Chiral anomaly in soft collinear effective theory
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Waalewijn, Wouter J. “Chiral anomaly in soft collinear effective
theory.” Physical Review D 79.3 (2009): 036003. (C) 2010 The
American Physical Society.
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http://dx.doi.org/10.1103/PhysRevD.79.036003
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Chiral anomaly in soft collinear effective theory
Wouter J. WaalewijnCenter 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.1Simply removing it would spoil the anomaly cancellation for the gauge sym-metries. 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 pertur-bation theory. By itself chiral perturpertur-bation theory would not include reactions such as 0 ! or K K ! þ0. 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 1
"2 divergences at one loop that are removed by renormalization. We will see that such diver-gences 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 Aus. The momenta of the collinear fields scale like pc ¼ ðn p; n p; p?Þ Qð2; 1; Þ and for usoft fields as pus Q2. Here Q is the hard scale, the SCET expansion parameter, and n and n are light-cone basis vectors satisfying n2¼ n2 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 2¼
QCD=Q. We will also need the power counting of the fields, which are listed in TableI.
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;pðxÞ. The label momenta are picked out by the label operatorP , e.g. P?n;p¼ p?n;p. The collinear covariant derivatives are then defined as
in D c¼ n P þ g n An;q; iD?c ¼ P?þ gA?n;q; in D ¼ in @ þ gn An;qþ gn Aus; (1) and the usoft covariant derivative as
iDus¼ i@þ gAus: (2) Note that these covariant derivatives are each homogenous in the power counting. We are now ready to write down the
1In 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.
leading order Lagrangian for a collinear quark n [10], Lð0Þ¼ nðxÞ in D þ i 6Dc ? 1 in D ci 6D c ? 6n 2nðxÞ; (3) where it is understood that the (suppressed) label momenta are summed over and that the label momenta of each term inL 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 next-to-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 com-puting 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
162 tr½F
F (4)
in the power counting, where J5¼
5. We use the 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),
J @J5; F g 2
162 tr½F
F ; (5)
and write their expansions as
J ¼ Jð4Þþ Jð5Þþ Jð6Þþ . . . ;
F ¼ Fð4Þþ Fð5Þþ Fð6Þþ . . . ; (6) where the order in is indicated in brackets. When calcu-lating 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 sim-plicity 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 ¼1
2in ðp þ qÞhggjn J 5j0i þ1
2in ðp þ qÞhggjn J 5j0i: (7) The tree level relation between the QCD and SCET (col-linear) field is [10] ¼ |{z}n þ 1 in D c i 6Dc ? 6n 2n |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 þ . . . ; (8)
with the power counting as indicated. This leads to
n Jð2Þ5 ¼ n6n 5n: (9) For n J5 we get an n6 . But n6n¼ 0, so the lowest non-vanishing contribution is atOð4Þ:
n J5ð4Þ¼ 1 in D c i 6Dc ? 6n 2n y 0n6 5 1 in D c i 6Dc ? 6n 2n ¼ ni 6D c ? 1 in D c 6n5 1 in D c i 6Dc ?n: (10)
However, it is important to note that n Jð4Þ5 contributes at the same order as the n J5term in the anomaly equation because of the power counting for collinear momenta,
hggjJð4Þj0i ¼1 2in ðp þ qÞhggjn J ð2Þ 5 j0i þ1 2in ðp þ qÞhggjn J ð4Þ 5 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 sub-sections. With these additional assumptions we have to
TABLE I. Power counting of the various fields and operators in SCET.
Collinear Usoft Covariant derivatives Operator n n A n A ? n n An qus A us in D c iD ? c in D iDus Power counting 0 2 3 2 0 2 2
keep gluons off shell, p2, q2 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 in D ¼ in @ þ gn An;qþ gn Aus¼ in @|ffl{zffl} 2 þ . . . ; in D ¼ i n D cþ Wi n DusWy ¼ n P|ffl{zffl} 1 þ i n @|ffl{zffl} 2 þ . . . ; iD? ¼ iD?c þ WiD?us Wy¼ gA?n;q |ffl{zffl} þ gA?us |ffl{zffl} 2 þ . . . : (12) W is the collinear Wilson line that only contains the n An component of the gluon field
W ¼ 1 g 1
n Pn A nþ . . . (13) and can be dropped because we are only looking at ? -polarized gluons. Writing
FF ¼ 4
g2 iDiDiDiD ; (14) the only contribution we need to consider atOð4Þ is Fð4Þ¼ g2 162 4 g2 n 2 n 2 tr½in @gA ? n;p n P gA? n;q in @gA? n;qn P gA ? n;p n P gA?n;pin @gA? n;qþ n P gA? n;qin @gA?n;p: (15) The corresponding matrix element is given by
hggjFð4Þj0i ¼ g2 82ðn p n q n q n pÞ AB? A ðpÞ BðqÞ; (16) where A ðpÞ,
B ðqÞ are the polarization vectors of the gluons and we assume the generators are normalized as tr½TATB ¼1
2
AB. The two-dimensional ?is defined as
? 1
2 nn¼18i tr½56nn6 ?? : (17) Having derived the explicit form of the SCET anomaly equation at LO
Jð4Þ¼ Fð4Þ; (18)
we will verify this result by calculating the one-loop diagrams.
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
hggjJð4Þj0i ¼1 2in ðp þ qÞhggjn J ð2Þ 5 j0i ¼1 2in ðp þ qÞhggj n6n 5nj0i; (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 pÞ2and ðl qÞ2. We will elaborate on this for the bubble diagram.
In dimensional regularization with d ¼ 4 2", the tri-angle diagram is given by
AT ¼ 1 2in ðp þ qÞð1Þ Z ‘l tr6n 5i n 6 2 n ðl qÞ ðl qÞ2 igTB ? 6l? n l þ l 6 ?? n ðl qÞ 6n 2i n 6 2 n l l2 igTA ? 6l? n ðl þ pÞþ l 6 ?? n l 6n 2i n 6 2 n ðl þ pÞ ðl þ pÞ2 þ ðp; ; AÞ $ ðq; ; BÞ; (20)
where R‘l ¼Rddlð2Þd. We will always suppress po-larization vectors. To treat 5in 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?!1l 2
?g? , where ¼ d 2 ¼ 2 2". We then use
?? ¼ ; ?? ? ¼ ð 2Þ? ¼ 2"? ; ?? ? ? ¼ 4g? ð4 Þ? ? : (21) This leads to
AT ¼ ig2n ðp þ qÞAB? ðI1þ "I2þ "I3 ð1 þ 2"ÞI4Þ þ ðp; ; AÞ $ ðq; ; BÞ ¼ g2 82n ðp þ qÞn p AB? ; (22) where I1 ¼ Z ‘ln ðl qÞ n ðl þ pÞl 2? n lðl qÞ2l2ðl þ pÞ2 ; I2 ¼ Z ‘l n ðl qÞl 2? ðl qÞ2l2ðl þ pÞ2; I3 ¼ Z ‘l n ðl þ pÞl 2? ðl qÞ2l2ðl þ pÞ2; I4 ¼ Z ‘l n l l 2? ðl qÞ2l2ðl þ pÞ2: (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 AB¼1 2in ðp þ qÞð1Þ Z ‘ltr6n 5in6 2 n ðl qÞ ðl qÞ2 ig2 TATB? ? n ðl þ p qÞþ TBTA? ? n l 6n 2i n 6 2 n ðl þ pÞ ðl þ pÞ2 ¼ ig2n ðp þ qÞAB? Z‘l 1 n l 1 n ðl þ p qÞ n ðl qÞ n ðl þ pÞ ðl qÞ2ðl þ pÞ2 : (24) 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 be-fore, 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
! 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 AB¼ 0.
We now add ATand ABand compare with hggjFð4Þj0i 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 withn 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 Jð4Þ5 :
hggjJð4Þj0i ¼1 2in ðp þ qÞhggj ni 6D c ? 1 in D c n5 1 in D ci 6D c ?nj0i: (25)
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 ð1Þ n 6l? n ðl þ pÞ 6n 5 l 6 ? n ðl qÞn; ð2Þ n 6l? n ðl þ pÞ 6n 5 gTB? n ðl qÞn; ð3Þ n gTA? n ðl þ pÞ 6n 5 l 6 ? n ðl qÞn; ð4Þ n gTA? n ðl þ pÞ 6n 5 gTB? n ðl qÞn: (26)
First of all we get a triangle [Fig. 3(a)] and bubble diagram [Fig.3(b)] coming from (1)
~
AAþ ~AB ¼ ig2n ðp þ qÞ AB? ðð1 þ 2"Þ~I1 "~I2 "~I3 ~I4 ð1 þ 2"Þ~I5Þ
þ ðp; ; AÞ $ ðq; ; BÞ; (27)
where
~ I1 ¼Z ‘l l 4 ? n lðl qÞ2l2ðl þ pÞ2; ~ I2 ¼Z ‘l l 4 ? n ðl þ pÞðl qÞ2l2ðl þ pÞ2; ~ I3 ¼Z ‘l l 4 ? n ðl qÞðl qÞ2l2ðl þ pÞ2; ~ I4 ¼ Z ‘l n l l 4? n ðl qÞn ðl þ pÞðl qÞ 2l2ðl þ pÞ2; ~ I5 ¼ Z ‘l l2? n lðl qÞ2ðl þ pÞ2: (28)
We also have diagrams [Fig.3(c1)with (2) and Fig.3(c2) with (3)] where one gluon comes out of the current
~ AC1þ ~AC2¼ 2ig2n ðp þ qÞ AB? ð~I6þ "~I7Þ þ ðp; ; AÞ $ ðq; ; BÞ; (29) with ~ I6 ¼Z ‘l n l l 2 ? n ðl qÞn ðl þ pÞl 2ðl þ pÞ2; ~ I7 ¼Z ‘l l 2 ? n ðl qÞl2ðl þ pÞ2: (30)
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, ~AD ¼ 0.
We simplify the remaining integrals using
l2? ¼ l2 n l n l ¼ ðl qÞ2 n ðl qÞn ðl qÞ ¼ ðl þ pÞ2 n ðl þ pÞn ðl þ pÞ; (31) where l2
?¼ ~l2?if g ¼ diagðþ Þ. For example, ~ I2¼ Z ‘l l2? n ðl þ pÞðl qÞ2l2 n ðl þ pÞl2 ? ðl qÞ2l2ðl þ pÞ2 ¼ ~I2Aþ ~I2B: (32)
Doing the math, we find
hggjJð4Þj0i ¼ ~A Aþ ~ABþ ~AC1þ ~AC2þ ~AD ¼ g2 82n ðp þ qÞn p AB? : (33) Again this agrees with hggjFð4Þj0i, as expected because the calculation so far only involved collinear fields.
D. LO general case
Finally we now drop the additional assumptions [e.g. n ðp þ qÞ ¼ 0] and study the general case. These assump-tions allowed us to look at one term of hggjJð4Þj0i 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
n p ¼ 0; n q ¼ 0: (34) After repeating the above analysis and performing the Feynman integrals we find
hggjJð4Þj0i ¼ g2
82n p n q AB?
¼ hggjFð4Þj0i: (35) Thus the SCET anomaly equation at LO Jð4Þ¼ Fð4Þ, which we derived by expanding, is correct at one loop. In the previous sections the bubble diagrams in Figs. 3(c1) and3(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
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
SCET fields to higher order in . Using the result from [14] we have ¼ n |{z} þ Wqus |ffl{zffl} 3 þ 1 in D c i 6Dc ? 6n 2n |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 þ W 1 n Pi 6D us ? 6n 2W y n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 3 þ . . . : (36) We find n J5ð2Þ¼ n6n 5n; (37) n Jð3Þ5 ¼ 0; (38) n Jð4Þ5 ¼ n6n 5Wqusþ H:c:; (39) and n J5ð4Þ¼ ni 6D c ? 1 in D c 6n5 1 in D ci 6D c ?n; (40) n Jð5Þ5 ¼ 2 ni 6D c ? 1 in D c 5Wqusþ W 1 n Pi 6D us ? 6n 2W y n þ H:c: (41) Expanding the other side of the QCD anomaly equation gives hggjFð4Þj0i ¼ g2 82ðn p n q n q n pÞ AB? A ðpÞ BðqÞ; (42) hggjFð5Þj0i ¼ g2 82n p n q AB? A ðpÞ B ðqÞ; (43) hggjFð6Þj0i ¼ g2 82ðn p n qr n q n prÞ AB? A ðpÞ BðqÞ; (44) where the subscript in pr denotes residual momentum, pr 2. We already checked at LO that the SCET anom-aly equation found this way is correct,Jð4Þ¼ Fð4Þ. 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 Jð5Þ5 . However, there are also contributions from diagrams
in-volving the leading current and insertion(s) of subleading Lagrangians. To be specific, the SCET anomaly equation at NLO and NNLO reads
T fJð4ÞiLð1Þg þ Jð5Þ¼ TfFð4ÞiLð1Þg þ Fð5Þ; (45) TfJð4Þ12iLð1ÞiLð1Þg þ TfJð4ÞiLð2Þg þ TfJð5ÞiLð1Þg þ Jð6Þ ¼ TfFð4Þ1 2iL ð1ÞiLð1Þg þ TfFð4ÞiLð2Þg þ TfFð5ÞiLð1Þg þ Fð6Þ; (46)
where TfJð4ÞiLð1Þg ¼Rd4xTJð4Þð0ÞiLð1ÞðxÞ, etc. Lð1Þand Lð2Þrefer 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 forFðiÞ. 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 toFð5ÞandFð6Þ.
We list the subleading Lagrangians we need below [14– 20]: Lð1Þ¼ ð nWÞi 6D?us 1 n P Wyi 6D?c 6n 2n þ ð ni 6D?cWÞ 1 n Pi 6D ? us Wy 6n 2n ; (47) Lð2Þ¼ ð nWÞi 6D?us 1 n Pi 6D ? us 6n 2ðW y nÞ þ ð ni 6D?cWÞ 1 ð n P Þ2in D us 6n 2ðW yi 6D? cnÞ; (48) Lð1Þq ¼ n 1 in D c igB6 ?cWqusþ H:c:; (49) where igB6 ?c ¼ ½i n Dc; i 6D?c.
B. NLO withn ðp þ qÞ ¼ 0
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 col-linear and the other usoft. The only additional assumption that makes sense here is
n ðp þ qÞ ¼ 0; (50)
because n p 1 2 n q. The axial current gives rise to
hggjTfJð4ÞiLð1Þg þ Jð5Þj0i ¼1 2in ðp þ qÞhggjðTfn J ð4Þ 5 iLð1Þg þ n J ð5Þ 5 Þj0i ¼1 2in ðp þ qÞhggj T ni 6D c ? 1 in D c 6n5 1 in D ci 6D c ?n iLð1Þ þ2 ni 6D c ? 1 in D c 5 Wqusþ W 1 n Pi 6D us ? 6n 2W y n þ H:c:j0i: (51)
Said in words, we get contributions from diagrams with n J5ð4Þ, one Lð1Þ vertex and Lð0Þ vertices, and diagrams with n Jð5Þ5 and onlyLð0Þvertices. There is no term inLð0Þ 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.4and5. 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 simplifi-cation that we can generally drop n q for the usoft mo-mentum q, because n q n p; n l.
The various diagrams lead to contributions ANLO
A1 þ ANLOA2 ¼ 2ig2n p AB ?
ð1 þ "Þð~I1 ~I2Þ; (52)
ANLOB1 þ ANLOB2 ¼ 2ig2n p AB?
ð1 þ "Þ~I2A; (53) ANLO C1 þ ANLOC2 ¼ 2ig2n p AB ? ð~I6þ "~I7Þ; (54) ANLO D1 þ ANLOD2 ¼ 0; (55)
where ~I2Awas defined in (32). Adding the pieces we get
hggjTfJð4ÞiLð1Þg þ Jð5Þj0i ¼ ANLO A1 þ ANLOA2 þ ANLOB1 þ ANLO B2 þ ANLOC1 þ ANLOC2 þ ANLO D1 þ ANLOD2 ¼ g2 82n p n q AB? ¼ hggjFð5Þj0i: (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 follow-ing reparametrization:
n pc! n pcþ n k; n p r! n pr n k; (57) where pcis the (collinear) label momentum, prthe (usoft) residual momentum, and k some constant usoft momen-tum. This ties togetherFð4ÞandFð6Þ, 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 Jð5Þ5 contributing to the NLO anomaly.
former. As can be seen from the expression forFð6Þin (44), we need to keep the residual momentum components of the external momenta
p ¼ pcþ pr¼ ð0; n pc; p?c ¼ 0Þ
þ ðn pr;n p r; p?r ¼ 0Þ (58)
to get a nonvanishing expression. Our assumptions are
n ðpcþ qcÞ ¼ 0; n ðp rþ qrÞ ¼ 0: (59)
The axial current gives hggjTfJð4Þ1 2iLð1ÞiLð1Þg þ TfJð4ÞiLð2Þg þ TfJð5ÞiLð1Þg þ Jð6Þj0i ¼1 2in ðp þ qÞhggjðTf n J ð2Þ 5 12iL ð1ÞiLð1Þg þ Tf n Jð2Þ 5 iLð2Þg þ Tf n J ð3Þ 5 iLð1Þg þ n J ð4Þ 5 Þj0i ¼1 2in ðp þ qÞhggjðTf n6n 5n12iL ð1ÞiLð1Þg þ Tf n6n 5niLð2Þg þ 0 þ ð n6n 5Wqusþ H:c:ÞÞj0i: (60)
We cannot make a diagram using n Jð4Þ5 and only Lð0Þ vertices, since the n J5ð4Þcurrent 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 for which we combined label and residual momenta P
lc R
‘lr!R‘l. This was possible because these dia-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?
rð. . .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 col-linear loops may still have nonzero contributions, for ex-ample, Pl c R ‘lrn ðl rþ prÞð. . .Þ ¼ n prPl c R ‘lrð. . .Þ. Because we take the external p?r ¼ q?r ¼ 0, many dia-grams vanish immediately.
First of all we get the same diagrams as at LO, but with an extraLð2Þinsertion [Figs.6(a1),6(a2),6(a3),6(b1), and 6(b2)]:
ANNLOA1 þ ANNLOA2 þ ANNLOA3 þ ANNLOB1 þ ANNLOB2 ¼ 2ig2n ðp þ qÞn p rAB? ð ^I1þ 2" ^I2 ð1 þ 2"Þ ^I4 ^I5Þ þ ðp; ; AÞ $ ðq; ; BÞ; (61) where ^ I1 ¼ Z ‘l n ðl þ pÞl 4? n lðl qÞ2l2ðl þ pÞ4; ^ I2¼ Z ‘l l4? ðl qÞ2l2ðl þ pÞ4; ^ I4¼ Z ‘l n l l 4? n ðl þ pÞðl qÞ2l2ðl þ pÞ4; ^ I5¼ Z ‘l n ðl þ pÞl 2? n lðl qÞ2ðl þ pÞ4: (62)
We also get a triangle [Figs.6(c1)and6(c2)] where one of the vertices isLð2Þand a bubble diagram [Fig.6(d)]with an Lð2Þvertex:
ANNLOC1 þ ANNLOC2 þ ANNLOD ¼ 2ig2n ðp þ qÞn p rAB? ð" ^I8 ð1 þ 2"Þ ^I9 þ ðp; ; AÞ $ ðq; ; BÞ þ ^I10Þ; (63) with ^ I8 ¼ Z ‘l l2? ðl qÞ2l2ðl þ pÞ2; ^ I9 ¼Z ‘l n l l 2 ? n ðl þ pÞðl qÞ2l2ðl þ pÞ2; (64) ^ I10¼ Z ‘l ð n ðl þ pÞÞ2 n lðl qÞ2ðl þ pÞ2: (65) Finally there is also a mixed collinear-usoft diagram [Fig.6 (e)] with a usoft loop momentum:
ANNLOE ¼ ig2n ðp þ qÞAB? Z ‘l n q n ðl qÞ n q n l l2 n p n p n ðl þ pÞþ ðp; ; AÞ $ ðq; ; BÞ ¼ 0: (66)
In a naive calculation where you forget to treat l as a residual momentum that is multipole expanded in the col-linear propagators, you would misleadingly find a nonzero result for this diagram.
Rewriting the loop integrals using (31) and d dðn pÞ 1 ðl þ pÞ2¼ nðl þ pÞ ðl þ pÞ4; d dðn pÞ 1 ðl þ pÞ2¼ nðl þ pÞ ðl þ pÞ4; (67) we get ^ I1 ¼ ^I2þ ^I5 ^I8 n p d dðn pÞ ^ I8; (68) ^ I4 ¼ ^I2 ^I8þ ^I9 n p d dðn pÞ I^8: (69)
This reduces the calculation to
hggjTJð4Þ1 2iL
ð1ÞiLð1Þþ TfJð4ÞiLð2Þg þ TfJð5ÞiLð1Þg þ Jð6Þj0i ¼ ANNLO
A1 þ ANNLOA2 þ . . . þ ANNLOD þ ANNLOE ¼ 2ig2n ðp þ qÞn p rAB? " ^I8 n p d dðn pÞI^8þ ð1 þ 2"Þ n p d dðn pÞ I^8 1 2I^10 þ ðp; ; AÞ $ ðq; ; BÞ: (70) Working out ^I10, 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
hggjTJð4Þ1 2iL ð1ÞiLð1Þþ TfJð4ÞiLð2Þg þ TfJð5ÞiLð1Þg þ Jð6Þj0i ¼ g2 82n ðp þ qÞn p r AB? ¼ hggjFð6Þj0i: (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-aly equations up to NLO read2
Jð4Þ¼ Fð4Þ; (72) T fJð4ÞiLð1Þg þ Jð5Þ¼ TfFð4ÞiLð1Þg þ Fð5Þ; (73) where Jð4Þ¼1 2n @½ n6n 5n þ1 2n @ ni 6D c ? 1 in D c 6n5 1 in D ci 6D c ?n þ @? n?5 1 in D c i 6Dc ? 6n 2nþ H:c: ; (74)
2We 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.
Jð5Þ¼1 2n @ 2 ni 6D c ? 1 in D c 5 Wqusþ W 1 n Pi 6D us ? 6n 2W y n þ @? n?5 Wqusþ W 1 n Pi 6D us ? 6n 2W y n þ H:c:; (75) Fð4Þ¼ 1 162 ? tr½in D i n DciD ? c iD? c permutations; (76) Fð5Þ¼ 1 82 ? tr½in D i n DciD ? c ðWiD? usWyÞ permutations: (77)
Here the sum over permutations means the sum of all different orderings of the operators multiplied by the sign of the permutation. Note thatFð4Þhas an additional factor of 12 compared to Fð5Þ, which comes from the redundant interchange of the iD?c.
3
The relevant terms ofLð1Þcan be 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 col-linear fields, and SCET with only colcol-linear 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, whereasJ has terms of arbitrary high order. This can be seen from replacing 1=ðin DcÞ ! 1=ðin Dcþ
Win DusWyÞ in (36) and expanding. If our result extends to higher orders, then certain matrix elements of the cur-rents will have to cancel each other beyond 8.
ACKNOWLEDGMENTS
I would like to thank Iain Stewart for many good dis-cussions 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
All the loop integrals needed for our calculations are given below: Z ‘dl 1 ðl2 m2Þ ¼ ið1Þ 162 ð1 þ " log4 þ . . .Þ ð 2 þ "Þ ð Þ ðm 2Þ2 "; (A1) Z ‘dl l 2 ? ðl2 m2Þ ¼ ið1Þ þ1 162 ð1 þ "ðlog4 1Þ þ . . .Þ ð 3 þ "Þ ð Þ ðm 2Þ3 "; (A2) Z ‘dl 1 n ðl þ aÞðl2 m2Þ ¼ ið1Þ 162 ð1 þ " log4 þ . . .Þ ð 2 þ "Þ ð Þ ðm2Þ2 " n a ; (A3) Z ‘dl l 2 ? nðlþaÞðl2m2Þ ¼ ið1Þ þ1 162 ð1þ"ðlog41Þþ...Þ ð 3þ"Þ ð Þ ðm2Þ3 " na ; (A4) where R‘dl ¼Rddlð2Þd and d ¼ 4 2". Note that l2
? ¼ ~l2? when g ¼ diagðþ Þ.
[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).
[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).
3In our derivation ofFð4Þin (16) we did not have this factor1 2,
[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).
[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).
