Table Of ContentA note on the implications of gauge invariance in QCD
Elliot Leader
Blackett laboratory
Imperial College London
Prince Consort Road
London SW7 2AZ, UK∗
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2 Enrico Predazzi
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Dipartimento di Fisica Teorica, Universit`a di Torino,
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J
and INFN, Sezione di Torino, Italy†
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(Dated: January 19, 2011)
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Abstract
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We compare and contrast the implications of gauge invariance for the structure of scattering
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amplitudes in QED and QCD. We derive the most general analogue for QCD of the famous QED
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1 rule that the scattering amplitude must be invariant if the polarization vectors ǫµj(kj) of any
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25 number of photons undergo the replacement ǫµj(kj) → ǫµj(kj) + ckjµj, where c is an arbitrary
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constant.
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.
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0 PACS numbers: 11.15.-q,12.38.-t, 12.38.Aw, 12.38.Bx
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∗ [email protected]
† [email protected]
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I. INTRODUCTION
InQEDtheamplitudeforageneralreactioninvolvingM incomingphotonswithmomenta
kµ1,kµ2.......kµM and N outgoing photons with momenta κν1,κν2.........κνN has the structure
1 2 M 1 2 N
A = ǫ∗ (κ )......ǫ∗ (κ )Mν1....νN ǫµ1(k )......ǫµM(k ) (1)
ν1 1 νN N µ1....µM 1 M
where the ǫµ(k) are the photon polarization vectors.
It is very well known that “as a consequence of gauge invariance”
κ Mν1...νj...νN = 0 Mν1.........νN kµi = 0 (anyi ∈ 1...M; j ∈ 1...N) (2)
j,νj µ1.........µM µ1...µi...µM i
irrespective of the value of (κ )2 or (k )2.
j i
It is perhaps less well known that the condition Eq. (2) does not hold for the QCD
amplitude for a reaction involving M incoming and N outgoing gluons.
The condition Eq. (2)is extremely important and useful:
• it simplifies the calculation of cross-sections from amplitudes;
• it controls the allowed tensorial structure of amplitudes; a topical example is deep
inelastic lepton-hadron scattering, where the interaction between the off-mass-shell
photon of momentum q and the hadron is described by the hadronic tensor Wµν,
which must satisfy the condition q Wµν = Wµνq = 0;
µ ν
• it provides a useful check on the correctness of calculations of scattering amplitudes.
It is therefore of interest to know what relation in QCD is closest to the QED condition
Eq. (2).
Sometime ago, inthe process ofwriting a textbookongaugetheories andparticle physics
[1] we discussed the difference between the implications of gauge invariance for amplitudes in
QEDandQCD, explaining that theresult Eq. (2)inQEDis farfromatrivial consequence of
electromagnetic gauge invariance in a quantum field theory and showing why such a relation
could not be derived in QCD. We ended up deriving what we believed was the closest
analogue in QCD to the QED relation Eq. (2). At the time, although we had not seen this
result in the literature, we assumed it was well known to workers in the field. Subsequently,
as a result of questions raised by colleagues and comments made at workshops, we have
2
come to realize that, in fact, the result is not generally known, and that it ought, therefore,
to be reported in the literature. In this paper we present a brief discussion of the differences
between QED and QCD and a derivation of our result for QCD. Our final result is presented
as a little Theorem at the end of Section IV.
II. QCD: A REMINDER
Ga , the SU(3) non-Abelian generalization of the standard QED field tensor Fa is given
µν µν
by
Ga = ∂ Aa −∂ Aa +gf Ab Ac, (3)
µν µ ν ν µ abc µ ν
where Aa is the gluon vector potential, a = 1,2,...8 is the octet colour label and f are the
µ abc
usual SU(3) structure constants , i.e. the SU(3) group generators T obey the commutation
a
relations
[T ,T ] = if T . (4)
a b abc c
The colour indices will be indifferently written as superscripts or subscripts. (Sometimes
−g is used instead of g. This is of no relevance here given that all perturbative QCD
calculations depend on g2.)
The covariant derivative operator is defined as
Dˆ ≡ ∂ −igT Aa. (5)
µ µ a µ
Acting on a given field which transforms according to a specific group representation, the
T are replaced by the corresponding representation matrices. As a consequence, acting on
a
quark fields, Eq. (5) becomes
(D ) = δ ∂ −igta Aa (6)
µ ij ij µ ij µ
where ta, a = 1,2,...8are3×3 Hermitianmatrices which forthetriplet SU(3) representation
are 1/2 the Gell-Mann matrices λa. Acting on the gluon fields, the T are represented by
a
the structure constants T → −if and Dˆ is then represented by
abc abc µ
(D ) = δ ∂ −igf Aa. (7)
µ bc bc µ abc µ
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III. THE QCD CURRENTS : DIFFERENCES FROM QED
The gauge invariant Lagrangian density L is written as
1
L = − Ga Gµν + iψ¯ γ (Dµ) ψ . (8)
4 µν a i µ ij j
Flavour summation, which is irrelevant for our discussion, is implied for the last term and
we have, as usual, assumed massless quarks.
From Eq. (8) one derives the equations of motion. For the gluon field we get
(Dµ) Gb = gJa (9)
ab µν ν
where the quark current is
Ja = ψ¯γ ta ψ (10)
ν i ν ij j
Now in QED the analogous electromagnetic current, the source of the photon field, is
conserved, whereas the QCD current Ja is not conserved in the usual sense, i.e. ∂µJa 6= 0.
ν µ
However, one finds that
(Dµ) Jb = 0. (11)
ab µ
It may appear puzzling that Ja is not conserved since the theory is invariant under the
µ
SU(3) groupofcolourgaugetransformations. Thereare indeedconserved Noethercurrents
C
but they do not coincide with Ja. The conserved Noether current is
µ
J˜a = Ja +f Gb Aν. (12)
µ µ abc µν c
The interesting feature is that using Eq.(7) in Eq.(9) one gets
∂µGa = gJ˜a (13)
µν ν
Notice, however, that, in contrast with QED, the current that appears in the above
equation is not just a quark current but contains G itself, This stems from the fact that
µν
while the photon field is electrically neutral, the gluons have a colour charge. This, of
course, is at the same time the richness and the complication of QCD compared with QED.
Ultimately, this is the effect of the gluon self coupling which will be discussed in the next
subsections.
Now in a gauge theory one has to choose a definite gauge in which to work and this
gauge fixing is carried out in such a way as to preserve the invariance of the theory under
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some variant of the global version of the original local gauge invariance. Concerning the two
currents we have introduced, the non-conserved Ja and the conserved J˜a, it is important to
µ µ
notice that neither is invariant under the QCD global transformations.
With standard notations, under an infinitesmal SU(3) non Abelian gaugetransformation,
the various QCD fields transform as
δψ = −itb ψ θ , (14)
j jk k b
δψ¯ = iψ¯ tb θ , (15)
j k jk b
δAa = f Ac θ , (16)
µ abc µ b
δGa = f Ga θ . (17)
µν abc µν b
Using these transformations, we see that both the currents Ja and J˜a are not gauge
µ
invariant. For example, for Ja we obtain
δJa = f Jcθ . (18)
µ abc µ b
and an analogous equation for J˜a, where θ is a constant phase.
µ b
In contrast, recall that in the Abelian case of QED. the electromagnetic current is invari-
ant under gauge transformations i.e.
δJem = 0. (19)
µ
Now it can be shown that the generators of any symmetry transformation are the con-
served charges associated with the Noether currents. Thus, in the em case the generator
is
Qˆ = d3xJem(x,t) (20)
0
Z
and, if the change induced in any function F(x) of the field operators by an infinitesimal
em gauge transformation is
F(x) → F(x)+δF(x), (21)
then
δF(x) ∝ [Qˆ,F(x)]. (22)
Owing to the gauge invariance of Jem, we then have
µ
d3x[Jem(x,t),Jem(y)] = 0. (23)
0 µ
Z
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IntheQCD case, thereareeight generatorsQˆa associatedwith theeight Noether currents
J˜a via
µ
Qˆa = d3xJ˜a(x,t) (24)
0
Z
and in contrast to QED, since neither Ja nor J˜a are invariant under gauge transformations,
µ µ
the analogue of Eq. (23) does not hold and
d3x[J˜a(x,t),Jb(y) or J˜b(y)] 6= 0. (25)
0 µ µ
Z
IV. CONSEQUENCES OF GAUGE INVARIANCE FOR MATRIX ELEMENTS
We shall now study the implications of the above on matrix elements in QED and QCD.
A. On the substitution ǫµ(k) → ǫµ(k) +ckµ for the photon polarization vector in
QED
Let us consider Compton scattering as a typical QED reaction
γ(k)+e(p) → γ(k′)+e(p′) (26)
whose scattering amplitude will be of the form
(S −1) = ǫµ⋆(k′)M ǫν(k) (27)
fi µν
It is often stated that, as a trivial consequence of gauge invariance, Eq. (27) is unchanged
when the photon polarization vector ǫµ(k) is replaced by
ǫµ(k) → ǫµ(k)+ckµ (28)
where c is an arbitrary constant, leading to the fundamental relations
k′µM = M kν = 0, (29)
µν µν
and strictly analogous relations hold for QED processes involving any number of external
photons (see Eq. (2) ).
In fact relation Eq. (29) is far from a trivial consequence of gauge invariance.
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In classical electrodynamics, the invariance under ǫµ(k) → ǫµ(k) +ckµ follows from the
gauge invariance of the theory under the local gauge transformation
1
A (x) → A (x) + ∂ Λ(x) (30)
µ µ µ
e
where Λ(x) is an arbitrary function of x. Eq. (28) is the Fourier transform of Eq. (30)
when A (x) is a plane wave. This is fine in classical electrodynamics but not in QED on
µ
two counts. First, the replacement Eq. (28) follows from Eq. (30) only if Λ(x) is not just
an arbitrary function of x but a scalar quantized field linear in the photon creation and
annihilation operators. Secondly, we have to make a choice of gauge and once this is done,
the theory is no longer invariant under local gauge transformations.
We can, however, always make a gaugechoice so that global gauge invariance is preserved.
In this case, the conditions Eq. (2) satisfied by the QED matrix elements follows from three
properties:
(i) The global gauge invariance implies the existence of a conserved Noether current
Jem(x);
µ
(ii) A (x) couples to this conserved current in the QED Lagrangian;
µ
(iii) The current itself is gauge invariant.
To prove that the above statements i.e. to prove that (i)-(iii) imply invariance under
the replacement Eq. (28), we resort to the LSZ reduction formalism (see ref [2]) applied to
Compton scattering as an example. We shall derive the first of the relations in Eq. (29) in
some detail.
The amplitude M in Eq. (27) is given by
µν
M = (−ie)2 d4xd4yexp(ik′y −ikx)hp′|T [Jem(y)Jem(x)]|pi (31)
µν µ ν
Z
where T is the time ordering operator, and where
(cid:3)A (x) = eJem(x), (32)
µ µ
which is a consequence of (ii), has been used.
From Eq. (31) we get
∂
k′µM = (1/i)(−ie)2 d4xd4y exp(ik′y) exp(−ikx)hp′|T [Jem(y)Jem(x)]|pi.
µν Z (cid:20)∂y (cid:21) µ ν
µ
(33)
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Integrating by parts and discarding the surface terms we obtain
∂
k′µM = i(−ie)2 d4xd4yexp(ik′y −ikx)hp′| T [Jem(y)Jem(x)]|pi. (34)
µν Z ∂y µ ν
µ
The spatial derivatives are not influenced by the time ordering so that
∂ ∂Jem(y)
T [Jem(y)Jem(x)] = T µ Jem(x) (35)
∂y µ ν (cid:20) ∂y ν (cid:21)
i i
while the time derivative gives
∂ ∂
T [Jem(y)Jem(x)] = [θ(y −x )Jem(y)Jem(x)+θ(x −y )Jem(x)Jem(y)]
∂y 0 ν ∂y 0 0 0 ν 0 0 ν 0
0 0
∂Jem(y)
+δ(y −x )[Jem(y)Jem(x) − Jem(x)Jem(y)] + T µ Jem(x) . (36)
0 0 0 ν ν 0 (cid:20) ∂y ν (cid:21)
0
Adding together the last two equations, we get
∂
T [Jem(y)Jem(x)] = δ(y −x )[Jem(y),Jem(x)]
∂y µ ν 0 0 0 ν
µ
∂Jem(y)
+ T µ Jem(x) . (37)
(cid:20) ∂y ν (cid:21)
µ
The last term of Eq. (37) is zero for the conserved electromagnetic current. The other
term is an equal-time commutator which can be evaluated explicitly taking Jem of the form
µ
¯
ψγ ψ, yielding
µ
[Jem(y),Jem(x)] = ψ¯(x)[γ γ γ −γ γ γ ]ψ(x)δ3(x−y). (38)
µ ν x0=y0 µ 0 ν ν 0 µ
It follows that the equal time commutator in Eq. (37) vanishes i.e.
[Jem(y),Jem(x)] = 0 (39)
0 ν x0=y0
and thus the entire RHS of Eq. (37) vanishes. Substituting this result into Eq. (34) we
obtain the desired result
k′µM = 0. (40)
µν
With inessential practical complications, this generalizes to reactions with anarbitrarynum-
ber of external photons, leading to the result Eq. (2).
The seemingly miraculous vanishing of the equal time commutator in Eq. (38) is actually
due to the gauge invariance of Jem and is consistent with Eq. (23). As we shall see shortly,
µ
this is exactly the point where the analogy with QED will break down in QCD.
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B. On the substitution ǫµ(k) → ǫµ(k)+ckµ for the gluon polarization vector in QCD
Let us now turn to the QCD analogue of Compton scattering i.e. elastic gluon-quark
scattering
Gb(k) + q(p) → Ga(k′) + q(p′). (41)
where for clarity we will suppress the colour labels of the quarks. The analogue of Eq. (31)
would now read
Mab = (−ig)2 d4xd4yexp(ik′y −ikx)hp′|T [Jˆa(y)Jˆb(x)]|pi (42)
µν µ ν
Z
where the current Jˆa(x) “drives” the gluon field i.e. the analogue of Eq. (32) is
µ
(cid:3)Aa(x) = gJˆa(x). (43)
µ µ
From Eqs.(3, 13) one finds
Jˆa(x) = J˜a(x) − f ∂ν(Ab Ac ). (44)
µ µ abc ν µ
The current Jˆa(x) is conserved because a) the Noether current J˜a(x) is and b) because
µ µ
the last term in Eq. (44) being antisymmetric in the colour indices vanishes when acted on
by ∂µ.
However, the current Jˆa(x), like Ja and J˜a(x) discussed earlier, is not gauge invariant.
µ µ µ
As a consequence, if, starting from Eq. (42), one repeats the steps leading from Eq. (31) to
Eq. (37) there is now no reason that the equal time commutator in the analogue of Eq. (37)
should vanish i.e. the QCD analogue of Eq. (39) is now
[Jˆa (y), Jˆb (x)] 6= 0. (45)
0 ν x0=y0
Thus we now find that
k′µMab 6= 0 (46)
µν
and similarly that Mabkν 6= 0.
µν
However, it is also possible to approach the problem from a slightly different angle and
write for the scattering amplitude, instead of Eq.(27),
(S −1) = ǫµ⋆(k′)Mab (47)
fi µ
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where, for QCD
Mab = (−ig) d4y exp(ik′y)hp′|Jˆa(y)|p;k,bi. (48)
µ µ
Z
A simple integration by parts now shows that
k′µMab = 0. (49)
µ
purely as a consequence of the conservation of Jˆa.
µ
Since, however,
Mab = Mab ǫν(k) (50)
µ µν
Eq. (49) implies that
k′µMab ǫν(k) = 0, (51)
µν
which is, of course, a weaker result than Eq. (40).
However, ǫν(k) is, apart from the condition k · ǫ = 0 and its normalization which is
irrelevant, an arbitrary 4-vector. So Eq. (51) must hold also if ǫν(k) is replaced by kν
provided that k2 = 0 i.e.
k′µMab kν = 0, for k2 = 0. (52)
µν
It is straightforward to generalize this result to a reaction involving M incoming gluons
with momentum kµj and colour b and N outgoing gluons, momentum κνi and colour a .
j j i i
Theorem: If the QCD amplitude, analogously to the QED amplitude Eq. (1), is written
A = ǫ⋆ (κ ).......ǫ⋆ (κ )Mν1.....νN;µ1.....µM ǫ (k ).......ǫ (k ) (53)
QCD ν1 1 νN N a1.....aN;b1.....bM µ1 1 µM M
then one gets zero if any number, ≥ 1, of the ǫ (k ) and/or ǫ∗ (k ) are replaced by k
µj j νi i j,µj
and/or κ respectively, provided that all these k ’s and κ ’s, with the exception of at most
i,νi j i
one of them, satisfy k2 = 0 and κ2 = 0.
j i
V. CONCLUSIONS
The powerful condition Eq. (2), which is valid in QED and which is extremely useful
in calculating cross-sections, and determining the tensorial structure of amplitudes, unfor-
tunately does not hold in QCD. The most general analogue of this, valid in QCD, is our
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