vac-uum expectation value. Vertices for ghost interactions with vector (middle) and scalar (right) fields. The dotted lines represent ghost propagators. Ghost-ghost-vector vertices involve one outgoing and one incoming momentum, with the outgo-ing one appearoutgo-ing in the vertex factor. (Some authors place a dot near the outgooutgo-ing line to indicate which momentum appears.)
and ghost propagators, is singled out. The ghost propagator, which is also anN×Nmatrix, is
iDG(k) = −i
k2−M2/ξ. (4.62)
The Higgs propagator is ann×nmatrix in the space of scalar field indices. It can be written as
i∆φ(k) = (I− P) i
k2−µ2 +P i
k2−M2/ξ, (4.63)
wherePis defined as the projection operator onto theN−M dimensional space of Goldstone fields spanned by the vectorsiLiν, i=M+ 1· · ·N. It is given explicitly by is the inverse of the vector mass-squared matrix in the subspace of massive vectors (and zero otherwise). The scalar propagator in (4.63) decomposes into two pieces. The first, proportional toI− P, represents the propagation of real physical scalars with mass-squared matrix µ2. The second term is the Goldstone boson contribution, which is present in an arbitraryξgauge. It is shorthand for
where the first term includes both the physical and Goldstone scalars.
We see that the propagators of gauge, ghost, and Higgs fields can be decomposed into pieces involving the propagation of physical particles, namely the first terms in (4.61) and (4.63), as well as pieces that involve unphysical poles atk2 = Mξ2 that depend upon the gauge. These unphysical poles do not correspond to physical particles, and are included only in internal lines in Feynman diagrams. They cancel in the final expressions for physical on-shell amplitudes, i.e.,S matrix elements, which are necessarily gauge invariant.
Rξ gauges for ξ 6= 0 are referred to as renormalizable gauges because the propagators are well-behaved fork2→ ∞, falling as 1/k2. In principle it is best to do calculations withξ arbitrary so that (a) the Feynman diagrams are well-behaved and manifestly renormalizable, and (b) so that the cancellation of theξdependence in the physicalSmatrix elements can be used as a check on the correctness of the calculation. In practice, however, it is messy to carry out calculations for an arbitraryξ, and therefore people often make use of particular gauges for calculations or formal arguments. The ξ = 1 gauge is known as the ’t Hooft-Feynman gauge. The gauge propagator takes the particularly simple form
iDVµν = −igµν
k2−M2, (4.68)
so theξ= 1 gauge is often convenient for carrying out concrete calculations. Another useful gauge isξ→ ∞, known as the renormalizable or Landau gauge. The propagators again take a relatively simple form in this limit
iDVµν(k) =−i gµν−kµkk2ν
k2−M2 , iDG(k) = −i
k2, i∆φ(k) = i
k2−µ2, (4.69) but this gauge still contains ghosts and unphysical scalar fields.
The above gauges are best for calculations of higher orders. However, the ξ → 0 or unitary gauge is particularly simple for displaying the physical particle content of the theory.
The propagators become iDVµν(k) = −i gµν−kMµk2ν
k2−M2 , iDG(k) = 0, i∆φ(k) = (I− P) i
k2−µ2. (4.70) That is, there are no ghost fields and only the physical, non-Goldstone scalar fields survive.
If one is only interested in calculating at tree level, the unitary gauge is very convenient.
However, the gauge boson propagator is badly behaved ask→ ∞; it approaches a constant rather than falling like 1/k2. It therefore induces severe ultraviolet divergences in higher-order calculations that must be handled very carefully.
The ghost fields do not entirely disappear from the theory in the unitary gauge (Wein-berg, 1973c,d). There is an effective multiscalar interaction
LJ =−iδ4(0)Tr ln(I+J), (4.71)
where
Jij=g2 1
M2
ik
hν|LkLj|φ0i. (4.72) The trace and matrices in (4.71) and (4.72) are restricted to theN−M dimensional subspace of broken generators ofG.LJ is a remnant of the ghost loops that survives asξ→0 because of the factors ξ−1 in the ghost-ghost-scalar vertices, which cancel the zeroes in the ghost propagators.LJ is necessary to cancel divergences due to gauge boson loops in multiscalar interactions.
Complex Scalars
It is straightforward to reexpress the results for the interaction vertices and propagators in a complex scalar basis, either by “starting from scratch” or by using the formal results in (3.71)–(3.73). Writingφ=v+φ0, where the fields and VEVs are now complex, the induced cubic and ghost-Higgs vertices in (4.54) and (4.59) are replaced by
LAAφ0 =g2
respectively, while the gauge boson mass matrix in (4.48) becomes
Mij2 =g2hv|LiLj+LjLi|vi. (4.74) The projection operator P onto the Goldstone subspace is usually obvious, but is best worked out in the Hermitian basis for complicated cases.
Let us illustrate the Rξ constructions for an SU(2) gauge theory involving a single complex scalar SU(2) doublet. (This is a simplified version of the standardSU(2)×U(1) model with the U(1) gauge coupling, and therefore electromagnetism, turned off.) The HermitianSU(2) gauge fieldsAiµ, i= 1· · ·3, will sometimes be re-expressed as
A±µ =A1µ∓iA2µ
√2 , A0µ=A3µ, (4.75)
where A+ = (A−)†, and the labels±and 0 are suggestive of the electric charges that will emerge when the model is promoted to SU(2)×U(1). The gauge self-interactions for the Afields are derived inProblem 4.8. Similarly, the scalar doublet can be written
φ= renormalizable gauge invariant potential forφ,
V(φ) = +µ2φ†φ+λ(φ†φ)2, (4.77) is identical to the Higgs potential in the standard model, and will be described in more detail inSection 8.2. Here we note that λ >0 is needed for vacuum stability. For µ2 >0 there is no SSB, i.e.,v≡ h0|φ|0i= 0, while v6= 0 forµ2<0.
In the unbroken case,v = 0, the complex scalar fields φ+ andφ0 are degenerate with massµ, and their self-interaction terms are
LI =−VI =−λ(φ†φ)2=−λ(φ−φ++φ0∗φ0)2. (4.78) The scalar gauge interactions6 can be obtained by writing the general form in (4.21) in terms of components, usingLi=τi/2 andAiτi=A0τ3+√ the product of two doublets transforms as 0+1 underSU(2) while that for two triplets is 0+1+2. However, the 1 is antisymmetric for the triplets and vanishes in this case, so only the singlet components can enter.
where A~µ·A~µ = 2A+µA−µ +A0µA0µ. The diagonal quantum number associated withT3 is conserved.
Forµ2<0 theSU(2) symmetry is completely broken. Similar to theU(1) example in (3.153) on page 119 the minimum occurs forv=ν/√
2, where|ν|2=−µ2/λ. Without loss of generality, we can choose ν to be real and in theφ0 direction: other orientations of the minimum differ bySU(2) transformations. Thus, we can write
v= 1
whereH andzare the Hermitian components ofφ00andw+≡φ+. Rewriting the potential in the new variables,
−µ2/λ and we have dropped the additive constant. We therefore recognize that H is a physical scalar field with mass MH = p
−2µ2 = √
2λν, while z, w+, and w− = (w+)† are the Goldstone bosons that should disappear in the unitary gauge. (The notation is chosen to coincide with the analogous standard model case.) The second term on the right is an induced cubic interaction.
In the notation following (3.71) on page 101,φcan be written in a Hermitian basis as
φh=
2 and the ordering of the components differs slightly from the convention in (4.52). Then, using theSU(2) representation matrices in (3.76),
L1hνh=−i
establishing that all three generators are broken and that the vectorsiLihνh span the Gold-stone subspace.
Substituting the representation in (4.80) forφ, the scalar gauge interactions become Lφ=1
where we have omitted the irrelevantφ0−Amixing terms. The three gauge bosonsAi (or A±, A0) have acquired a common mass
MA=gν
2 (4.85)
by the Higgs mechanism. They are degenerate because the Lagrangian has an unbroken O(3) global symmetry after the breaking, as is discussed in Section 8.2.2. The derivative cubic terms always connectH to a Goldstone field (or two Goldstone fields to each other), so they disappear in the unitary gauge. (There are important analogs involving physical Higgs fields in models with extended Higgs sectors, such as in theminimal supersymmetric extension of the standard model (MSSM).) The induced cubic term νH ~A2 involves only the physical scalar field. The projection operatorP in (4.63) projects ontow± andz, while I− P projects ontoH.
As a simple example, let us consider the amplitude forHH →A+A− in anRξ gauge, with the tree-level diagrams shown in Figure 4.3. Using the gauge and scalar propagators in (4.61) and (4.63), and the interaction vertices in (4.81) and (4.84), the amplitude is
M =
where the 3! in the last term is combinatoric. The propagators are iDAµν+(k) =−i part of theA+ term, leaving theA+ andH exchange diagrams evaluated in unitary gauge.
This illustrates the general result that physical on-shell amplitudes are independent ofξ.
A+
additional
u-channel diagrams obtained from the first two by p1↔p2,
4.5 ANOMALIES
Anomaliesrefer to quantum effects that break the symmetries associated with the classical equations of motion. In particular, they may occur when the diverences in a theory cannot be
regularized in a way that is consistent with the original symmetries. The Adler-Bell-Jackiw anomalies (Adler, 1969; Adler and Bardeen, 1969; Bell and Jackiw, 1969; Bardeen, 1969;
Adler, 2004) are singularities associated with the fermion triangle diagram contributions to the vertex of three currents, as shown inFigure 4.4. If one or three of the vertices involve
Jµi
Jνj
Jρk
i
k j
Figure 4.4
Left: The anomalous triangle diagram for the
JµiJνjJρkvertex. Right: The analogous diagram for the triple gauge boson vertex.
an axial vector (γ5) coupling, as can occur for a chiral symmetry, the diagram diverges linearly, leading to an anomalous divergence of the currents in perturbation theory that is not revealed by formal manipulation of the field equations. If one or more of the currents is associated with a global symmetry of the theory, the anomalous divergence does not cause any particular problems, and it can even be useful (Adler, 1969; ’t Hooft, 1976a), as we will see inSections 5.2 and5.8.3. If the currents are all associated with gauge symmetries, however, then the diagram contributes to the triple gauge vertex and cannot be regularized in a way consistent with gauge invariance, implying that the renormalizability of the theory is lost.
The anomaly coefficientAijk in the vertex of currentsi, j,andkis (Georgi and Glashow, 1972)
Aijk = 2TrLiL{LjL, LkL} −2TrLiR{LjR, LkR}, (4.88) independent of the fermion masses. We require that eachAijk must vanish for a renormal-izable gauge theory, both when the three currents are all associated with the same group factor and when they are associated with two or three factors in a direct product group.
Another constraint comes from the (universal) interaction of fermions with gravity (see, e.g., Weinberg, 1995), which leads to a breaking of gauge invariance in the presence of a gravitational field, proportional to the trace anomaly
Ti= TrLiL−TrLiR. (4.89)
We therefore require for this to vanish as well.
Aijk and Ti vanish for chiral theories (LiL = LiR), and also if both LiL and LiR are real (equivalent to −LiTL,Rin a Hermitian basis) since TrM = TrMT. The only simple Lie algebras in Table 3.3 that admit complex representations are SU(m) for m ≥ 3 (which includesSO(6)∼SU(4)),SO(4m+ 2) form≥2, andE6, so anomalies can occur only for gauge groups that include these factors7or U(1)’s.
There are no anomalies for pure QED or QCD, since they are non-chiral. The full SM is
7The anomalies associated with threeSO(4m+ 2) form≥2 or threeE6factors vanish even for complex representations (Georgi and Glashow, 1972; Okubo, 1977).
based onSU(3)×SU(2)×U(1), withSU(2)×U(1) chiral, but as will be seen in Section 8.1 the anomalies cancel between quarks and leptons or (in non-trivial ways) betweenLandR.
We discussed in Section 2.10 that instead of working in terms of the L and R-chiral particle fields ψL,R, one could just as well express the theory in terms of the L-chiral particle and antiparticle fieldsψLandψcL, whereψRandψLc are related by (2.301). The 2n fields (ψLψcL)T transform under the reducible 2n×2n-dimensional representation matrices
Li =
LiL 0 0 LciL
=
LiL 0 0 −LiTR
, (4.90)
where theψcLtransform as−LiTR since they are basically the adjoints of theψR. The anomaly conditions in this basis are
Aijk= 2TrLi{Lj,Lk}= 0, Ti= TrLi= 0, (4.91) which are equivalent to (4.88) and (4.89).
4.6 PROBLEMS
4.1 The spontaneous breaking of a global or local U(1) symmetry allows one-dimensional cosmic string classical solutions, analogous to the two-dimensional domain walls considered in Section 3.3.2. Consider the complex scalarφin aU(1) gauge theory, with
L=−1
4FµνFµν+ (Dµφ)†Dµφ−V(φ), V(φ) =µ2φ†φ+λ φ†φ2 ,
where Dµφ = (∂µ+igAµ)φ, Fµν is the field strength tensor, λ > 0, and µ2 < 0. It is convenient to rewrite
V(φ) =λ
φ†φ−1 2ν22
, ν = r−µ2
λ ,
where we have dropped the irrelevant constant−λν4/4. It was shown (Nielsen and Olesen, 1973) that there is a classical solution for φ(x) and Aµ(x) corresponding to a vortex or string running along the z axis from −∞ to +∞. In cylindrical coordinates (r, θ, z) the solution exhibits
φ−−−→r→∞ ν
√2e−inθ,
so thatV →0 forr→ ∞. Single-valuedness requires thatnis an integer. Find the asymp-totic expression for the vector potential Aµ for the string solution, for which Fµν → 0, Dµφ→0, and calculate the magnetic fluxR B~ ·d ~S =R ∇ ×~ A~
·d ~S for the solution.
4.2 As mentioned in Section 3.2.5 it often occurs that the spontaneous breaking of a continuous symmetry leaves a discrete subgroup unbroken. If the original symmetry was local, the remaining subgroup is known as adiscrete gauge symmetry(e.g., Ib´a˜nez and Ross, 1992). As a simple example, consider a model involving three complex scalarsφi, i= 1· · ·3, with aU(1) gauge symmetry. Suppose the potential is of the form
V(φ1, φ2, φ3) =
3
X
i=1
Vi(φ†iφi) +Vcubic,
where Vi(φ†iφi) are quadratic functions of φ†iφi (i.e., quadratic and quartic in φi and φ†i) and
Vcubic=σ1φ1φ2φ3+σ2φ21φ†2+h.c.
(a) Find the U(1) charges of theφi for which the theory isU(1) invariant.
(b) Show thatσ1,2 can be taken to be real w.l.o.g.
(c) Suppose theViare such that the minimum ofV occurs forh0|φ3|0i 6= 0 buth0|φ1,2|0i= 0.
Show that a discreteZ3symmetry remains unbroken.
4.3 Let Φi, i= 1· · ·m2−1, bemHermitian scalars transforming according to the adjoint representation ofSU(m). Define them×mmatrix Φ =P
iΦiLi as in (3.31), whereLi are the fundamental representation matricesLim. Show that the gauge covariant derivative is
DµΦ =∂µΦ +igh
A~µ·~L,Φi .
4.4 From (3.19) the defining (vector) representation ofSO(m) consists of them(m−1)/2 antisymmetric imaginary m×m matrices. These are conveniently labeled by two indices, with
Lij
ab=−i(δaiδbj−δbiδja),
where i, j, a, and b all range from 1 to m. Clearly, Lij = −Lji and Lii = 0. One could restrict the indices so that i < j, but it is convenient not to do so, provided one is careful about double counting. The trace is
Tr LijLkl
= 2(δikδjl−δilδjk).
From the vector representation, one has the Lie algebra Tij, Tkl
=i −δjkTil−δilTjk+δikTjl+δjlTik ,
where the generators Tij have the same labelling convention as Lij. (It is instructive to specialize these relations to m= 3.)
(a) Calculate the gauge covariant derivative (DµΦ)a for a scalar or fermion field Φa, a= 1· · ·m, transforming as a vector underSO(m), labeling the gauge bosons by Aijµ =−Ajiµ and the gauge coupling asg.
(b) Calculate the field strength tensorFµνij forAijµ.
4.5 The gauge transformation for a non-abelian gauge field is given in (4.24).
(a) Verify that (4.24) reduces to (4.26) for small|βi|. (b) Verify the transformation (4.22) for a matter field.
(c) Use (4.24) to prove (4.30) for all ~β (i.e.,notjust small|βi|). This implies thatLKEA is invariant.
(d) Rederive (4.30) by the simpler method of first proving ig ~Fµν·L~ = [Dµ, Dν], with Dµ
from (4.17), and then showing thatU DµU−1=D0µ≡∂µ+ig ~Aµ0 ·~L.
4.6 Derive the gauge vertices in Figure 4.1 for Hermitian scalars.
4.7 Derive the triple gauge vertex rule shown in Figure 4.1.
4.8 Specialize the 3- and 4-point gauge vertices in Figure 4.1 to the case of SU(2). De-fine the fields A± and A0, as in (4.75). Show that the vertices for A+µ(p)A0ν(q)A−σ(r), A+µA0νA0σA−ρ, andA+µA+νA−σA−ρ are, respectively,
igCµνσ(p, q, r), −ig2Qµρνσ, ig2Qµνρσ,
and that the others vanish. All of the particles and momenta flow into the vertices.
Cµνσ(p, q, r) is defined in Figure 4.1, while
Qµνρσ ≡2gµνgρσ−gµρgνσ−gµσgνρ.
Note thatQis symmetric inµ↔ν orρ↔σ, thatQµνρσ=Qρσµν, and that Qµνρσ+Qµσνρ+Qµρνσ= 0.
4.9 Prove that the gauge boson mass matrix in (4.48) is real, symmetric, has non-negative eigenvalues, and that N −M eigenvalues are non-zero. Hint: define an eigen-value and normalized eigenvector of M2 as λ and w, i.e.,M2w = λw. Use the fact that λ=wTM2w=wiMij2wj. Recall that the vectorsLiν span anN−M dimensional space.
4.10 Derive the effective multiscalar interaction in (4.71).
4.11 Extend theSU(2) model on pages 149-151 (with µ2<0) by the addition of a non-chiral fermion doubletψ=
ψ+ ψ0
with massmψ.
(a) Find the interaction vertices for ψ+→ψ0A+,ψ+→ψ+A0, andA0→A+A−.
(b) Write the tree-level amplitude for ψ+(p1)ψ−(p2) → A+(p3)A−(p4) in the Rξ gauge.
Show that it is independent ofξ.
(c) Show that the individual diagrams (for the longitudinal polarizations) grow ∝ s for sMA2, m2ψ, but that the leading term cancels in the full amplitude. (The implications for unitarity and renormalizability will be discussed inChapter 7.1.)
4.12 (a) Calculate the anomaly coefficient for a left-chiral fundamental representation of SU(m) in terms of thedijk defined in (3.26).
(b) Consider a field ψab that transforms as the (reducible) direct product of two SU(m) fundamentals. It follows from (3.118) that the corresponding representation matrices are
LiD
ab;cd= Liacδbd+δacLibd
, which can conveniently be written in direct product notation asLiD=Li⊗I+I⊗Li, whereIis them×midentity. Calculate the anomaly coefficient for a left-chiral field transforming asψab. Hint: TrA⊗B = TrATrB, and (A⊗B)(C⊗D) = AC⊗BC.
DOI: 10.1201/b22175-5
C H A P T E R