Chiral anomaly in soft collinear effective theory
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Waalewijn, Wouter J. “Chiral anomaly in soft collinear effective
theory.” Physical Review D 79.3 (2009): 036003. (C) 2010 The
American Physical Society.
As Published
http://dx.doi.org/10.1103/PhysRevD.79.036003
Publisher
American Physical Society
Version
Final published version
Accessed
Thu May 26 22:10:00 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/51029
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
PHYSICAL REVIEW D 79, 036003 (2009)
Chiral anomaly in soft collinear effective theory
Wouter J. Waalewijn
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 10 September 2008; published 5 February 2009)
Anomalies have infrared and ultraviolet ingredients, and are often realized in effective theories in a
nontrivial way. We study the chiral anomaly in soft collinear effective theory (SCET), where the anomaly
equation has terms contributing at different orders in the power expansion. The chiral anomaly equations
in SCET are computed up to next-to-next-to-leading order in the power counting with external collinear
and/or ultrasoft gluons. We do this by expanding the QCD anomaly equation, using the tree level (leading
order in s ) relations between QCD and SCET fields. The validity of this correspondence between the
anomaly equations is confirmed by direct computation of the one-loop diagrams in SCET.
DOI: 10.1103/PhysRevD.79.036003
PACS numbers: 11.30.Rd, 12.38.Aw
I. INTRODUCTION
In calculations with a hierarchy of scales it is often
useful to work with an effective field theory, in which
ultraviolet degrees of freedom have been integrated out.
Examples of this are heavy quark effective theory and soft
collinear effective theory (SCET). Or as a more well
known example, integrating out a heavy quark.
Anomalies [1,2] come from ultraviolet divergences, but
they are also an infrared effect in the sense that only
massless particles contribute [3,4]. If masses are generated
through a Higgs mechanism as in the standard model, fields
are massless above the symmetry breaking scale and can
contribute to anomalies too.
Because of the infrared nature of anomalies one would
expect effective field theories to reproduce the anomalies
of the full theory in some way, but this is not always a
trivial matter. As an illustration we consider the case of
integrating out a heavy quark doublet.1 Simply removing it
would spoil the anomaly cancellation for the gauge symmetries. D’Hoker and Farhi show in a detailed analysis that
the effective action contains additional Wess-Zumino
terms and terms involving the Goldstone-Wilczek current
that restore the gauge symmetry [5,6].
Another example is the axial anomaly in chiral perturbation theory. By itself chiral perturbation theory would
not include reactions such as 0 ! or KK !
þ 0 . They do not show up because of the symmetry
M ! M, where M is the matrix of Goldstone bosons,
which is not a symmetry of QCD. Again one needs to add
Wess-Zumino terms and additional terms that couple to the
gauge fields [7,8].
The goal of this paper is to study how the chiral anomaly
is realized in SCET. SCET has been introduced to describe
the dynamics of energetic light hadrons and provides a
systematic way to separate the hard, collinear, and soft
scales [9–12]. It captures the long-distance physics in
terms of collinear, soft, and ultrasoft (usoft) fields. The
short-distance physics is absorbed into Wilson coefficients.
So in SCET a quark field gets replaced by these different
fields, which can each run around in loops. Currents in
SCET often generate "12 divergences at one loop that are
removed by renormalization. We will see that such divergences show up in individual SCET anomaly diagrams, but
cancel when they are summed. Also in SCET there are
vertices with two quark fields and more than one gluon,
which lead to nontriangle anomaly diagrams. These graphs
turn out to be important.
Which SCET fields one needs to consider depends on
the physical process one has in mind. We will be looking at
SCETI with one collinear direction n . The fields in this
theory are collinear quarks n and gluons A
n and usoft
quarks qus and gluons A
.
The
momenta
of
the
collinear
us
2
fields scale like p
¼
ðn
p;
n
p;
p
Þ
Qð
;
1;
Þ and
c
?
2
for usoft fields as p
us Q . Here Q is the hard scale, the SCET expansion parameter, and n and n are lightcone basis vectors satisfying n2 ¼ n 2 ¼ 0 and n n ¼ 2.
One can, for example, take n ¼ ð1; 0; 0; 1Þ and n ¼
ð1; 0; 0; 1Þ. For a typical process 2 ¼ QCD =Q. We
will also need the power counting of the fields, which are
listed in Table I.
To get a systematic expansion in the power counting,
collinear fields have both a label momentum p containing
the collinear momentum and a (usoft) coordinate x,
n;p ðxÞ. The label momenta are picked out by the label
operator P , e.g. P ? n;p ¼ p? n;p . The collinear covariant
derivatives are then defined as
in Dc ¼ n P þ gn An;q ;
in D ¼ in @ þ gn An;q þ gn Aus ;
iD
us ¼ i@ þ gAus :
In the standard model only the top would get integrated out
before W and Z do. For simplicity we assume the existence of a
heavy electroweak doublet.
(1)
and the usoft covariant derivative as
1
1550-7998= 2009=79(3)=036003(11)
?
iD?
¼ P
c
? þ gAn;q ;
(2)
Note that these covariant derivatives are each homogenous
in the power counting. We are now ready to write down the
036003-1
Ó 2009 The American Physical Society
WOUTER J. WAALEWIJN
PHYSICAL REVIEW D 79, 036003 (2009)
TABLE I. Power counting of the various fields and operators in SCET.
Collinear
Operator
Power counting
n
n An
0
A?
n
Usoft
n An
2
qus
3
leading order Lagrangian for a collinear quark n [10],
n ðxÞ in D þ i6Dc
¼
L ð0Þ
?
1
6n
c
ðxÞ; (3)
i6
D
?
c
in D
2 n
where it is understood that the (suppressed) label momenta
are summed over and that the label momenta of each term
in L are conserved.
This paper is organized as follows: We start in Sec. II A
by matching the chiral anomaly equation in QCD onto
SCET, using the tree level relations between fields. For
simplicity we first restrict ourselves to leading order (LO)
in the power counting, postponing the matching up to nextto-next-to-leading order (NNLO) to Sec. III A. Whenever
we speak of e.g. LO and next-to-leading order (NLO) in
this paper, we will always be referring to the order in the
power expansion and not to radiative corrections. No
contributions are expected beyond one loop due to the
Adler and Bardeen theorem [13].
We then verify these SCET anomaly equations by computing the chiral anomaly from one-loop graphs in SCET.
This is done at LO in Secs. II B, II C, and II D, NLO in
Sec. III B, and NNLO in Sec. IV. At LO only collinear
fields play a role and so we expect agreement, since SCET
with only collinear fields is just a boosted version of QCD.
Things become more interesting at higher orders when
diagrams with both collinear and usoft fields show up,
but we find that the derived SCET anomaly equations are
correct. We end with some concluding remarks in Sec. V
and list the full SCET anomaly equations up to NLO. The
appendix contains a table of the loop integrals we used.
II. THE SCET ANOMALY AT LO
A. Matching the QCD anomaly onto SCET
In this section we will calculate the chiral anomaly in
SCET at LO. We start by expanding the left- and right-hand
side of the QCD anomaly equation
@ J5 ¼ g2
tr½F F 162
(4)
5 . We use the
in the power counting, where J5 ¼ tree level relations between fields, effectively matching
QCD onto SCET at tree level. In the next subsections we
will explicitly check that the SCET anomaly equation we
find holds at one loop.
We introduce the following notation for the left- and
right-hand side of Eq. (4),
Covariant derivatives
A
us
2
iD?
c
in Dc
0
J @ J5 ;
F in D
2
iD
us
2
g2
tr½F F ; (5)
162
and write their expansions as
J ¼ J ð4Þ þ J ð5Þ þ J ð6Þ þ . . . ;
F ¼ F ð4Þ þ F ð5Þ þ F ð6Þ þ . . . ;
(6)
where the order in is indicated in brackets. When calculating the one-loop anomaly diagrams, we will (as usual)
restrict ourselves to two outgoing gluons. Gauge invariance
then determines anomaly matrix elements for more gluons.
Let p, q be the momenta of these two gluons. For simplicity we assume that they have no component in the ?
direction, p? ¼ q? ¼ 0. We will also assume that the
gluons are ? polarized. The matrix element of J is then
given by
hggjJ j0i ¼ 12in ðp þ qÞhggjn J5 j0i
þ 12in ðp þ qÞhggjn J5 j0i:
(7)
The tree level relation between the QCD and SCET (collinear) field is [10]
1
6n
i6Dc þ . . . ;
¼ n þ
|{z} in Dc ? 2 n
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
(8)
2
with the power counting as indicated. This leads to
n J5ð2Þ ¼ n 6n 5 n :
(9)
For n J5 we get an 6n. But 6nn ¼ 0, so the lowest nonvanishing contribution is at Oð4 Þ:
y
1
6n
1
6n
i6Dc? n 0 6n5
i6Dc? n
n J5ð4Þ ¼
in Dc
2
in Dc
2
c
1
1
¼ n i6D?
6n 5
i6Dc? n :
(10)
i
n
D
c
in D
c
However, it is important to note that n J5ð4Þ contributes at
the same order as the n J5 term in the anomaly equation
because of the power counting for collinear momenta,
hggjJ ð4Þ j0i ¼ 12in ðp þ qÞhggjn J5ð2Þ j0i
þ 12in ðp þ qÞhggjn J5ð4Þ j0i:
(11)
We will first study these terms separately by imposing
n ðp þ qÞ ¼ 0 or n ðp þ qÞ ¼ 0 in the following subsections. With these additional assumptions we have to
036003-2
CHIRAL ANOMALY IN SOFT COLLINEAR EFFECTIVE THEORY
PHYSICAL REVIEW D 79, 036003 (2009)
keep gluons off shell, p , q Þ 0, because everything
would otherwise trivially vanish. At the end we will drop
these additional assumptions, effectively combining the
two cases.
Now we consider the right-hand side of the anomaly
equation. With our assumptions the important terms are
Having derived the explicit form of the SCET anomaly
equation at LO
in D ¼ in @ þ gn An;q þ gn Aus ¼ |ffl{zffl}
in @ þ . . . ;
we will verify this result by calculating the one-loop
diagrams.
2
2
1
?
n ¼ 18i tr½5 6n 6n?
?
2 n
:
J
ð4Þ
¼ F ð4Þ ;
(17)
(18)
2
in D ¼ in Dc þ Win Dus
Wy
¼ |ffl{zffl}
n P þ |ffl{zffl}
in @ þ . . . ;
2
1
?
?
?
y
iD
þ WiD?
us W ¼ gAn;q þ gAus þ . . . :
? ¼ iDc
|ffl{zffl} |ffl{zffl}
B. LO with n ðp þ qÞ ¼ 0
We start by considering the simplest case, namely n ðp þ qÞ ¼ 0 for the momenta of the external gluons. The
axial current is then given by
2
hggjJ ð4Þ j0i ¼ 12in ðp þ qÞhggjn J5ð2Þ j0i
(12)
W is the collinear Wilson line that only contains the n An
component of the gluon field
W ¼1g
1
n An þ . . .
n P
(13)
and can be dropped because we are only looking at ?
-polarized gluons. Writing
F F ¼ 4
iD iD iD iD ;
g2
(14)
the only contribution we need to consider at Oð4 Þ is
F ð4Þ ¼ n n
g2
4
tr½in @gA?
2 n;p
2
2 2
g
16
? P gA?
n P gA?
n;p
n;q in @gAn;q n
?
?
P gA?
n P gA?
n;p in @gAn;q þ n
n;q in @gAn;p :
(15)
The corresponding matrix element is given by
g2
ðn p n q n q n pÞ
82
AB ?
A ðpÞB ðqÞ;
hggjF ð4Þ j0i ¼ (16)
where A ðpÞ, B ðqÞ are the polarization vectors of the
gluons and we assume the generators are normalized as
tr½T A T B ¼ 12 AB . The two-dimensional ? is defined as
¼ 12in ðp þ qÞhggjn 6n 5 n j0i;
(19)
and there are two diagrams that contribute at this order,
shown in Fig. 1. As mentioned before, taking the gluons on
shell would make everything vanish because of n ðp þ
qÞ ¼ 0. We therefore keep the gluons off shell, which
generally takes care of any IR divergences as well.
Because we assume n ðp þ qÞ ¼ 0, we still need to be
careful when combining denominators as ðl þ pÞ2 and ðl qÞ2 . We will elaborate on this for the bubble diagram.
In dimensional regularization with d ¼ 4 2", the triangle diagram is given by
Z
1
6n n ðl qÞ
AT ¼ in ðp þ qÞð1Þ ‘l tr 6n 5 i
2
2 ðl qÞ2
?
6 l?
6 l? ?
6n 6n n l
i
þ
igT B
n l
n ðl qÞ 2 2 l2
?6 l?
6 l? ? 6n 6n n ðl þ pÞ
A
i
þ
igT
n ðl þ pÞ
n l 2 2 ðl þ pÞ2
þ ðp; ; AÞ $ ðq; ; BÞ;
(20)
R
R d
where ‘l ¼ d lð2Þd . We will always suppress polarization vectors. To treat 5 in dimensional regularization
correctly requires some care, for example, using the
’t Hooft-Veltman scheme. We avoid such complications
by never (anti)commuting with 5 in our calculations.
Since the external momenta have no ? component, we
can immediately replace l? l? ! 1 l2? g
? , where ¼ d 2 ¼ 2 2". We then use
FIG. 1. Triangle and bubble diagram contributing to the anomaly at LO, for n ðp þ qÞ ¼ 0.
036003-3
WOUTER J. WAALEWIJN
?
? PHYSICAL REVIEW D 79, 036003 (2009)
? ?
¼ ;
? ¼
? ? ?
?
? ¼ 4g
ð 2Þ? ¼ 2"? ;
ð4 Þ? ? :
(21)
This leads to
AT ¼ ig2 n ðp þ qÞ
AB ?
ðI1 þ "I2 þ "I3
ð1 þ 2"ÞI4 Þ þ ðp; ; AÞ $ ðq; ; BÞ
¼
g2
n ðp þ qÞn p
82
AB ?
;
(22)
where
I1 ¼
I2 ¼
I3 ¼
I4 ¼
Z
Z
‘l
n ðl qÞn ðl þ pÞl2?
;
n lðl qÞ2 l2 ðl þ pÞ2
‘l
n ðl qÞl2?
;
ðl qÞ2 l2 ðl þ pÞ2
Z
n ðl þ pÞl2?
‘l
;
ðl qÞ2 l2 ðl þ pÞ2
Z
n l l2?
‘l
:
ðl qÞ2 l2 ðl þ pÞ2
! 0 and " ! 0 matters, and we should first take " ! 0
with Þ 0 to regulate any IR divergences. Proceeding
along this way, we still find AB ¼ 0.
We now add AT and AB and compare with hggjF ð4Þ j0i as
given in (16). They agree. This was expected since at this
order only collinear fields are involved, which is just a
boosted version of QCD.
C. LO with n ðp þ qÞ ¼ 0
We move on to the next case: n ðp þ qÞ ¼ 0. This time
the relevant term of the axial current comes from n J5ð4Þ :
c
1
1
hggjJ ð4Þ j0i ¼ in ðp þ qÞhggj n i6D?
2
in D
c
1
5
n
i6Dc j0i:
in Dc ? n
(23)
We computed these integrals using an appropriate
Feynman parametrization. All the necessary integrals can
be found in the appendix.
The contribution from the bubble diagram (right panel of
Fig. 1) is
Z
1
6n n ðl qÞ
AB ¼ in ðp þ qÞð1Þ ‘l tr 6n 5 i
2
2 ðl qÞ2
A B ? ?
T T T B T A ? ? 6n 6n n ðl þ pÞ
i
þ
ig2
2 2 ðl þ pÞ2
n ðl þ p qÞ
n l
It can be part of a diagram in various different ways; for
example, a gluon can now come out of the current. These
different ways are pictured in Fig. 2, and the corresponding
expressions are
n ðl qÞn ðl þ pÞ
1
1
‘l
:
n l n ðl þ p qÞ ðl qÞ2 ðl þ pÞ2
By sending l ! l p þ q we see that the two terms are
the same (giving a factor of 2, rather than canceling each
other). We do not have a second contribution from
ðp; ; AÞ $ ðq; ; BÞ for this diagram. As mentioned before, we need to be careful in dealing with IR divergences
here. Writing ¼ n ðp þ qÞ ! 0, we encounter "
when doing this integral. Obviously the order of taking
FIG. 2.
6 l?
6 l?
6n 5
;
n ðl þ pÞ
n ðl qÞ n
ð2Þ n
6 l?
gT B ?
6n 5
;
n ðl þ pÞ
n ðl qÞ n
(26)
gT A ?
gT B ?
6n 5
:
ð4Þ n
n ðl þ pÞ
n ðl qÞ n
(24)
ð1Þ n
gT A ?
6 l?
6n 5
;
ð3Þ n
n ðl þ pÞ
n ðl qÞ n
¼ ig2 n ðp þ qÞ AB ?
Z
(25)
First of all we get a triangle [Fig. 3(a)] and bubble
diagram [Fig. 3(b)] coming from (1)
A~A þ A~B ¼ ig2 n ðp þ qÞ AB ? ðð1 þ 2"ÞI~1 "I~2
"I~3 I~4 ð1 þ 2"ÞI~5 Þ
þ ðp; ; AÞ $ ðq; ; BÞ;
where
n J5ð4Þ can be inserted into diagrams in various ways.
036003-4
(27)
CHIRAL ANOMALY IN SOFT COLLINEAR EFFECTIVE THEORY
PHYSICAL REVIEW D 79, 036003 (2009)
FIG. 3. Diagrams contributing to the anomaly at LO for n ðp þ qÞ ¼ 0.
I~ 1 ¼
I~2 ¼
I~3 ¼
I~4 ¼
I~5 ¼
Z
Z
Z
Z
Z
‘l
‘l
‘l
hggjJ ð4Þ j0i ¼ A~A þ A~B þ A~C1 þ A~C2 þ A~D
l4?
;
n lðl qÞ2 l2 ðl þ pÞ2
¼
l4?
n ðl þ pÞðl qÞ2 l2 ðl þ pÞ
;
2
l4?
;
n ðl qÞðl qÞ2 l2 ðl þ pÞ2
(28)
l2?
:
n lðl qÞ2 ðl þ pÞ2
We also have diagrams [Fig. 3(c1) with (2) and Fig. 3(c2)
with (3)] where one gluon comes out of the current
A~C1 þ A~C2 ¼ 2ig2 n ðp þ qÞ
I~ 6 ¼
I~7 ¼
Z
n l
‘l
;
n ðl qÞn ðl þ pÞl2 ðl þ pÞ2
n p ¼ 0;
(29)
‘l
n ðl qÞl ðl þ pÞ2
hggjJ ð4Þ j0i ¼
Finally there is also a diagram [Fig. 3(d)] where both
gluons come out of the current, which involves (4). This
turns out to be zero, A~D ¼ 0.
We simplify the remaining integrals using
AB ?
¼ hggjF ð4Þ j0i:
Thus the SCET anomaly equation at LO J ð4Þ ¼ F ð4Þ ,
which we derived by expanding, is correct at one loop. In
the previous sections the bubble diagrams in Figs. 3(c1)
and 3(c2) contributed, but the bubble diagram with the two
gluon vertex in Figs. 1 and 3(b) turned out to be zero.
However, in this general case with on-shell momenta we do
get a genuine contribution from them.
III. THE SCET ANOMALY AT NLO
(31)
where l2? ¼ l~2? if g ¼ diagðþ Þ. For example,
Z l2?
n ðl þ pÞl2?
I~2 ¼ ‘l
n ðl þ pÞðl qÞ2 l2 ðl qÞ2 l2 ðl þ pÞ2
(32)
¼ I~2A þ I~2B :
(34)
(35)
l2? ¼ l2 n l n l ¼ ðl qÞ2 n ðl qÞn ðl qÞ
Doing the math, we find
g2
n p n q
82
(30)
:
¼ ðl þ pÞ2 n ðl þ pÞn ðl þ pÞ;
n q ¼ 0:
After repeating the above analysis and performing the
Feynman integrals we find
l2?
l2?
2
(33)
Finally we now drop the additional assumptions [e.g. n ðp þ qÞ ¼ 0] and study the general case. These assumptions allowed us to look at one term of hggjJ ð4Þ j0i in (11)
at the time, so we basically have to add the anomalies of
Secs. II B and II C. The only other place these assumptions
were used is in simplifying the Feynman integrals. We can
now take the gluons on shell, and we will assume
ðI~6 þ "I~7 Þ
with
Z
:
Again this agrees with hggjF ð4Þ j0i, as expected because
the calculation so far only involved collinear fields.
AB ?
þ ðp; ; AÞ $ ðq; ; BÞ;
AB ?
D. LO general case
n l l4?
‘l
;
n ðl qÞn ðl þ pÞðl qÞ2 l2 ðl þ pÞ2
‘l
g2
n ðp þ qÞn p
82
A. Matching the QCD anomaly onto SCET at higher
orders
We will now move on to calculating the chiral anomaly
in SCET beyond LO in the power counting, by taking into
account the higher order terms in Eq. (6). Again we start by
matching the QCD anomaly equation onto SCET at tree
level. The derivation is similar to that in Sec. II A, but this
time we need the tree level relation between the QCD and
036003-5
WOUTER J. WAALEWIJN
PHYSICAL REVIEW D 79, 036003 (2009)
SCET fields to higher order in . Using the result from [14]
we have
1
6n
¼ n þ Wqus þ
i6Dc? n
|{z} |ffl{zffl} in Dc
2
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
3
volving the leading current and insertion(s) of subleading
Lagrangians. To be specific, the SCET anomaly equation at
NLO and NNLO reads
T fJ ð4Þ iLð1Þ g þ J ð5Þ ¼ TfF ð4Þ iLð1Þ g þ F ð5Þ ;
(45)
2
1
6n y
i6Dus
W n þ . . . :
þW
?
n P
2
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
(36)
¼ TfF ð4Þ12iLð1Þ iLð1Þ g þ TfF ð4Þ iLð2Þ g þ TfF ð5Þ iLð1Þ g
3
þ F ð6Þ ;
We find
n J5ð2Þ
¼ n 6n 5 n ;
(37)
n J5ð3Þ ¼ 0;
(38)
n J5ð4Þ ¼ n 6n 5 Wqus þ H:c:;
(39)
and
1
c
n J5ð4Þ ¼ n i6D?
c
n J5ð5Þ ¼ 2n i6D?
in Dc
TfJ ð4Þ12iLð1Þ iLð1Þ g þ TfJ ð4Þ iLð2Þ g þ TfJ ð5Þ iLð1Þ g þ J ð6Þ
6n 5
1
i6Dc ;
in Dc ? n
(40)
1
in Dc
1
6n y
i6Dus
W
5 Wqus þ W
þ H:c:
n
n P ? 2
(41)
Expanding the other side of the QCD anomaly equation
gives
g2
ðn p n q n q n pÞ
82
AB ?
A ðpÞB ðqÞ;
R
where TfJ ð4Þ iLð1Þ g ¼ d4 xTJ ð4Þ ð0ÞiLð1Þ ðxÞ, etc. Lð1Þ and
Lð2Þ refer to all terms in the quark and gluon action of the
respective order.
Let us consider the right-hand side of these equations.
Calculating the anomaly at one loop corresponds to tree
level diagrams for F ðiÞ . The only nontrivial diagrams come
from insertions on external gluon lines. If one would
include insertions on external gluon lines, one would get
contributions on both sides of these equations, which are
equal because these insertions are not part of the loop. We
will not consider such diagrams, and so the right-hand side
of (45) and (46) reduces to F ð5Þ and F ð6Þ .
We list the subleading Lagrangians we need below [14–
20]:
1
6n
W y i6D?
þ ðn i6D?
c
c WÞ
n P
2 n
1
y 6n
i6D?
W
;
(47)
n P us
2 n
?
Lð1Þ
¼ ðn WÞi6Dus
1
6n y
i6D?
ðW n Þ þ ðn i6D?
us
c WÞ
n P
2
1
6n
in Dus ðW y i6D?
(48)
c n Þ;
2
2
ðn P Þ
?
Lð2Þ
¼ ðn WÞi6Dus
hggjF ð4Þ j0i ¼ hggjF ð5Þ j0i ¼
g2
n p n q
82
AB ?
(42)
A ðpÞB ðqÞ; (43)
g2
hggjF ð6Þ j0i ¼ 2 ðn p n qr n q n pr Þ
8
AB ?
A ðpÞB ðqÞ;
(46)
Lð1Þ
q ¼ n
1
igB
6 ?
c Wqus þ H:c:;
in Dc
(49)
Dc ; i6D?
where igB
6 ?
c ¼ ½in
c .
B. NLO with n ðp þ qÞ ¼ 0
(44)
where the subscript in pr denotes residual momentum,
2
p
r . We already checked at LO that the SCET anomaly equation found this way is correct, J ð4Þ ¼ F ð4Þ . In the
remainder of this paper we will do the NLO and NNLO
calculations.
When we go beyond LO, we obviously get contributions
to the anomaly from higher order currents such as n J5ð5Þ .
However, there are also contributions from diagrams in-
At NLO things become more interesting, because both
collinear and ultrasoft fields are involved. To get the right
power counting one of the outgoing gluons must be collinear and the other usoft. The only additional assumption
that makes sense here is
n ðp þ qÞ ¼ 0;
because n p 1 2 n q.
The axial current gives rise to
036003-6
(50)
CHIRAL ANOMALY IN SOFT COLLINEAR EFFECTIVE THEORY
FIG. 4.
PHYSICAL REVIEW D 79, 036003 (2009)
Diagrams from n J5ð4Þ contributing to the NLO anomaly. The usoft gluon vertex is Lð1Þ .
1
hggjTfJ ð4Þ iLð1Þ g þ J ð5Þ j0i ¼ in ðp þ qÞhggjðTfn J5ð4Þ iLð1Þ g þ n J5ð5Þ Þj0i
2
c
1
1
1
6n 5
i6Dc? n iLð1Þ
¼ in ðp þ qÞhggj T n i6D?
2
in Dc
in Dc
c
1
1
6n y
i6Dus
W
þ 2n i6D?
5 Wqus þ W
þ
H:c:
j0i:
n
n P ? 2
in Dc
Said in words, we get contributions from diagrams with
n J5ð4Þ , one Lð1Þ vertex and Lð0Þ vertices, and diagrams
with n J5ð5Þ and only Lð0Þ vertices. There is no term in Lð0Þ
for a ? -polarized usoft gluon, so the usoft gluon has to
come out of the Lð1Þ vertex or the n J5ð5Þ current. This
leads to the diagrams in Figs. 4 and 5. Obviously we do not
have ðp; ; AÞ $ ðq; ; BÞ terms in this case, since one
gluon is collinear and one usoft. As it turns out, we already
computed all the necessary integrals in the LO n ðp þ
qÞ ¼ 0 calculation. Here we have the additional simplification that we can generally drop n q for the usoft momentum q, because n q
n p; n l.
The various diagrams lead to contributions
NLO
2
ANLO
p
A1 þ AA2 ¼ 2ig n
AB ?
NLO
2 p
ANLO
B1 þ AB2 ¼ 2ig n
AB ?
NLO
2
ANLO
p
C1 þ AC2 ¼ 2ig n
AB ?
ð1 þ "ÞðI~1 I~2 Þ; (52)
ð1 þ "ÞI~2A ;
(53)
ðI~6 þ "I~7 Þ;
(54)
NLO
ANLO
D1 þ AD2 ¼ 0;
(55)
where I~2A was defined in (32). Adding the pieces we get
(51)
NLO
NLO
hggjTfJ ð4Þ iLð1Þ g þ J ð5Þ j0i ¼ ANLO
A1 þ AA2 þ AB1
NLO
NLO
þ ANLO
B2 þ AC1 þ AC2
NLO
þ ANLO
D1 þ AD2
g2
n p n q
82
¼ hggjF ð5Þ j0i:
¼
AB ?
(56)
So the SCET anomaly equation is correct at NLO too.
IV. THE SCET ANOMALY AT NNLO
We will pursue our calculation to one higher order in .
At this order the loop momentum can have either collinear
or ultrasoft scaling.
Let us start by observing that there is a freedom in what
you call label and residual momentum. Consequently
SCET should be invariant under (for example) the following reparametrization:
n pc ! n pc þ n k;
n pr ! n pr n k; (57)
where pc is the (collinear) label momentum, pr the (usoft)
residual momentum, and k some constant usoft momentum. This ties together F ð4Þ and F ð6Þ , basically predicting
the latter. It is easy to check that (42) + (44) satisfies this.
We will also calculate the loop diagrams, to verify the
SCET anomaly equation at this order. Since both gluons
are collinear this time, either n ðp þ qÞ ¼ 0 or n ðp þ
qÞ ¼ 0 can be imposed. We will restrict ourselves to the
FIG. 5. Diagrams coming from n J5ð5Þ contributing to the NLO anomaly.
036003-7
WOUTER J. WAALEWIJN
PHYSICAL REVIEW D 79, 036003 (2009)
former. As can be seen from the expression for F ð6Þ in (44),
we need to keep the residual momentum components of the
external momenta
to get a nonvanishing expression. Our assumptions are
n ðpc þ qc Þ ¼ 0;
n ðpr þ qr Þ ¼ 0:
(59)
¼ 12in ðp þ qÞhggjðTfn J5ð2Þ 12iLð1Þ iLð1Þ g þ Tfn J5ð2Þ iLð2Þ g þ Tfn J5ð3Þ iLð1Þ g þ n J5ð4Þ Þj0i
¼ 1in ðp þ qÞhggjðTfn 6n 5 n 1iLð1Þ iLð1Þ g þ Tfn 6n 5 n iLð2Þ g þ 0 þ ðn 6n 5 Wqus þ H:c:ÞÞj0i:
(60)
p ¼ pc þ pr ¼ ð0; n pc ; p?
c ¼ 0Þ
þ ðn pr ; n pr ; p?
r ¼ 0Þ
(58)
The axial current gives
hggjTfJ ð4Þ12iLð1Þ iLð1Þ g þ TfJ ð4Þ iLð2Þ g þ TfJ ð5Þ iLð1Þ g þ J ð6Þ j0i
2
2
We cannot make a diagram using n J5ð4Þ and only Lð0Þ
vertices, since the n J5ð4Þ current has a collinear and a usoft
quark coming out of it. Therefore the only contributions
come from n J5ð2Þ .
There is a bit of a technical issue with loop integrals that
we need to discuss here. At LO and NLO we had collinear
loops
R we combined label and residual momenta
P R for which
‘l
!
‘l. This was possible because these diar
lc
grams only involved n lc , l?
c , and n lr . Now we can
get diagrams that also depend on other components of
the residual loop momenta e.g. l?
r . This requires us to do
R
‘l?
r ð. . .Þ separately, which yields zero for a collinear loop
since all factors of l?
r are in the numerator. This is not the
case when the loop momentum is usoft, because then l?
r
appears in the denominator as well. Diagrams with collinear loops
R still have nonzero contributions,
P may
P R for example,
‘lr n ðlr þ pr Þð. . .Þ ¼ n pr lc ‘lr ð. . .Þ.
lc
?
Because we take the external p?
r ¼ qr ¼ 0, many diagrams vanish immediately.
First of all we get the same diagrams as at LO, but with
an extra Lð2Þ insertion [Figs. 6(a1), 6(a2), 6(a3), 6(b1), and
6(b2)]:
ANNLO
þ ANNLO
þ ANNLO
þ ANNLO
þ ANNLO
A1
A2
A3
B1
B2
¼ 2ig2 n ðp þ qÞn pr
AB ?
ðI^1 þ 2"I^2
ð1 þ 2"ÞI^4 I^5 Þ þ ðp; ; AÞ $ ðq; ; BÞ; (61)
where
I^ 1 ¼
I^2 ¼
I^4 ¼
I^5 ¼
Z
Z
Z
Z
‘l
‘l
n ðl þ pÞl4?
;
n lðl qÞ2 l2 ðl þ pÞ4
l4?
;
ðl qÞ2 l2 ðl þ pÞ4
n l l4?
‘l
;
n ðl þ pÞðl qÞ2 l2 ðl þ pÞ4
‘l
n ðl þ pÞl2?
:
n lðl qÞ2 ðl þ pÞ4
FIG. 6. Diagrams contributing to the anomaly at NNLO for n ðp þ qÞ ¼ 0.
036003-8
(62)
CHIRAL ANOMALY IN SOFT COLLINEAR EFFECTIVE THEORY
We also get a triangle [Figs. 6(c1) and 6(c2)] where one of
the vertices is Lð2Þ and a bubble diagram [Fig. 6(d)]with an
Lð2Þ vertex:
ANNLO
þ ANNLO
þ ANNLO
D
C1
C2
¼ 2ig2 n ðp þ qÞn pr
PHYSICAL REVIEW D 79, 036003 (2009)
In a naive calculation where you forget to treat l as a
residual momentum that is multipole expanded in the collinear propagators, you would misleadingly find a nonzero
result for this diagram.
Rewriting the loop integrals using (31) and
AB ?
ð"I^8 ð1 þ 2"ÞI^9
þ ðp; ; AÞ $ ðq; ; BÞ þ I^10 Þ;
d
þ pÞ
nðl
1
¼
;
dðn pÞ ðl þ pÞ2
ðl þ pÞ4
(63)
with
I^ 8 ¼
I^9 ¼
Z
Z
‘l
ðl l2?
2
qÞ l2 ðl
þ pÞ2
;
n l l2?
‘l
;
n ðl þ pÞðl qÞ2 l2 ðl þ pÞ2
Z
d
1
nðl þ pÞ
¼
;
2
dðn pÞ ðl þ pÞ
ðl þ pÞ4
(64)
we get
ðn ðl þ pÞÞ2
:
(65)
n lðl qÞ2 ðl þ pÞ2
Finally there is also a mixed collinear-usoft diagram [Fig. 6
(e)] with a usoft loop momentum:
Z
n q
¼ ig2 n ðp þ qÞ AB ? ‘l
ANNLO
E
n q n ðl qÞ
n p
nl
þ ðp; ; AÞ $ ðq; ; BÞ ¼ 0:
2
l n p n ðl þ pÞ
I^ 10 ¼
(67)
‘l
(66)
I^ 1 ¼ I^2 þ I^5 I^8 n p
d
I^ ;
dðn pÞ 8
(68)
I^ 4 ¼ I^2 I^8 þ I^9 n p
d
I^ :
dðn pÞ 8
(69)
This reduces the calculation to
1
þ ANNLO
þ . . . þ ANNLO
þ ANNLO
hggjT J ð4Þ iLð1Þ iLð1Þ þ TfJ ð4Þ iLð2Þ g þ TfJ ð5Þ iLð1Þ g þ J ð6Þ j0i ¼ ANNLO
D
E
A1
A2
2
d
d
1
I^8 þ ð1 þ 2"Þn p
I^8 I^10 þ ðp; ; AÞ $ ðq; ; BÞ:
¼ 2ig2 n ðp þ qÞn pr AB ? "I^8 n p
dðn pÞ
dðn pÞ
2
(70)
Working out I^10 , one finds zero. The remainder is fairly
easy to evaluate if you take ðp; ; AÞ $ ðq; ; BÞ before
performing the integral over Feynman parameters. We find
aly equations up to NLO read2
J
1 ð1Þ ð1Þ
ð4Þ
iL iL
hggjT J
þ TfJ ð4Þ iLð2Þ g þ TfJ ð5Þ iLð1Þ g
2
þ J ð6Þ j0i
¼
2
ð4Þ
¼ F ð4Þ ;
T fJ ð4Þ iLð1Þ g þ J ð5Þ ¼ TfF ð4Þ iLð1Þ g þ F ð5Þ ;
(72)
(73)
where
g
n ðp þ qÞn pr
82
AB ?
¼ hggjF ð6Þ j0i:
(71)
Once again we see by direct computation that the SCET
anomaly equation is correct.
V. CONCLUSIONS
We derived the chiral anomaly equations in SCET up to
NNLO by matching the QCD anomaly equation onto
SCET, using the tree level relations between fields in
QCD and SCET. Gathering all the expressions, the anom-
1
J ð4Þ ¼ n @½n 6n 5 n 2
c
1
1
1
c
þ n @ n i6D?
6n 5
i6D 2
in Dc ? n
in Dc
1
c 6n
þ @?
i6
D
þ
H:c:
;
(74)
n ? 5
in Dc ? 2 n
2
We only studied some terms of the NNLO anomaly equations
and would need the higher order terms in (36) to write down the
full expression at NNLO.
036003-9
WOUTER J. WAALEWIJN
PHYSICAL REVIEW D 79, 036003 (2009)
c
1
1
J ð5Þ ¼ n @ 2n i6D?
2
in Dc
1
6n y
us
i6D
W n
5 Wqus þ W
n P ? 2
1
6n y
i6Dus
W
þ @?
n
n ? 5 Wqus þ W
n P ? 2
þ H:c:;
(75)
1 ?
?
tr½in D in Dc iD?
c iDc
162
permutations;
(76)
1 ?
?
y
tr½in D in Dc iD?
c ðWiDus W Þ
82
permutations:
(77)
F ð4Þ ¼
F ð5Þ ¼
y
Win Dus W Þ in (36) and expanding. If our result extends
to higher orders, then certain matrix elements of the currents will have to cancel each other beyond 8 .
ACKNOWLEDGMENTS
I would like to thank Iain Stewart for many good discussions and helpful suggestions regarding this work. This
work is supported by the DOE Office of Nuclear Physics
under Grant No. DE-FG02-94ER40818.
APPENDIX: USEFUL INTEGRALS
Here the sum over permutations means the sum of all
different orderings of the operators multiplied by the sign
of the permutation. Note that F ð4Þ has an additional factor
of 12 compared to F ð5Þ , which comes from the redundant
3
ð1Þ
interchange of the iD?
can be
c . The relevant terms of L
found at the end of Sec. III A.
By explicitly calculating one-loop graphs we verified
these equations. Although we have not checked every
possible term in the operator relations, our work suggests
their correctness. At LO the anomaly only involves collinear fields, and SCET with only collinear fields is just a
boosted version of QCD, so we expected agreement. It was
not a priori clear how the anomaly equations would work
out beyond that order.
Of course there are still higher orders to consider, which
might be worth pursuing if one has in mind applying the
anomaly in some specific process. F contains terms up to
order 8 , whereas J has terms of arbitrary high order. This
can be seen from replacing 1=ðin Dc Þ ! 1=ðin Dc þ
3
In our derivation of F ð4Þ in (16) we did not have this factor 12 ,
because we were distinguishing the outgoing gluons.
[1] S. L. Adler, Phys. Rev. 177, 2426 (1969).
[2] J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).
[3] Y. Frishman, A. Schwimmer, T. Banks, and S.
Yankielowicz, Nucl. Phys. B177, 157 (1981).
[4] A. D. Dolgov and V. I. Zakharov, Nucl. Phys. B27, 525
(1971).
[5] E. D’Hoker and E. Farhi, Nucl. Phys. B248, 59 (1984).
[6] E. D’Hoker and E. Farhi, Nucl. Phys. B248, 77 (1984).
[7] J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971).
All the loop integrals needed for our calculations are
given below:
Z
1
ið1Þ
‘d l 2
¼
ð1 þ " log4 þ . . .Þ
2
ðl m Þ
162
ð 2 þ "Þ 2 2 "
ðm Þ
; (A1)
ð Þ
Z
‘d l
Z
‘d l
ðl2
l2?
ið1Þ þ1
¼
ð1 þ "ðlog4 1Þ þ . . .Þ
2
m Þ
162
ð 3 þ "Þ 2 3 "
ðm Þ
;
(A2)
ð Þ
1
ið1Þ
¼
ð1 þ " log4 þ . . .Þ
2
2
n ðl þ aÞðl m Þ
162
ð 2 þ "Þ ðm2 Þ2 "
;
ð Þ
n a
(A3)
Z
l2?
ið1Þ þ1
¼
ð1þ"ðlog41Þþ...Þ
2 m2 Þ
nðlþaÞðl
162
ð 3þ"Þ ðm2 Þ3 "
; (A4)
ð Þ
na
R
R
where ‘d l ¼ dd lð2Þd and d ¼ 4 2". Note that
l2? ¼ l~2? when g ¼ diagðþ Þ.
‘d l
[8] E. Witten, Nucl. Phys. B223, 422 (1983).
[9] C. W. Bauer, S. Fleming, and M. E. Luke, Phys. Rev. D 63,
014006 (2000).
[10] C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart,
Phys. Rev. D 63, 114020 (2001).
[11] C. W. Bauer and I. W. Stewart, Phys. Lett. B 516, 134
(2001).
[12] C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys. Rev. D 65,
054022 (2002).
036003-10
CHIRAL ANOMALY IN SOFT COLLINEAR EFFECTIVE THEORY
[13] S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 1517
(1969).
[14] C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys. Rev. D 68,
034021 (2003).
[15] J. Chay and C. Kim, Phys. Rev. D 65, 114016 (2002).
[16] A. V. Manohar, T. Mehen, D. Pirjol, and I. W. Stewart,
Phys. Lett. B 539, 59 (2002).
PHYSICAL REVIEW D 79, 036003 (2009)
[17] M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann,
Nucl. Phys. B643, 431 (2002).
[18] T. Feldmann, Nucl. Phys. B, Proc. Suppl. 121, 275 (2003).
[19] D. Pirjol and I. W. Stewart, Phys. Rev. D 67, 094005
(2003).
[20] M. Beneke and T. Feldmann, Phys. Lett. B 553, 267
(2003).
036003-11