Effective field theory for Nambu–Goldstone bosons
When a continuous symmetry is spontaneously broken, the low-energy physics of the system is dominated by the ensuing Nambu–Goldstone (NG) bosons. Examples include phonons in crystalline solids, spin waves in ferromagnets, or pions in QCD. Effective field theory (EFT) constitutes a suitable tool for the analysis of NG bosons, as it exploits the underlying symmetry, leaving just a few unknown low-energy coupling constants to be determined by experiment. The aim of this text is to give a concise overview of the construction of effective Lagrangians for NG bosons of internal symmetries in quantum many-body systems, whether relativistic or nonrelativistic. I put emphasis on subtleties that arise when the assumption of Lorentz invariance, common in textbook treatments, is dropped. I first summarize the main result and afterwards outline its derivation. The material presented here bases largely on, and further develops, the pioneering work of Leutwyler (Physical Review D 49, 3033–3043, 1994).
Effective Lagrangian in the derivative expansion
The degrees of freedom of the EFT are the NG fieldsπa(x), which represent coordinates on the
coset space of broken symmetry, denoted as G/H. In the following, I will follow the notation
Ti,j,... for unspecified generators of G, Tρ,σ,... for generators of H, and Ta,b,... for the remaining,
broken generators. It is convenient to couple the EFT to a background gauge field, Ai µ, for
each generatorTi. These allow one to determine correlation functions of the conserved currents
of the theory. At sufficiently low energies, the effective action can be organized in a covariant gradient expansion in the joint number of derivatives acting on the NG fields and of gauge fields. Owing to the lack of Lorentz invariance, spatial and temporal derivatives have to be counted separately. Denoting the part of the effective Lagrangian withsspatial andttemporal derivatives asL(s,t), the expansion takes the form L=L(0,1)+L(0,2)+L(2,0)+· · ·, assuming in
addition rotational invariance. The first terms of the expansion can then be written as
L(0,1)
=ca(π) ˙πa+ei(π)Ai0, L (0,2)
= 1
2g¯ab(π)D0π
a
D0πb, L(2,0) =−
1
2gab(π)Drπ
a
Drπb, (1)
where Dµπa = ∂µπa −Aiµhai(π) is a covariant derivative of the NG field. Each part of the
Lagrangian is invariant, possibly up to a surface term, with respect to the simultaneous gauge transformation of the NG and gauge fields,
δπa(x) = i(x)hai(π), δAiµ(x) =∂µi(x) +fjki A j µ(x)
k
(x). (2)
Hereiare the infinitesimal parameters of the transformation andfi
jk are the structure constants
of the symmetry algebra. The gauge invariance allows one to determine the form of the coupling functionsca(π), ei(π), gab(π) and ¯gab(π) in terms of a few unknown parameters. The derivation
follows four separate steps:
• Write down all operators with a given number of derivatives and gauge fields, compatible with rotational invariance.
• Use the gauge invariance to eliminate the NG fields, thus expressing the coupling functions in terms of their values at π = 0.
• Substitute the solution to the differential equations obtained in the second step, and from here obtain a set of constraints for the low-energy coupling constants.
More details of the derivation will be given below; here I just summarize the final result. First, define the element of the coset space G/H as U(π) = eiπaT
a and subsequently the
Lie-algebra-valued Maurer–Cartan (MC) form as ω(π) = ωa(π)dπa = Tiωia(π)dπa = −iU(π)
−1dU(π).
Finally, define a bilinear form on the Lie algebra of G as Tr(TiTj) = Ωij. It is assumed to
be nondegenerate but not necessarily proportional to δij as is often the case, and not even
positive-definite. It can be used to raise and lower indices of generators as in Ti = ΩijT j. The
coupling functions can then be cast as
gab(π) =gcd(0)ωca(π)ω d
b(π), ei(π) = ej(0)
U(π)−1TiU(π)
j
, ca(π) = −ei(0)ωai(π),
(3) and analogously for ¯gab(π). The effective Lagrangian is now fully specified in terms of the
low-energy coupling constants gab(0),g¯ab(0), determining decay constants of the NG modes,
and ei(0), associated with charge densities in the ground state. In order to be compatible with
G-invariance, these have to satisfy further constraints,
fσac gcb(0) +fσbc gac(0) = 0, fσji ei(0) = 0, (4)
and similarly for ¯gab(0).
Differential equations for the coupling functions
The set of Ward–Takahashi identities for the Green’s functions of the conserved currents is equivalent to the requirement that the generating functional of the theory, obtained by func-tional integration over the NG fields πa, is invariant with respect to the gauge transformation
of the background fields, given in Eq. (2). This is in turn equivalent to the requirement that the effective action isG-invariant with respect to the simultaneous transformation of the gauge and NG fields, or that the effective Lagrangian density is invariant up to a surface term. Let us first focus onL(0,2). Allowing for all operators compatible with rotational invariance, we write
it as
L(0,2)
= 1
2g¯ab(π) ˙π
a
˙
πb−h¯ai(π)Ai0π˙
a
+1
2 ¯
kij(π)Ai0A
j
0, (5)
and subject it to the transformation (2). Defining the symbols ∂a = ∂π∂a and di = hai∂a, the
infinitesimal transformation of the LagrangianL(0,2) reads
δL(0,2) =1 2
i
(dig¯ab) ˙πaπ˙b+ ¯gabπ˙a( ˙πci∂chbi +h b i˙
i
)−i(di¯haj)A j
0π˙
a−¯
haj( ˙j +f j kiA
k
0
i
) ˙πa
−¯hajA j
0( ˙π
b
i∂bhai +h a i˙
i
) + 1 2
k
(dk¯kij)Ai0A
j
0+ ¯kijAi0( ˙
j
+fk`j Ak0`)
=i
1 2π˙
aπ˙bd
ig¯ab+ ¯gabπ˙aπ˙c∂chbi −A j
0π˙
ad
i¯haj−Ak0π˙
afj
ki¯haj−Aj0π˙
b¯h aj∂bhai
+1
2A
j
0A
k
0di¯kjk +Aj0A
`
0¯kjkf`ik
+ ˙iπ˙a(¯gabhbi −¯hai)−haih¯ajAj0+ ¯kjiAj0
.
Vanishing of the terms proportional to ˙i requires that ¯
hai= ¯gabhbi and k¯ij = ¯gabhaih b
j. (7)
Similarly, vanishing of the terms containing two powers of ˙πa leads to the condition
di¯gab+ ¯gac∂bhic+ ¯gbc∂ahci = 0. (8)
The coefficient proportional to Aj0π˙a reads d
i¯haj+fjik¯hak + ¯hbj∂ahbi, which gives with the help
of Eq. (8) the condition
dihaj −djhai =f k ijh
a
k. (9)
Finally, the term proportional to Aj0Ak
0 disappears if di¯kjk + ¯kj`fki` + ¯kk`fji` = 0. Plugging in
from Eq. (7) and using both Eq. (8) and Eq. (9), we find that this condition is already satisfied, and L(0,2) (and analogously L(2,0)) is gauge-invariant provided Eqs. (7), (8) and (9) hold.
The variation of L(0,1) is calculated similarly,
δL(0,1) =i(dica) ˙πa+ca( ˙πbi∂bhai +h a i˙
i) +i(d
iej)Aj0+ei( ˙i+fjki A j
0
k)
= ˙i(ei+cahai) + i
( ˙πadica+caπ˙b∂bhai +A j
0diej +fjikekAj0)
=∂0[i(ei+cahai)] + i
˙
πa(dica−∂aei−hbi∂acb) +iAj0(diej +fjikek).
(10)
From the requirement of vanishing of the latter two terms, we can at once read off the conditions that have to be satisfied in order that L(0,1) is gauge-invariant up to a surface term,
diej =fijkek, (11)
and
hbi(∂bca−∂acb) =∂aei. (12)
Equations (8), (9), (11) and (12) have an elegant geometrical interpretation. Namely, Eq. (8) says that the functions hai define Killing vectors of the G-invariant metrics gab and ¯gab on the
coset space G/H. The transformation of the NG fields, Eq. (2), defines infinitesimal motion on the coset space generated by the group G. Moreover, Eq. (9) ensures that the Lie-algebraic structure of the symmetry transformations is respected by these geometric transformations on the coset space. Finally, Eq. (11) simply states that ei transforms as an adjoint field of the
symmetry group.
Putting together all the constraints leads to the Lagrangian (1). Note that while L(0,2) and L(2,0) can both be expressed solely in terms of the covariant derivatives, without an explicit
dependence on the gauge field, this is not the case for L(0,1), namelyL(0,1) =c
aD0πa+Ai0(ei+
cahai). This is deeply related to the fact that the former are strictly invariant under the gauge
transformation (2), whereas that latter changes by a surface term. The term L(0,1) is naturally particular to systems lacking Lorentz invariance. By the same token, only with the help of Lorentz invariance can one construct the EFT in terms of a G-invariant Lagrangian density.
Coupling functions from the Maurer–Cartan form
The differential equations obtained above are valid regardless of the choice of coordinates πa
coset space in the exponential form, U(π) = eiπaTa. In this case, an explicit solution for the coupling functions can be given. As the first step, recall that the group action on the coset space is given by multiplying U(π) with an element of Gfrom the left,
eiiTiU(π) =U(π0(π, ))eiTσkσ(π,), (13)
where the last exponential represents the compensating transformation that keeps U(π) in the coset space. Multiplying this equation from the left by U(π)−1, expanding to first order in i
and setting kσ(π, ) = ikiσ(π), appropriate for infinitesimal transformations, we obtain
U−1TiU =ωahai +Tσkiσ. (14)
We would now like to derive expressions for the coupling functions ca(π), ei(π), gab(π) in terms
of the MC form ω(π). To that end, we use a trick. So far, we discussed the transformation properties of the fields and the Lagrangian under infinitesimal gauge transformations. However, the Lagrangian must change just by a surface term also under a finite gauge transformation. In such a case, the transformation rule (2) for the gauge field has to be modified to
A0µ=gAµg−1+ ig∂µg−1, (15)
where Aµ = AiµTi and g = ei
iT
i. From Eq. (13) we can see that choosing i(x) = −δi
aπa(x),
that is, g = U(π)−1, we can eliminate the NG fields altogether. At the same time, the gauge field acquires the value
A0µ=U−1AµU −ωa∂µπa=ωa(Aiµh a
i −∂µπa) +TσkiσA i
µ =−ωaDµπa+TσkiσA i
µ, (16)
where we used Eq. (14) in the second step.
The metric ¯gab(π)
Since every part of the Lagrangian L(s,t) scales differently in spatial and temporal derivatives,
each piece of the action must be gauge-invariant on its own. Let us first see how this can be used to derive an explicit expression for the metric ¯gab(π). Applying the above gauge transformation,
L(0,2) becomes
L0(0,2) = 1
2¯gab(0)h
a i(0)h
b j(0)A
0i
0A 0j
0. (17)
This can in principle differ from L(0,2) by a surface term, but we know from the previous section that this part of the Lagrangian is actually gauge-invariant. Using finally the fact that ha
σ(0) = 0 and hab(0) = δba, which follows from ωa(0) = Ta and from Eq. (14), we obtain
L(0,2) = 1
2g¯ab(0)ω
a
c(π)ωdb(π)D0πcD0πd, whence we identify
¯
gab(π) = ¯gcd(0)ωac(π)ω d
b(π). (18)
This fixes the π-dependence of the metric, but it does not yet guarantee its G-invariance. To ensure this, we must check that the Killing equation (8) is satisfied. It is sufficient to check
global G-invariance in the absence of the gauge field, taking L(0,2) = 1
2g¯ab(0)ω
a(π)ωb(π), with
a slightly abusive notation for ωi(π) = ωia(π) ˙πa. It is straightforward to show that under a global symmetry transformation, the MC form transforms as ω → hωh−1 + i(dh)h−1, where
h=eiTσkσ. For infinitesimal i, we thus get
δiωj =kiσf j kσω
k−δj σπ˙
a∂
As long as the set of all generators spans a fully reducible representation of the unbroken subgroup H, the basis of generators can always be chosen so that fρ
σa = 0. The broken part of
the MC form then transforms covariantly, and the variation of the Lagrangian is
δL(0,2) =ig¯ab(0)ωa(π)kiσ(π)fcσb ωc(π) =
1 2
ikσ
i(π)ωa(π)ωc(π)[¯gab(0)fcσb + ¯gcb(0)faσb ], (20)
which reproduces the first of the constraints (4).
The coefficient ei(π)
Let us now apply the same trick to L(0,1). Upon the gauge transformation (16), it becomes
L0(0,1) =e
i(0)A00i =−ei(0)ωai(π) ˙π a+e
i(0)
U(π)−1A0U(π)
i
. (21)
This can again differ fromL(0,1) by a surface term. However, the second piece does not contain
any time derivatives so it is free from such ambiguity and we can identify
ei(π) = ej(0)
U(π)−1TiU(π)
j
=ej(0) Tr
TjU(π)−1TiU(π)
. (22)
As before, it is not a priori clear that this provides an invariant action without further con-straints. Let us check whetherei(π) defined by this equation satisfies Eq. (11). To that end, it is
convenient to define a matrixE(π) = Tie
i(π) and express Eq. (22) asE(π) =U(π)E(0)U(π)−1.
A simple calculation then gives [introducing a shorthand notation E0 for E(0)]
diE =U U−1(diU)E0U−1+U E0U−1U(diU−1) = iU(haiωa)E0U−1−iU E0(haiωa)U−1
= iU[haiωa, E0]U−1 = i[Ti, E]−ikiσU[Tσ, E0]U−1,
(23)
where we used Eq. (14). The first term can be rewritten with the help of Eq. (30) as
i[Ti, E] = iekΩjk[Ti, Tj] =−ekΩjkfij`T` =ekΩj`fijkT`=Tjfijkek. (24)
We thus arrive at the simple expression diE =Tjfijkek−ikσiU[Tσ, E0]U−1, which is equivalent
to Eq. (11) provided that [Tσ, E0] = 0 for all unbroken generators Tσ. This commutator can be
in turn expressed using once more Eq. (30) as
[Tσ, E0] =ei(0)Ωji[Tσ, Tj] = iei(0)ΩjifσjkTk =−iei(0)Ωjkfσji Tk =−iTjfσji ei(0). (25)
The invariance condition therefore takes the form of the second constraint in Eq. (4).
The coefficient ca(π)
The Lagrangian (21) can in principle differ from L(0,1) by a surface term, but we can at least
guess from it that
ca(π) =−ei(0)ωai(π). (26)
Let us prove that this satisfies Eq. (12) and thus represents a correct expression. We start from the global symmetry transformation of ca(π) ˙πa. On the one hand, we have δi(caπ˙a) =
˙
πa(d
ica+cb∂ahbi). On the other hand, writing ca(π) ˙πa =−ei(0)ωi(π), we find using Eq. (19)
−e0jδiωj =−e0jk σ if
j kσω
k+e0
jδ j σπ˙
a∂
akiσ =e
0
jδ j σπ˙
a∂
since the first term vanishes thanks to the invariance condition on ei(0). Comparing these two
expressions, we finally get
dica−hbi∂acb =−∂a(cbhbi) +e
0
jδ j
σ∂akiσ =e
0
j∂a(ωjbh b i +δ
j σk
σ i) = e
0
j∂a(U−1TiU)j =∂aei, (28)
which is equivalent to Eq. (12). This completes the desired derivation of the NG field depen-dence of the effective couplings.
Appendix: Normalization of generators
The normalization of symmetry generators is defined by Tr(TiTj) = Ωij. Note that in the
adjoint representation, this bilinear form is exactly the Killing form ofG. However, the choice of representation is immaterial here; all expressions for the effective Lagrangian are independent of the representation. For compact and semisimple Lie algebras, Ωij can be made proportional
to δij by a suitable choice of basis of generators. However, since such a choice is not possible
for non-compact algebras, and since it may not be practical for the compact ones, it is useful to set up a formalism which does not make any assumption on Ωij, apart from its regularity.
Taking the trace of the product [Ti, Tj]Tk and using the definition of the structure constants of
G, one immediately finds that these satisfy the relations
fij`Ω`k =fjk` Ω`i =fki`Ω`j, (29)
which generalize the full antisymmetry of fk
ij in the case that Ωij ∝ δij. Multiplying twice by
the inverse of Ω, we moreover infer the useful identity
fk`i Ω`j +fk`j Ω`i= 0. (30)
Both these identities have a simple interpretation. Observing that bothfk
ij and Ωij are invariant
under adjoint group action and restoring the proper ordering of indices by setting fk ij = f
k ij ,
Eq. (29) expresses the cyclic symmetry of the invariant tensor fijk and Eq. (30) the
