Low-scale SUSY breaking and the (s)goldstino physics

0.2 MB·English

by  I. Antoniadis

CERN-PH-TH/2012-231
November 1, 2012
Low-scale SUSY breaking and the (s)goldstino physics.
2
1
0
2 I. Antoniadisa†, D. M. Ghilenceaa,b ‡

t
c
O aCERN - Theory Division, CH-1211 Geneva 23, Switzerland.
1 bTheoretical Physics Department, National Institute of Physics
3
and Nuclear Engineering (IFIN-HH) Bucharest, MG-6 077125,Romania.

]
h Abstract
t
-
p For a 4D N=1 supersymmetric model with a low SUSY breaking scale (f) and gen-
e eral Kahler potential K(Φi,Φ†) and superpotential W(Φi) we study, in an effective
h j
[ theory approach, the relation of the goldstino superfield to the (Ferrara-Zumino) su-

perconformal symmetry breaking chiral superfield X. In the presence of more sources
1
v of supersymmetry breaking, we verify the conjecture that the goldstino superfield is
6 the (infrared) limit of X for zero-momentum and Λ (Λ is the effective cut-off
3 → ∞
scale). We then study the constraint X2 = 0, which in the one-field case is known to
3
8 decouple a massive sgoldstino and thus provide an effective superfield description of
.
0 the Akulov-Volkov action for the goldstino. In the presence of additional fields that
1 contribute to SUSY breaking we identify conditions for which X2 = 0 remains valid,
2
in the effective theory below a large but finite sgoldstino mass. The conditions ensure
1
: that the effective expansion (in 1/Λ) of the initial Lagrangian is not in conflict with
v
i the decoupling limit of the sgoldstino (1/msgoldstino Λ/f, f <Λ2).
X ∼
r
a
†on leave from CPHT (UMRCNRS 7644) Ecole Polytechnique, F-91128 Palaiseau, France.
‡e-mail addresses: [email protected]; [email protected].
1 Introduction
Supersymmetry, if realised in Nature, must be broken at some high scale. In this work we
consider the case of SUSY breaking at a (low) scale √f M in the hidden sector,
Planck
≪
an example of which is gauge mediation. In this case the transverse gravitino couplings
( 1/M ) can be neglected relative to their longitudinal counterparts ( 1/√f) that
Planck
∼ ∼
areduetoits goldstino component. Ifso, onecan then work in thegravity decoupled limit,
with a massless goldstino. The auxiliary field of the goldstino superfield breaks SUSY
spontaneously, while the goldstino scalar superpartner (sgoldstino) can acquire mass and
decoupleatlowenergy,similartoSMsuperpartners,toleaveanon-linearSUSYrealisation.
To describe this regime one can work with component fields and integrate out explicitly
the sgoldstino and other superpartnerstoo (if massive), to obtain the effective Lagrangian.
Alternatively one can use a less known but elegant superfield formalism endowed with
constraints, see [1] for a review. If applied to matter, gauge and goldstino superfields,
these constraints project out the massive superpartners, giving a superfield action for the
light states. Such constraint may be applied only to the goldstino superfield which can be
coupled to the linear multiplets of themodel (e.g. MSSM), to parametrize SUSYbreaking.
In this work we study SUSY breaking in the hidden sector and its relation to the
goldstino superfield in the presence of more sources of SUSY breaking and the connection
ofthegoldstinosuperfieldtothesuperconformalsymmetrybreakingchiralsuperfieldX [2].
Itwasnoticedlongago[3](seealso[1])thataLagrangian, functionofΦ1 = (φ1,ψ1,F1)
= d4θ Φ†Φ1+ d2θfΦ1+h.c. , with a constraint: (Φ1)2 = 0, (1)
L 1
Z nZ o
providesanonshellsuperfielddescriptionoftheAkulov-Volkov actionforthegoldstinofield
[4]. Indeed,theconstraint(whichgeneratesinteractions)hasasolutionΦ1 = ψ1ψ1/(2F1)+
√2θψ1+θθF1, which “projects out” the sgoldstino φ1. When this Φ1 is used back in (1),
one obtains the onshell-SUSY Lagrangian for a massless goldstino ψ1. If more fields are
present and for general K, W, the situation is more complicated and was little studied.
Further, it was only recently conjectured [1] thatthe goldstino superfieldis the infrared
(i.e. zero-momentum) limit of the superconformal symmetry breaking chiral superfield X,
that breaks the conservation of the Ferrara-Zumino current [2]. In the light of the above
discussion, one then also expects that in such limit X2 (Φ1)2 = 0, and this conjecture
∼
was verified in very simple examples. We address these two problems, more exactly:
a) the convergence of the field X to the goldstino superfield, in the limit of vanishing
momentum. We note that one must also take Λ .
→ ∞
b) the validity and implications of the constraint X2 = 0 [1, 3]. This is supposed to
decouple (project out) the sgoldstino. In particular this can mean an infinite sgoldstino
mass[5],howeverondimensionalgroundsthisisactuallyproportionaltof/Λ. Itisdifficult
to satisfy a) and b) simultaneously in general cases, because of opposite limits, of large
1
Λ and large sgoldstino mass, while f < Λ2. This situation is further complicated by the
presenceofmorefields,someofwhichcanalsocontributeto(spontaneous)SUSYbreaking.
Theabove problems were studied for simple superpotentials, like linear superpotentials
[1, 6] or with only one field breaking SUSY [5], with many phenomenological applications
studied in [7]. We investigate below problems a), b) for general K and W, with more fields
contributing to SUSY breaking in the hidden sector. This helps a better understanding of
SUSY breaking and its transmission to the visible sector via the coupling [1] of the X field
to models like the Minimal Supersymmetric Standard Model (MSSM).
X
2 (s)Goldstino and its relation to the chiral superfield .
2.1 Goldstino and sgoldstino eigenstates for arbitrary K, W.
The starting point is the general Lagrangian
L = d4θ K(Φi,Φ†)+ d2θ W(Φi)+ d2θ W†(Φ†)
j i
Z nZ Z o
i
= K j ∂ φi∂µφ† + ψiσµ ψ ψiσµψ +FiF†
i µ j 2 Dµ j −Dµ j j
1 h (cid:0) 1 (cid:1)1 i
+ Kklψiψjψ ψ + W Kijψ ψ Fk W ψiψj +h.c. (2)
4 ij k l k − 2 k i j − 2 ij
h i
(cid:0) (cid:1)
where we ignored a ( 1/4)✷K in the rhs. Here K ∂K/∂φi, Kn ∂K/∂φ†, Kn
− i ≡ ≡ n i ≡
∂2K/(∂φi∂φ†), W = ∂W/∂φj, Wj = (W )†, etc, with W = W(φi), K = K(φi,φ†).
n j j j
Terms with more than two derivatives of K are suppressed by powers of Λ which is the
UV cutoff of the model, Kk 1/Λ, Kij 1/Λ2, etc. We also used the notation
ij ∼ km ∼
ψl ∂ ψl Γl (∂ φj) ψk, Γl = (K−1)l Km
Dµ ≡ µ − jk µ jk m jk
ψ ∂ ψ Γjk(∂ φ†) ψ , Γjk = (K−1)mKjk (3)
Dµ l ≡ µ l− l µ j k l l m
Eq.(2) is the offshell form of the Lagrangian. The eqs of motion for auxiliary fields
F† = (K−1)i W +(1/2)Γlj ψ ψ
m − m i m l j
Fm = (K−1)mWi+(1/2)Γmψlψj (4)
− i lj
can be used to obtain the onshell form of L:
i
L = Kj ∂ φi∂µφ† + ψiσµ ψ ψiσµψ Wk(K−1)i W
i µ j 2 Dµ j −Dµ j − k i
h i
1 (cid:0) 1 (cid:1)
W ΓmW ψiψj +h.c. + Rklψiψjψ ψ , Rkl = Kkl Kn Γkl(5)
− 2 ij − ij m 4 ij k l ij ij − ij n
h i
(cid:0) (cid:1)
2
Here Rkl is the curvature tensor and the potential of the model is
ij
V = W (K−1)i Wj (6)
i j
The derivatives of K, W are scalar fields-dependent. In the following we always work in
normal coordinates, in which case ki = δj, ki = kjk.... = 0, where ki... are the values of
j i jk... i j....
Ki.... evaluated on the ground state (denoted φk , Fk , ψk =0). We denote the field
j... h i h i h i
fluctuations by δφi=φi φi . In normal coordinates kkl used below is actually kkl=Rkl.
−h i ij ij ij
From the eqs of motion for Fi, φi, after taking the vev’s, then
kj F† +f =0, kj Fi F† +f Fm = 0 (7)
i h ji i imh ih ji kmh i
We denote by f , f , f , the values of corresponding, field dependent W , W , W ,
i ik ijk i ik ijk
evaluated on the ground state, so
f = W ( φm ), f = W ( φm ), f = W ( φm ), fi = Wi( φm ), etc. (8)
i i ij ij ijk ijk
h i h i h i h i
Eq.(7) then becomes
F† = f , f Fm = 0. (9)
h ji − j kmh i
Tobreaksupersymmetry,anon-vanishingvevofanauxiliaryfieldisneeded,whichrequires
detf = 0. The goldstino mass matrix is (M ) = W ΓmW evaluated on the ground
ij F ij ij− ij m
state, giving (M ) = f in normal coordinates. A consequence of the last eq in (9) is
F ij ij
that the goldstino eigenvector (normalised to unity) is
F† ψm f ψm
ψ˜1 = h mi = m , m = 0. (10)
− F† Fi 1/2 f fi 1/2 ψ˜1
h i ih i i
(cid:2) (cid:3) (cid:2) (cid:3)
Further, regarding the scalar sector, the mass matrix has the form
(V)k (V) fikf kjk fif f fj
M2 = l kl = il− il j jkl , (11)
b " (V)kl (V)kl # " fjklfj filfik −kijkl fifj #
where Vk = ∂2V/(∂φl∂φ†), V = ∂2V/(∂φl∂φk), etc, is evaluated on the ground state.
l k kl
The two real components of the complex sgoldstino are mass degenerate only if in
(11) the off-diagonal (holomorphic or anti-holomorphic) blocks vanish. For simplicity we
assume that this is indeed the case. This restricts the generality of our superpotential by
the condition
f fk = 0, (12)
ijk
3
that we assume to be valid in this paper1. In this case, the block (V)k determines the
l
massspectrumandeigenstates. Themassof (complex) sgoldstinoobtained fromthisblock
must involve Kahler terms (their derivatives), it cannot acquire corrections from f and it
ij
must be proportional to SUSY breaking, thus it depends on f , fi. The only possibility in
i
ij ij
normal coordinates is to contract the only non-trivial, non-vanishing tensor k = R with
kl kl
f , fi and ensure the correct mass dimension and sign. The result for this (mass)2 is given
i
in the equation below and a discussion can be found in [8]. Finally since SUSY is broken
spontaneously, the sgoldstino mass eigenvector is expected to have a form similar to that
of goldstino itself in eq.(10). Indeed, in thelimit of ignoring theKahler partof (V)k which
l
is sub-dominant, of order (1/Λ2), the sgoldstino is the (massless) eigenvector of fikf
il
O
matrix, and has the form:
f δφm kijf f fkfl
φ˜1 = m + (1/Λ2), m2 = kl i j (13)
[fif ]1/2 O φ˜1 − f fm
i m
where δφm = φm φm is the field fluctuation about the ground state. The mass of
− h i
sgoldstino φ˜1 comes from D-terms, which are (1/Λ2).
O
Spontaneous SUSY breaking suggests the auxiliary of goldstino superfield should have
a similar structure:
f Fm
F˜1 = m (14)
[fif ]1/2
i
This is verified onshell, when F† = W + (1/Λ2) is expanded about the ground state to
i − i O
linear order fluctuations F† = f f δφm+ (1/Λ2). One then finds from (14)2
i − i− im O
f ( fm)
F˜1 = m − + (1/Λ2) (15)
[fif ]1/2 O
i
ForillustrationletusnowconsiderindetailthecaseofonlytwofieldspresentinLagrangian
(2), and also present the expression of the second mass eigenvector. For the fermions, the
mass eigenvectors (normalised to unity) are given below, with ψ˜1 the goldstino field:
1
ψ˜1 = f ψ1+f ψ2 , m2 = 0.
[f fi]1/2 1 2 ψ˜1
i
h i
1 f
ψ˜2 = (f /ρ)ψ1 +f ρψ2 , m2 = fijf , ρ= | 1|. (16)
[f fi]1/2 − 1 2 ψ˜2 ij f
i 2
h i | |
1 AnexamplewhensuchconditionisrespectedisforasuperpotentialofthetypeW =f1Φ1+λ/6(Φ2)3,
with Φ1 breaking SUSYand Φ2 a matter field. This example will beconsidered later.
2 Also at minimum, V should bejust F˜1 2 which is respected.
|h i|
4
For the scalars sector, we find after some algebra the mass eigenstates3 of (M2) = Vk:
b l
1
φ˜1 = f (1+ξk˜ )δφ1 +f (1+ξk˜ )δφ2 ,
1 11 2 12
[f fi]1/2
i
h i
1
φ˜2 = (f /ρ)(1+ξk˜ )δφ1 +f ρ(1+ξk˜ )δφ2 , (17)
1 21 2 22
[f fi]1/2 −
i
h i
with the notation
k˜ = kijρk f f fkfl, k˜ = kijνk f f fkfl, ξ [2(f fk)(fijf )]−1
11 kl i i j 12 kl i i j ≡ k ij
k˜ = kijσk f f fkfl, k˜ = kijδk f f fkfl, (18)
21 kl i i j 22 kl i i j
where
ρ1 = 2/ρ2, ρ2 = 1+ρ2+2/ρ2, ρ1 = 3 ρ2, ρ2 = 2,
1 1 2 − − 2 −
ν1 = 2, ν2 = (1 ρ2), ν1 = (1 ρ2), ν2 = 2ρ2,
1 − 1 − − 2 − − 2
σ1 = 2, σ2 = 1+1/ρ2+2ρ2, σ1 = 3 1/ρ2, σ2 = 2ρ2,
1 − 1 2 − − 2
δ1 = 2/ρ2, δ2 = (1 1/ρ2), δ1 = (1 1/ρ2), δ2 = 2. (19)
1 1 − − 2 − − 2 −
In our normal coordinates kkl=Rkl. We also find the masses:
ij ij
kij f f fkfl kij f f fkfl
m2 = kl i j , m2 = fijf kjk fif + kl i j . (20)
φ˜1 − f fm φ˜2 ij − ik j fmf
m m
We identify φ˜1 of (17) as the sgoldstino, since its mass should not receive corrections from
f , in the limit of ignoring the curvature tensor corrections in (17), and it has a form
ij
similar to that of goldstino eigenstate (10), (16). Regarding the auxiliary fields, one can
show that F˜2 = O(1/Λ2) and that F˜1 is that in (15).
We conclude that the goldstino superfield has the onshell SUSY form
f δΦk
Φ˜1 = k + (1/Λ2)
on−shell [fif ]1/2 on−shell O
i
(cid:12) (cid:12)
δΦk(cid:12) δφk +√2(cid:12)θψk +θθ ( fk). (21)
on−shell ≡ −
(cid:12)
(cid:12)
where we used that kij = (1/Λ2) and that auxiliary fields are on-shell.
kl O
Eqs.(17) to (21) are valid under the assumption that corrections suppressed by powers
of Λ are sub-leading to the superpotential SUSY corrections, proportional to f , see also
ij
3 eqs.(16) and (17) are multiplied in therhs by f2 /f2 which is set to unityby phase rescaling Φ2.
| |
5
(11). Let us introduce a parameter ζ equal to the ratio of the Kahler curvature tensor
contracted by the SUSY breaking scale(s) f to the SUSY “mass term” (f ):
i ij
kkl fifjf f m2
ζ = ξk˜ ij k l sgoldstino 1. (22)
ij ∼ (fpf )(f fmn) ∼ f fmn ≤
p mn mn
If ζ 1 the results of this section such as (17) and (21) are valid and terms suppressed
≤
by high powers of Λ can be neglected, as we actually did. For ζ 1 the eigenvectors have
∼
a more complicated form (easily obtained) and is not presented here. The limit ζ 1
≫
corresponds to decoupling a massive sgoldstino and is discussed in Section 2.4
We shall compare eq.(21) to the chiral superfield X that breaks superconformal sym-
metry, conjectured in [1] to be equal, in the infrared limit to the goldstino superfield Φ˜1.
2.2 The chiral superfield X and its low-energy limit.
Let us explore the properties of the superconformal symmetry breaking chiral superfieldX
and examine its relation to the goldstino superfield found earlier. The definition of X is
α˙
D = D X, X (φ ,ψ ,F ) (23)
αα˙ α X X X
J ≡
where is the Ferrara-Zumino current [2]. For a review of this topic, see for example
J
section 2.1 in [1]. ψ is related to the supersymmetry current and F to the energy-
X X
momentum tensor. For the general, non-normalizable action in (2), this equation has a
solution [9]
1 1
X = 4W D2K D2Y†(Φ†) (24)
− 3 − 2
We find the component fields of X to be (ignoring the improvement term D2Y†(Φ†)):
4 1
φ = 4W(φi)+ KjF† Kijψ ψ
X 3 j − 2 i j
h i
∂φ 4i
ψ = ψk X σµ Kj∂ ψ +Kijψ ∂ φ†
X ∂φk − 3 µ j j µ i
(cid:0) (cid:1)
∂φ 1 ∂2φ 4 i
F = Fi X ψiψj X + Kj ∂ φi∂µφ† + ψiσµD ψ D ψiσµψ
X ∂φi −2 ∂φi∂φj 3 i µ j 2 µ j− µ j
n h i
i (cid:0) (cid:1)
∂ Kj∂ φ† Kjψiσµψ (25)
− µ µ j − 2 i j
(cid:16) (cid:17)o
In these relations all derivatives are scalar-fields dependent quantities. As a side-remark,
one also notices that the integer powers n 1 of these components have a nice compact
≥
structure:
6
φXn = (φX)n, (n 1)
≥
ψXn = n(φX)n−1ψX = ψj ∂∂φφXjn +O(∂µ)
FXn = n(φX)n−2 φX FX − n−2 1 ψXψX =Fj ∂∂φφXjn − 12ψiψj ∂∂φ2φi∂Xφnj +O(∂µ),(26)
h i
where the terms (∂ ) vanish in the infrared limit of zero momenta.
µ
O
Notice that in the leading (zero-th) order in 1/Λ, the only dependence of these compo-
nents on the Kahler comes through φ via its term KjF†, with additional contributions,
X j
fermionic dependent being4 (1/Λ).
O
From (25) we expand X about the ground state and denote w = W( φk ). Keeping
h i
linear fluctuations in fields, one obtains from eq.(25) that
8
φ = 4w+ f δφj + (1/Λ),
X j
3 O
8
ψ = f ψk + (1/Λ),
X k
3 O
8 4 8
F = f ( fk) 4fkf δφm fkδF† + f δFk + (1/Λ). (27)
X 3 k − − km − 3 k 3 k O
Up to a constant we can write, using eq.(21), that onshell-SUSY:
8
X = f δφk +√2θψk +θθ( fk) + (1/Λ)
on−shell 3 k − O
(cid:12)(cid:12) = 8 f (cid:2)δΦk + (1/Λ) (cid:3) (28)
3 k on−shell O
(cid:12)
(cid:12)
Comparing this result against that for the goldstino superfield of (21), one has
8
X = f fiΦ˜1 + (1/Λ). (29)
on−shell 3 i on−shell O
(cid:12) p (cid:12)
(cid:12) (cid:12)
Note that the X field goes to the (onshell) goldstino field in the limit of vanishing mo-
mentum and in addition Λ when higher dimensional terms in the Kahler potential
→ ∞
decouple. This clarifies the relation between the goldstino and the superconformal sym-
metry breaking superfields for general K and W, in the presence of more sources of SUSY
breaking, and is one of the results of this work. All directions of supersymmetry breaking
contribute to the relation between these two superfields.
4 The bracket in FXn is SUSYinvariant for n=2, and then FX2 =0 is invariant.
7
X
2.3 Further properties of the field .
Let us compute the onshell form of X by eliminating the auxiliary fields Fk in (25) and
then examine under what conditions X2 could vanish. The results below are valid up to
(∂ ), where all terms Kj, W ... etc are actually scalar-fields dependent. One has
µ k
O
φ = σ+σmn ψ ψ ,
X m n
8
ψ = ψkW ,
X k
3
F = β+β ψmψn +βkl (ψmψn) (ψ ψ ), (30)
X mn mn k l
where
σ = 4W (4/3) Kl(K−1)k W ,
− l k
σmn = (2/3) KlΓmn Kmn = (1/Λ),
l − O
β = (8/3(cid:2)) W (K−1)m W(cid:3) k,
− m k
β = 2 W Γk W = 2W + (1/Λ),
mn k mn− mn − mn O
βkl = (1(cid:2)/3) Kkl Kj (cid:3)Γkl (1/3)Rkl = (1/Λ2). (31)
mn mn− mn i ≡ mn O
(cid:0) (cid:1)
Here we made explicit the terms which are suppressed by powers of the cutoff scale.
In [1, 3] it was used that the constraint X2 = 0 projects out the sgoldstino field5.
In a strict sense this constraint is valid only in the limit of an infinite sgoldstino mass.
So the problem is that one has an expansion in 1/Λ of the initial Lagrangian which can
conflict with an expansion in the inverse sgoldstino mass, 1/m2 1/(f2/Λ2) = Λ2/f2,
φ˜1 ∼ i i
that decouples the sgoldstino. The effective Kahler terms must give a mass to sgoldstino
(which would otherwise be massless at tree level in spontaneous Susy breaking), and must
simultaneously be large enough for the sgoldstino to decouple at low energy. The two
expansions may have only a very small overlap region of simultaneous convergence.
To have X2 = 0 it is necessary and sufficient to have FX2 = 0. This can be checked
directly. If for example FX2 = 0 then one immediately shows that φX ψXψX so X2 = 0.
∼
Let us compute the value of FX2 in general, using (30). One has
FX2 = α+αmn(ψmψn)+λmn(ψmψn)+νmkln(ψmψn)(ψkψl)+ξmn(ψmψn)(ψ1ψ1)(ψ2ψ2)
= α+α (ψmψn)+ (1/Λ) (32)
mn
O
with
64
α = 2σβ, α = 2σβ W W , (33)
mn mn m n
− 9
5Additional constraints for matter superfieldscan be used to decouple superpartnersat low energy.
8
while the remaining coefficients are suppressed, of order (1/Λ) or higher6 and vanish at
O
large Λ. In this limit only, expanding (32) about the ground state (or using (27)) we find
64
FX2 = − 9 (2fkfkfl δφl +fkfl ψkψl)+O(1/Λ) (34)
up to a constant ( w). To see if FX2 and thus X2 vanish (hereafter this is considered
∝
up to (1/Λ) terms) after decoupling massive scalar fields, one should integrate out δφk,
O
k = 1,2..., via the eqs of motion. With δφk expressed in terms of the light fermionic
and other scalar degrees of freedom, one checks in this way if X2 = 0, without the need
of computing the mass eigenstates7. In general, upon integrating out the sgoldstino and
additional massive scalars, X2 necessarily contains terms suppressed by the sgoldstino
mass. In most cases X2 does not vanish anymore if this mass is finite, except in specific
ij
cases, dueto additional simplifying assumptions(symmetries, etc) for the terms (e.g. k )
mn
of the Lagrangian. In these cases the convergence problem mentioned earlier is not an
issue. We discuss such a case in the next two sections.
2.4 Decoupling all scalar fields, for vanishing SUSY mass terms.
Let us consider the special case of a vanishing SUSY term, i.e. f = 0 or assume it is
ij
much smaller than the Kahler terms in the mass matrix of eq.(11). We consider only two
fields present in the Lagrangian (2), both of which can contribute to the SUSY breaking.
This can be generalised to more fields. For f = 0 we have two massless fermions. As a
ij
result, both scalar fields, which are massive (via Kahler terms), can be integrated out and
expressedintermsofthesemasslessfermions,withoutspecialrestrictionsforscalespresent.
We shall do this and then examine under what conditions X2 vanishes after decoupling.
RegardingtherelationofX tothegoldstinosuperfield,inthiscaseitisdifficulttodefine
the latter, as both fields contribute to SUSY breaking and they can also mix. In previous
sections, see eq.(22), (29) the superpotential (f ) terms were dominant and Kahler terms
ij
were a small correction ( 1/Λ2) to the scalars mass matrix (11), while here the situation
∝
is reversed. Eq.(11) with vanishing off-diagonal blocks and vanishing fikf gives a mass
il
matrix M2=(V)k= kjkfif in basis δφ1,2 with eigenvalues
b l − il j
1
m2 = kmj fkf √∆ + (fijf ), ∆=(kmj fkf )2 4 det(kpj fkf ) (35)
φ˜1,2 2 − mk j ± O ij mk j − mk j
h i
where all indices take values 1 and 2, and the determinant is over the free indices. This
is the counterpart to the result in (20). The eigenvectors can also be found8. How do we
6The expressions of these terms are λmn = 2βσmn = O(1/Λ), νmkln = 2(cid:2)σβmkln+βmnσkl(cid:3) = O(1/Λ),
ξmn78FT=ohr2eyo(cid:2)nσae1r1efiβeφ˜m2l1d2n,2b+=re(σa12k/2i|nφβ˜gm11,12nS|U)−{S(2Y2σk,112f2jk1βfkm16=f2nj0)=,−(1Of(cid:2)j((6=k1111/mn=Λ)−0(cid:3)).ka22nmnd)fifmwfn=±0√,∆FX(cid:3)δ2φ=1+0δφif2δ}φ+1O=(ψf1ijψf1ij/)(−(|2φ˜1f,12)|:sneeor[m1])..
9