Conformal Invariance for Non-Relativistic Field Theory

12 Pages·0.13 MB·English

by  Thomas Mehen

CALT-68-2242
UCSD/PTH 99-14
Conformal Invariance for Non-Relativistic Field Theory
Thomas Mehena, Iain W. Stewartb, and Mark B. Wisea
aCalifornia Institute of Technology, Pasadena, CA 91125
0 bDepartment of Physics, University of California at San Diego,
0
0 9500 Gilman Drive, La Jolla, CA 92099
2

n
a
J

9
Abstract
2

3
v
5
2 Momentum space Ward identities are derived for the amputated n-point
0 Green’s functions in 3 + 1 dimensional non-relativistic conformal field the-
0
ory. For n = 4 and 6 the implications for scattering amplitudes (i.e. on-shell
1
9 amputated Green’s functions)are considered. Any scale invariant 2-to-2 scat-
9
teringamplitudeis also conformally invariant. However, conformalinvariance
/
h
imposes constraints on off-shell Green’s functions and the three particle scat-
t
-
tering amplitude which are not automatically satisfied if they are scale in-
p
e variant. As an explicit example of a conformally invariant theory we consider
h
non-relativistic particles in the infinite scattering length limit.
:
v
i
X
r
a
1
Poincar´e invariant theories that are scale invariant usually have a larger symmetry group
called the conformal group1. A similar phenomena happens for 3+1 dimensional non-
relativistic systems. These are invariant under the extended Galilean group, which con-
sists of 10 generators: translations (4), rotations (3), and Galilean boosts (3). The largest
space-timesymmetry groupofthefreeSchr¨odinger equationiscalledtheSchr¨odinger ornon-
relativistic conformal group [2]. This group has two additional generators corresponding to
a scale transformation, and a one-dimensional special conformal transformation, sometimes
called an “expansion”. The infinitesimal Galilean boost, scale and conformal transforma-
tions are
boosts: ~x′ = ~x+~vt , t′ = t ,
scale: ~x′ = ~x+s~x , t′ = t+2st , (1)
conformal: ~x′ = ~x ct~x , t′ = t ct2 ,
− −
where ~v, s and c are the corresponding infinitesimal parameters. (The finite scale transfor-
mation is ~x′ = es~x, t′ = e2st, and the finite conformal transformation is ~x′ = ~x/(1+ ct),
1/t′ = 1/t+c.)
In this letter we explore the implications of non-relativistic conformal invariance for 3+1
dimensional physical systems. In relativistic theories, conformal invariance can be used to
constrain the functional form of n-point correlation functions [1], however, on-shell scat-
tering amplitudes are typically ill-defined because of infrared divergences associated with
massless particles. In non-relativistic theories scattering amplitudes are well defined even
in the conformal limit. We show how conformal invariance can be used to gain informa-
tion about scattering amplitudes by deriving Ward identities for the amputated momentum
space Green’s functions. While the off-shell Green’s functions can be changed by field redef-
initions, the scattering amplitudes (on-shell Green’s functions) are physical quantities and
are therefore unchanged. We find that any 2-to-2 (identical particle) scattering amplitude
that satisfies the scale Ward identity automatically satisfies the conformal Ward identity.
However, this is not the case for the corresponding off-shell Green’s function or for the 3-to-3
scattering amplitude. We construct a field theory that has a four point function which obeys
1Exceptions are known to exist, however, these theories suffer from pathologies, such as
non-unitarity. A detailed discussion of scale and conformal invariance in relativistic theories
can be found in Ref. [1].
2
the scale and conformal Ward identities and conjecture that the higher point functions in
this theory also obey these Ward identities. On-shell it gives S-wave scattering with an
infinite scattering length.
For the interaction of two nucleons, the scattering lengths in the 1S and3S channels are
0 1
large (a(1S0) = 23.7fm and a(3S1) = 5.4fm) compared to the typical length scales in nuclear
−
physics. In the limit that these scattering lengths go to infinity (and higher terms in the
effective range expansion are neglected) we show that the four point Green’s function obeys
the scale and conformal Ward identities. Thus, two body nuclear systems at low energies are
approximately scale and conformal invariant. It is likely that in some spin-isospin channels
the higher point functions will also obey these Ward identities. Whether this conformal
invariance can lead to new predictions for many body nuclear physics is presently unclear,
but seems worthy of further study.
The action for a free non-relativistic field N(~x,t) is
2
S = dtd3x N† i∂ + ∇ N , (2)
0 t
Z (cid:16) 2M(cid:17)
whereM ismassoftheparticlecorrespondingtothefieldN. Under aninfinitesimalGalilean
transformation N′(~x′,t′) = (1+iM~v ~x)N(~x,t) or equivalently
·
δ N(~x,t) = N′(~x,t) N(~x,t) = D N(~x,t) =~v iM ~x t~ N(~x,t). (3)
g g
− ·(cid:16) − ∇(cid:17)
The action in Eq. (2) is invariant under the infinitesimal scale transformation in Eq. (1)
with N′(~x′,t′) = (1 3s/2)N(~x,t) or equivalently
−
3
δ N(~x,t) = D N(~x,t) = s +~x ~ +2t∂ N(~x,t), (4)
s s t
− (cid:16)2 ·∇ (cid:17)
and under the infinitesimal conformal transformation provided N′(~x′,t′) = (1 + 3ct/2
−
iM c~x2/2)N(~x,t) or equivalently
3t iM~x2
δ N(~x,t) = D N(~x,t) = c +t~x ~ +t2∂ N(~x,t). (5)
c c t
(cid:16) 2 − 2 ·∇ (cid:17)
Now consider adding interactions that preserve these invariances (an explicit example
will be considered later). The position space Green’s functions for the interacting theory,
G(2n)(~x ,t ) = G(2n)(~x ,t ;...;~x ,t ), are defined by2
i i 1 1 2n 2n
2In non-relativistic theories particle number is conserved so there must be the same number
of N’s as N†’s.
3
G(2n)(~x ,t ) = Ω T N(~x ,t ) N(~x ,t )N†(~x ,t ) N†(~x ,t ) Ω , (6)
i i 1 1 n n n+1 n+1 2n 2n
h | n ··· ··· o| i
where Ω is the vacuum of the interacting theory and is assumed to be invariant under the
| i
Schr¨odinger group. Under the infinitesimal transformations in Eqs. (3-5)
δ G(2n)(~x ,t ) = Ω T δ N(~x ,t )N(~x ,t ) N†(~x ,t ) Ω +...
(g,s,c) i i (g,s,c) 1 1 2 2 2n 2n
h | n ··· o| i
+ Ω T N(~x ,t ) N†(~x ,t )δ N†(~x ,t ) Ω
1 1 2n−1 2n−1 (g,s,c) 2n 2n
h | n ··· o| i
n 2n
= Dk + Dk† Ω T N(~x ,t ) N†(~x ,t ) Ω , (7)
hkX=1 (g,s,c) k=Xn+1 (g,s,c)ih | n 1 1 ··· 2n 2n o| i
where Dk is the differential operator for coordinates (~x ,t ). Invariance under Galilean
(g,s,c) k k
boosts, scale, and conformal symmetry implies that
δ G(2n)(~x ,t ) = 0. (8)
(g,s,c) i i
The momentum space Green’s functions G(2n)(p~ ,E ) = G(2n)(p~ ,E ;...;~p ,E ) are
i i 1 1 2n 2n
the Fourier transform of the position space Green’s functions
2n dE d3p
G(2n)(~x ,t ;...;~x ,t ) = k ke−iηk(Ektk−p~k·~xk)
1 1 2n 2n (cid:20) Z (2π)4 (cid:21)
kY=1
2n 2n
(2π)4 δ η E δ(3) η p~ G(2n)(E ,~p ;...;E ,~p ), (9)
k k k k 1 1 2n 2n
× (cid:16)kX=1 (cid:17) (cid:16)kX=1 (cid:17)
where η is 1 for incoming particles (subscripts 1,...,n) and 1 for outgoing particles
j
−
(subscripts n+1,...,2n). ThedeltafunctionsinEq.(9)ariseduetotranslationalinvariance.
Using Eq. (8) with ~x = 0 and t = 0 it is straightforward to show that invariance under
2n 2n
Galilean boosts, scale transformations, and conformal transformations implies the Ward
identities
G(2n)(E ,~p ;...;E ,~p ) = 0, (10)
(g,s,c) 1 1 2n 2n
D
where
2n−1 ∂
= M~ +p~ , (11)
Dg jX=1 (cid:16) ∇pj j∂Ej(cid:17)
2n−1 ∂
= 7n 5+ ~p ~ +2E ,
Ds − jX=1 (cid:16) j ·∇pj j ∂Ej(cid:17)
2n−1 7 ∂ M ∂2 ∂
= η + ~ 2 +E +~p ~ .
Dc jX=1 j (cid:16)2 ∂Ej 2 ∇pj j ∂Ej2 j ·∇pj∂Ej(cid:17)
4
In deriving Eq. (10) we have integrated over the delta functions in Eq. (9) so that
2n−1 2n−1
E = η E , p~ = η p~ . (12)
2n j j 2n j j
jX=1 jX=1
The S-matrix elements are related to the amputated Green’s functions (2n)(p~ ,E ) =
i i
A
(2n)(p~ ,E ;...;~p ,E ) defined by3
1 1 2n 2n
A
2n ~p 2
(2n)(E ,p~ ) = E j G(2n)(E ,~p ), (13)
A i i (cid:20)jY=1(cid:16) j − 2M(cid:17)(cid:21) con. i i
where E and p~ are given by Eq. (12). G(2n) is the connected part of G(2n) and also
2n 2n con.
satisfies Eq. (10). Applying the Galilean boost and scale Ward identities in Eq. (10) to
Eq. (13) gives
˜ (2n)(E ,~p ) = 0, (14)
(g,s) i i
D A
˜
where = and
g g
D D
2n−1 ∂
˜ = 3n 5+ ~p ~ +2E . (15)
Ds − jX=1 (cid:16) j ·∇pj j ∂Ej(cid:17)
Applying the conformal Ward identity in Eq. (10) to Eq. (13) gives
1 1
˜ (2n) + η p~ ˜ (2n) = 0, (16)
c j j g s
D A jηjEj −( jηjp~j)2/(2M)(cid:20)M(cid:16)Xj (cid:17)·D −D (cid:21)A
P P
where
2n−1 3 ∂ M ∂2 ∂
˜ = η + ~ 2 +E +~p ~ . (17)
Dc jX=1 j(cid:16)2 ∂Ej 2 ∇pj j ∂Ej2 j ·∇pj ∂Ej(cid:17)
Therefore, amputated Green’s functions satisfying Eq. (14) also satisfy
˜ (2n) = 0. (18)
c
D A
The leading term in the effective field theory for non-relativistic nucleon-nucleon scat-
tering corresponds to a scale invariant theory in the limit that the S-wave scattering lengths
go to infinity (see for e.g. Ref. [3]). As we will see below, this limit corresponds to a fixed
3Neglecting relativistic corrections to S , Eq. (13) is exact because adding interactions to
0
Eq. (2) does not effect the two point function since there is no pair creation in the non-
relativistic theory.
5
C C C C C
C 0 0 0 0 0
0
+ + + ...
FIG. 1. Terms contributing to (4) from the interaction in Eq. (19).
A
point of the renormalization group. Since in nature the S-wave scattering lengths are very
large, it is the unusual scaling of operators at this non-trivial fixed point [4] that controls
their importance in this effective field theory [5,6]. Motivated by this we add to Eq. (2) the
interaction
S = dtd3x C (NTPN)†(NTPN), (19)
1 0
−Z
where N is now a two component spin-1/2 fermion field and P = iσ /2. Higher body non-
2
derivative interaction terms are forbidden by Fermi statistics. The interaction in Eq. (19)
only mediates spin singlet S-wave NN scattering. The NN scattering amplitude arises from
the sum of bubble Feynman diagrams shown in Fig. 1. The loop integration associated with
a bubble has a linear ultraviolet divergence and consequently the values of the coefficients
C depend on the subtraction scheme adopted. In minimal subtraction, if p 1/a where
0
≫
p is the center of mass momentum and a is the scattering length, then successive terms
in the perturbative series represented by Fig. 1 get larger and larger. Subtraction schemes
have been introduced where each diagram in Fig. 1 is of the same order as the sum. One
such scheme is PDS [5], which subtracts not only poles at D = 4, but also the poles at
D = 3 (which correspond to linear divergences). Another such scheme is the OS momentum
subtraction scheme [4,7]. In these schemes the coefficients are subtraction point dependent,
C C (µ). Calculating the bubble sum in PDS or OS gives
0 0
≡
C (µ)
(4) = − 0 , (20)
A 1+MC (µ) 2µ 4M(E +E )+(p~ +p~ )2 iǫ /(8π)
0 1 2 1 2
h −q− − i
where
4π 1
C (µ) = . (21)
0
−M µ 1/a
−
Note that Eq. (20) holds in any frame and we have not imposed the condition that the
external particles be on-shell. It is easy to see that the limit a corresponds to
→ ±∞
a nontrivial ultraviolet fixed point in this scheme. If we define a rescaled coupling Cˆ
0
≡
MµC (µ)/(4π), then
0
6
+ + ...
FIG. 2. Terms contributing to (6) from the interaction in Eq. (19). The filled circle denotes
A
the sum of diagrams in Fig. 1.
d
ˆ ˆ ˆ
µ C (µ) = C (µ) 1+C (µ) . (22)
0 0 0
dµ h i
ˆ
The limit a corresponds to the fixed point C = 1. At a fixed point one expects
0
→ ±∞ −
the theory to be scale invariant. In fact, it can be easily verified that in the a limit
→ ±∞
8π 1
(4) = (23)
A M 4M(E +E )+(p~ +p~ )2 iǫ
1 2 1 2
q− −
satisfies both the scale and conformal Ward identities in Eqs. (14) and (18). In the case of
(4) the conformal Wardidentity gives non-trivial informationabout the off-shell amplitude.
A
For instance the amplitude
8π 1
(4) = (24)
A M (p~ ~p )2 (p~ p~ )2 iǫ
1 3 2 3
q− − − − −
is scale and Galilean invariant but not conformally invariant. The expressions for (4) in
A
Eqs. (23) and (24) agree on-shell, where E = p~ 2/(2M).
i i
The interaction in Eq. (19) also induces non-trivial amputated Greens functions, (2n),
A
for n > 2. (For n = 3 see Fig. 2.) It is believed that non-perturbatively the higher point
functions are finite and we speculate that with C at its critical fixed point the action S +S
0 0 1
defines a non-relativistic conformal field theory.
We will now derive scale and conformal Ward identities for the on-shell amplitudes since
these are the physical quantities of interest. Consider the four point function for a scalar
field4. After imposing translation invariance it is a function of 12 variables
(4)(p~ ,p~ ,p~ ,E ,E ,E ). (25)
1 2 3 1 2 3
A
4Eqs. (25) through (27) are valid for fermions, but when imposing rotational invariance
we will assume the particles are scalars. For fermions (4) has spin singlet and spin triplet
A
parts, and the expressions in Eq. (29) are valid for the spin singlet component.
7
The Ward identity ˜ (4) = 0 is solved by the function (4)(p~ ,p~ ,U ,U ,U ) where
g A B 1 2 3
D A A
~p 2
i
p~ = ~p ~p , ~p = p~ ~p , U = ME . (26)
A 1 3 B 2 3 i i
− − − 2
Therefore, using the Galilean boost invariance gives three constraints on (4) leaving 9
A
variables. For this function the scale and conformal identities are
3 ∂
˜ = 1+p~ ~ +p~ ~ +2 U ,
Ds A ·∇pA B ·∇pB j∂U
jX=1 j
3 ∂2
˜ ~ ~
= M + η U . (27)
Dc (cid:18)−∇pA ·∇pB j j∂U2(cid:19)
jX=1 j
Three more constraints are given by rotation invariance leaving a function of 6 variables,
(4)(x,y,γ,U ,U ,U ), where
1 2 3
A
x = ~p 2, y = ~p 2, γ = ~p ~p . (28)
A B A B
·
In terms of these variables we have
∂ ∂ ∂ 3 ∂
˜ = 1+2x +2y +2γ +2 U ,
s j
D ∂x ∂y ∂γ ∂U
jX=1 j
∂ ∂2 ∂2 3 ∂2 3 ∂2
˜ = M ˜ +4γ γ 2 U η U . (29)
Dc − (cid:18)∂γDs ∂x∂y − ∂γ2 − j∂γ∂U − j j∂U2(cid:19)
jX=1 j jX=1 j
On-shell the four point function has an additional four constraints U = U = U = 0 and
1 2 3
γ = 0, where the last condition follows because U = U +U U γ = 0. The operator
4 1 2 3
− −
˜
can be defined consistently on-shell since all derivatives with respect to U and γ are
s 1,2,3
D
multiplied by coefficients which vanish in the on-shell limit. In taking the on-shell limit
we are assuming that derivatives of (4) with respect to the off-shell parameters are not
A
singular. This is true of the explicit example in Eq. (23) as long as the momentum of the
nucleons in the center of mass frame is nonzero. Finally, from Eq. (29) we see that on-shell
a scale invariant (4) is automatically conformally invariant.
A
Solving ˜ (4) = 0, the most general scattering amplitude consistent with Schr¨odinger
s
D A
group invariance is
1 y x 1
A(4) = F − = F(cosθ), (30)
os √x+y (cid:16)y +x(cid:17) 2p
where F is an arbitrary function, and θ is the scattering angle in the center of mass frame.
Conformal invariance does not restrict the angular dependence of the scattering amplitude.
8
Additional physical criteria can be used to provide further constraints. The condition that
the S-wave scattering length goes to infinity corresponds to a fine tuning that produces a
boundstateatthreshold. Assuming thatthisistheonlyfinetuningandthattheinteractions
are short range the threshold behavior of the phase shift in the ℓth partial wave is δ p2ℓ+1
ℓ
∼
for ℓ > 0. It is easy to see that the only partial wave obtained from Eq. (30) with acceptable
threshold behavior is the S-wave, so F can be replaced by a constant. In the limit a
→ ∞
the interaction in Eq. (19) provides an explicit example of a scale invariant theory which
has this behavior.
In the case of the 3-to-3 scattering amplitude, conformal invariance will provide a new
constraint independent from that of scale invariance. We proceed exactly as in the case of
the 2-to-2 scattering amplitude. After imposing energy and momentum conservation the 6
point function has 20 coordinates
(6)(p~ ,...~p ,E ,...,E ). (31)
1 5 1 5
A
Using the Galilean boost invariance leaves 17 coordinates
(6)(p~,~k,p~′,~k′,U ,...,U ), (32)
1 5
A
where U = ME p~ 2/2 and
i i i
−
2p~ p~ p~
3 2 1 ~
p~ = − − , k = ~p ~p , (33)
2 1
3 −
2(p~ +~p +p~ )
p~′ = 1 2 3 p~ p~ , ~k′ = p~ p~ .
4 5 5 4
3 − − −
In terms of these variables
5 ∂
˜s = 4+p~ ~ p +~k ~ k +~p′ ~ p′ +~k′ ~ k′ +2 Uj
D ·∇ ·∇ ·∇ ·∇ ∂U
jX=1 j
M 5 ∂2
˜ = ~2 +3~ 2 ~2 3~ 2 +M η U . (34)
Dc 3 h∇p ∇k −∇p′ − ∇k′i jX=1 j j ∂Uj2
Next consider imposing rotational invariance. For simplicity we specialize to the case of a
scalar field. Rotational invariance implies that (6) should be a function of 14 variables. We
A
have chosen
(6)(z ,...,z ,γ,U ,...,U ), (35)
1 8 1 5
A
where
9
z = p~2, z =~k2, z = p~′2, z = ~p ~k, (36)
1 2 3 4
·
z = p~ p~′, z = p~ ~k′, z =~k p~′, z = ~p′ ~k′,
5 6 7 8
· · · ·
γ =~k2 ~k′2 +3p~ 2 3p~ ′2.
− −
The coordinates U and γ vanish on-shell since U = 5 η U +γ/4. For the function in
i 6 j=1 j j
P
Eq. (35) the scale and conformal derivatives are
8 ∂
˜ = 4+2 z +... ,
s j
D ∂z
jX=1 j
∂ ∂ ∂ M 8 ∂2 ∂
˜ ˜
= 2M +3 + A +4M D +... . (37)
c jk s
D (cid:18)∂z ∂z − ∂z (cid:19) 3 ∂z ∂z ∂γ
1 2 3 jX,k=1 j k
The ellipses are terms with factors of U or γ and therefore vanish on-shell,
i
4z 0 0 2z 2z 2z 0 0
 1 4 5 6 
 0 12z 0 6z 0 0 6z 0 
 2 4 7 
 
 
 0 0 4z 0 2z 0 2z 2z 
 − 3 − 5 − 7 − 8 
 
 
 2z 6z 0 3z +z z z 3z 0 
 4 4 1 2 7 9 5 
A =   , (38)
jk  
 2z 0 2z z z +z z z z 
 5 − 5 7 − 1 3 8 − 4 − 6 
 
 
 2z 0 0 z z z 3z 0 3z 
 6 9 8 2 − 3 − 5 
 
 
 0 6z 2z 3z z 0 z +3z z 
 7 − 7 5 − 4 − 2 3 − 9 
 
 
 0 0 2z 0 z 3z z 3z z 
 − 8 − 6 − 5 − 9 − 1 − 2 
 
and z = ~k ~k′. It is possible to express z in terms of z ,...,z . For scale invariant
9 9 1 8
·
amputated Green’s functions the conformal operator can be defined on-shell because terms
that involve derivatives with respect to the off-shell parameters (U and γ) have coefficients
i
which vanish on-shell.
Even after demanding scale invariance the conformal Ward identity still imposes a non-
trivial constraint on the amplitude. It is easy to find examples of boost and scale invariant
functions which do not satisfy ˜ (6) = 0. Due to the complexity of Eq. (37) we have not
c
D A
attempted to find its general solution.
The effective field theory for the strong interactions of nucleons is more complicated than
the toy model given by S +S , because nucleons have isospin degrees of freedom. The in-
0 1
clusion of internal degrees of freedom does not change the Ward identities that correlations
10