R/r)4+...?
where g1∼10−2and g2∼ 10−4for many homonuclear diatomic systems. Thus one expects
that even for kR ∼ 1 higher order corrections are negligible despite the strong divergence at
short distance. These expectations are indeed met a posteriori on the light of about a hun-
dred calculations based on phenomenological potentials [6]. This result not only favours the
view that these rather simple approaches based on the leading vdW forces are phenomeno-
logically sound but also shows that a huge reduction of parameters takes place suggesting
that atoms in the ultra-cold regime can indeed be handled without much explicit reference to
the underlying electronic structure of atoms. The scaling universal relation, Eq. (3), allows
for a quite general discussion on effective interactions in vdW units, as we advance here.
(4)
3 Effective Short Distance Potentials
In the ultra-cold regime, i.e. for extremely long de Broglie wavelengths much larger than the
vdW scale, λ =1/k ≫R , one expects the long range character to become largely irrelevant,
keeping the first two terms in Eq. (2). Thus, one might want to represent the vdW potential
by an effective potential with a finite range, rc, featuring the truncated ERE, Eq. (2), and
dismissing any explicit reference to the underlying photon exchange. However, even at very
low energies, causality arguments provide the shortest possible value for rc, which for vdW
forces yields rc> 0.6R [2]. Using for illustration a square well (SW) potential with range rc
and depth V0, Veff(r) = −V0θ(rc−r), one obtains
α0= rc−tan√MV0rc
√MV0
,
r0= rc
?
1−
1
α0rcMV0−
r2
c
3α2
0
?
.
(5)
Reproducing Eq. (3) is not possible for a common potential. Indeed, the sign of the 1/α2
term is just opposite, so that for small α0we cannot represent the interaction by this short
range potential. On the contrary, for large scattering lengths α0≫ R we obtain rc= 1.395R
and V0= π2/(4r2
C2= −1
one gets MCSW
potential Veff(r) = −V0rcδ(r−rc) we get for α0≫ R the results MCDS
MCDS
directly on the constants C0and C2. Note that while a C4exists for these short distance
potentials, the original vdW potential yields a divergence, in harmony with the observation
that the ERE for short range potentials differs at O(k4) from the vdW expression, Eq. (3).
0
cM). In terms of volume integrals,
C0=
?
d3xVeff(x),
6
?
d3xr2Veff(x)
(6)
0/R = −14.41 and MCSW
2/R3= 2.80. If we use instead a delta-shell (DS)
0/R = −13.15 and
2/R3= 2.40, not far from the SW estimate. This suggests using a formulation based

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4 Effective Field Theory
The EFT approach has often been invoked to highlight universal features of ultra-cold few
atoms systems (for reviews see e.g. [8,9]). We re-analyze it on the light of the universal and
extremely successful scaling relation, Eq. (3). For definiteness, we consider the Galilean
invariant Lagrangian density [10] expanded in composite Bosonic spinless field operators
with increasing energy dimensions and including multi-particle interactions,
L = ψ†
?
i∂t+∇2
2m
?
ψ −C0
2(ψ†ψ)2−C2
2[∇(ψ†ψ)]2−D0
6(ψ†ψ)3+...
(7)
Here C0, C2and D0are low energy constants which are fixed from few body dynamics.
Using Feynman rules in the two-body sector one derives a scale dependent and momentum
truncated self-adjoint pseudo-potential in the CM system (k and k′are relative momenta)
?k′|V|k? =?C0+C2(k2+k′2)+...?θ(Λ −k)θ(Λ −k′).
The cut-off Λ is introduced here to handle the power divergent integrals arising in the scat-
tering problem, which in terms of the Lippmann-Schwinger (LS) equation becomes
(8)
?k′|T|k? = ?k′|V|k?+M
?
d3q
(2π)3
?k′|V|q??q|T|k?
p2−q2+i0+
,
(9)
implementing unitarity for p ≤Λ. Using the potential of Eq. (8) the LS Eq. (9) reduces to a
system of algebraic equations which solution is well known (see e.g. Ref. [11]) yielding
−
1
α0Λ=4?−2c2
r0Λ =16?c2
2+90π4+15(3c0+2c2)π2?
9π?c2
2+12π2c2+9π4?
π(c2+6π2)2
2−10c0π2?
,
(10)
−12c2
?c2+12π2?
(c2+6π2)2
1
α0Λ+3c2π?c2+12π2?
(c2+6π2)2
1
α2
0Λ2,
where c0= MΛC0, c2= MΛ3C2. By eliminating C0in terms of α0we have written r0in a
form similar to Eq. (3). This leads for any cut-off Λ to the mapping (α0,r0) → (C0,C2). For
C2= 0 one gets r0= 4/πΛ which for α0≫ R yields ΛR = 0.91 and MC0/R = −21.6 from
matching the scattering length and the effective ranges rvdW
for C2?= 0 can be looked up at Fig. 1 in vdW units and for the specific case α0/R = 10
where a weakly bound state takes place. As we see there is a clear stability plateau in the
region Λ ∼ π/(2R) illustrating the basic point of the EFT; low energy physics is cut-off
independent within a given cut-off window which does not resolve length scales shorter
than the vdW scale. Numerically we get MC0/R ∼ −15 and MC2/R3∼ 2 for Λ ∼ π/(2R),
in agreement with the previous SW and DS analysis. The values of Λ where the EFT low
energy parameters diverge correspond to an upper bound above which C0and C2become
complex, violating the self-adjointness of the potential [2] and the Lagrangian, L(x) ?=
L†(x). Thus, off-shell two-body unitarity and hence three-body unitarity are jeopardized
for ΛR ≥ 4, despite the phase shift being real and on-shell unitarity being fulfilled.
Direct inspection shows that a perfect matching between the vdW and the EFT effec-
tive ranges, Eq. (3) and Eq. (10) for any α0is not possible. So we try out including re-
dundant operators which are usually discarded [10] but are needed to guarantee off-shell
renormalizability of the LS equation [12]. A Galilean invariant term of the form ∆L =
−1
0
=rEFT
0
. The cut-off dependence
2C′
2(ψ†ψ)?ψ†?i∂t+∇2/2m?ψ?is formally redundant since it can be eliminated by a

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0.5 1.01.52.0 2.53.03.54.0
?30
?20
?10
0
10
? R
MC0?R
0.51.0 1.52.02.53.03.54.0
?10
?8
?6
?4
?2
0
2
4
? R
MC2? R3
Fig. 1 Cut-off dependence of the EFT coefficients MC0/R when C2= 0 (dashed red) and MC0/R3and
MC2/R3after Eqs. (10) (full blue) for the case α0/R = 10. R is the vdW scale defined as R = (MC6/¯ h2)
1
4.
field transformation ψ → ψ +1
well. The new term adds a correction ∆V =C′
vanishing on-shell. Solving the LS equation and eliminating C0in terms of α0yields
4C′
2ψ(ψ†ψ) which generates additional three body forces as
2(2p2−k2−k′2)/2 to the potential, Eq. (8),
r0Λ =16?(c′
2−2c2)2+36π4+6(8c2−c′
π?−2c2+c′
−12?(c′
?−2c2+c′
2= MΛ3C′
eliminated by making C2→ C2+1
second and the third coefficients holding regardless on the particular regularization method.
Perfect matching can only be achieved with complex coefficients. Minimizing the difference
between rEFT
0
and rVdW
0
provides a reasonable range ΛR = 1.6−1.8 ∼ π/2. As we can see,
universal two-body scaling features encoded in Eq. (3) and exhibiting the underlying vdW
(two photon exchange) nature of interactions impose severe restrictions on the EFT solu-
tion with no explicit photonic degrees of freedom and distinguish between naively unitarily
equivalent Hamiltonians mixing different particle number (see e.g. Ref. [13]). Therefore,
these limitations are expected to play a role in the EFT analysis of three-body forces.
2)π2?
2−12π2?2
2−2c2)2+48c2π2?
2−12π2?2
2appears through the combination C′
2C′
1
Λα0+3π?(c′
2−2c2)2+48c2π2?
?−2c2+c′
2−2C2which cannot be completely
2. Note the accidental correlation −4/π between the
2−12π2?2
1
Λ2α2
0
,
(11)
where c′
Acknowledgements I thank A. Calle Cord´ on for collaboration in [2,6] . Work supported by Ministerio de
Ciencia y Tecnolog´ ıa under Contract no. FIS2008-01143/FIS and Junta de Andaluc´ ıa grants no. FQM225-05
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