научная статья по теме ANOMALIES (SELECTED TOPICS) Физика

Текст научной статьи на тему «ANOMALIES (SELECTED TOPICS)»

ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 5, с. 926-938

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

ANOMALIES (SELECTED TOPICS)

©2010 V. I. Zakharov*

Institute for Theoretical and Experimental Physics, Moscow, Russia

Received July 6, 2009

The presentation is intended to be mainly pedagogical and covers the chiral anomaly and its most significant applications. Finally, we comment on recent issues, related to the lattice measurements.

1. INTRODUCTION

For the purposes of this review, by anomalies one can understand violations of the Ehrenfest theorem according to which all matrix elements of the classical equations of motion (considered as operators) vanish. Thus, anomaly is established if we find, for example, a nonvanishing matrix element of the Dirac equation for a charged particle:

{ip(D + гт)ф) = G,

(1)

where D = y^D^ and D^ is the covariant derivative and m is the mass of the charged particle.

The first and most famous example of the anomaly is the chiral anomaly [1,2]:

— ta -

daO'a = 2m?/075'0 + — FßVFßV,

(2)

where = ipjaYs^, FßV is the electromagnetic field strength tensor, ^ is the operator of the electron field (1/2)eaßlS FaßFjs = FßvFßV. The use of the classical equations of motion would give only the first term in the r.h.s. of Eq. (2). The fF term is the anomaly.

The anomaly (2) was derived in terms of ultraviolet (UV) regulators. We will use another derivation, in terms of the infrared (IR) divergences which was originally suggested in [3]. The both approaches have advantages and relative disadvantages. In fact, the most interesting is their interplay which is a kind of UV—IR connection intensely discussed in other contexts recently.

The literature on the anomalies is enormous which is well justified: it is the anomalies which make the difference between the classical and quantum field theories. The selection of topics is a matter of a great extent. I was asked to emphasize educational side and, therefore, concentrate on the most famous and mathematically simple case of the chiral anomaly (2) and its immediate generalizations. The same is true

for the uses of the anomalies included into this review: we mention only very few examples. This material takes about 50% of the review. Then we will discuss more advanced topics, in particular, nonrenormal-ization theorems (started with [4]) for higher-order contributions to the chiral anomaly and the role of the classical solutions which are crucial to make the anomalies effective, as observed first in [5, 6]. Finally, we turn to the recent topic of localization of the classical vacuum solutions by high-frequency perturbative fields.

2. CHIRAL ANOMALY DERIVATION

By the time when Dolgov and myself started to work on paper [3] the anomaly (2) already captured attention of theorists. However, the derivation of the anomaly did not look completely satisfactory to us since we could not rule out (for ourselves) that another way of regularizing the theory in the UV would allow to avoid the anomaly. The anomaly is associated with the triangle graph of Fig. 1 and we decided to evaluate the graph keeping as close as

E-mail: vzakharov@itep.ru

ki

Fig. 1. Anomalous triangle graph. Momenta q, k1, k2 are carried by the axial current a® and two photons, respectively. The triangle corresponds to a fermion of small mass m.

possible to the classical equations of motion and avoiding in this way any possible anomaly.

It is well known, how to "keep closest to the classical equations of motion". First, the Born graphs (without loops) are just equivalent to the classical field theory and there can be no anomaly associated with the Born graphs. Proceeding to loop graphs one observes that the imaginary part of the corresponding invariant amplitude, Im A is given by Born graphs (Cutkosky rules). Finally, using the dispersion relations

Re A(q2) = - f ^ Im A(s) + n J (s - q2)

+ (possible subtractions),

(3)

(4)

where we imposed the condition of gauge invariance by using the electromagnetic field strength tensor rather than the gauge potential AM. The matrix element from the pseudoscalar density is described by another function g2(q2):

(Yl\iipY5^\0) = mg2(q )F^UF^v,

(5)

one apparently can avoid any violation of the classical equations of motion (since they are valid for the imaginary part identically). Note that (3) does not use any explicit UV regulators. True, the dispersion integral might diverge and one has to make subtractions but the subtraction constants cannot enforce any anomaly by themselves.

In particular, since the axial current entering Eq. (2) is classically conserved in the chiral limit of m = 0, we expect that for the amplitudes describing the divergence of the current Im A = 0 in this limit and the dispersion relation (3) would trivially lead to Re A = 0 as well. We will see now how this logic, on the absence of anomalies in fact fails in case of the triangle graph of Fig. 1.

At first, a few words on the kinematics. The triangle graph depends on three invariants, q2, kf, kf, where q, ki, k2 are 4-momenta carried by the axial current and photons, respectively. We will consider q2 = 0, kf = kf = 0. Then the matrix element in point is controlled by a single invariant function

gi (q2):

(Tf\a5a I0) = gi(q2)qaF^v ,

h

Fig. 2. Imaginary part of the anomalous triangle graph. The intermediate fermions are on mass shell. (The signs are the same as in Fig. 1.)

is obtained from the triangle graph by putting the intermediate e+e- pair on the mass shell (see Fig. 2):

Im gi(q2 ) = J (phase space)

(6)

x A(current — e+e )A(e+e — 77).

-

Trying to implement (6) we encounter a difficulty: the amplitude A(e+e- _ 77) is not defined for the forward scattering if m = 0. Indeed in this kinematics we hit the pole associated with the electron exchange:

+ o _^ .-v^A (7)

(phase space) • A(e+e — 77) ~ sin 9d9

r^j -

EeEY — \ pe \\ pY \ • cos d + m2 '

where we introduced the factor m in the r.h.s. for convenience (since the trace over the 7 matrices associated with the triangle graph is trivially proportional to m).

Note that conservation of the a® current in the limit m = 0 requires gi(s) = 0 in this limit. Let us check this first for the imaginary part which

where we omitted all the factors but the pole and the sensitive part of the phase space, Ee, EY are the energies of the initial electron and final photon, pe, pY are their 3-momenta, 9 is the scattering angle. We see that if we put the electron mass m = 0 the amplitude is not defined at 9 = 0.

Thus, we are introducing an infinitesimal mass m = 0, m _ 0 to regularize the imaginary part in the IR. Note, however, that the phase space sin 9d9 is vanishing in the dangerous limit 9 _ 0 and the integration over 9 introduces only a logarithmic divergence in the charged-particle mass m. Thus we have the following estimate for the imaginary part:

1 ^fl;2

Im gi(s) ~---ln(s/m2) (s> m2). (8)

s s

Let us explain the origin of various factors here. The log stems from the integration over the scattering angle 9, see above. The proportionality to m2 is a reflection of the chirality conservation in the limit m _ 0. Indeed, the imaginary parts associated with

k

2

x

the matrix elements of the axial current and of the pseudoscalar density satisfy the classical equation of motion:

Im gi ~ m2 ■ (Im g2).

(9)

And, the last but not the least, the overall factor 1/s in Eq. (8) is reconstructed from dimensional considerations. Indeed, the current a^ = has

dimension of mass to the third power, while the invariant amplitude (5) has the dimension of the mass to the fifth power. Hence, the proportionality factor 1/s.

Now we come to the culmination. The imaginary part (8) does vanish in the limit m — 0, in accordance with the classical equations of motion. Substitute, however, the imaginary part into the dispersion relations (3):

oo

Reg\{q2) ~ m2— / -^-\n(s/m2) ~ (10)

k J s — q2 s2

4m2

m2 1 1 2 2

--o o ~ ~2 W >m )'

q2 m2 q2

where the crucial factor 1/m2 arises as a result of integrating over s because the integral is saturated in the IR (the imaginary part Im gi is nonvanishing for s > 4m2).

Keeping all the constants one reproduces the anomaly (2). The reason for the anomaly now is the infrared divergence of the dispersive integral.

3. UV APPROACH

The use of the UV regularizations to derive anomalies is based on the observation that the classical equations of motion are valid for the loop graphs as well provided that the integrals over the virtual momenta are made convergent well enough by introducing heavy regulator fields1).

Let us consider the trace of energy—momentum tensor 9W as an example. The operator 9W generates change of all scales and can be obtained by differentiating the Lagrangian with respect to all the mass parameters M^:

9w = S Mt

d

dMj

L.

(11

1)A subtle point is that the loop integral can give a finite answer and still the anomaly is there. The reason is that the equations of motion involve derivatives and to check the Ehrenfest theorem one has to start with matrix elements of the derivatives which diverge worse in the UV than the matrix elements of the currents.

Consider pure Yang—Mills theory with the La-grangian

where g2 is the coupling and Gis the non-Abelian field strength tensor, a is the color index.

Classically, there is no mass parameter since the coupling is dimensionless and

(9^^)class =

The same is true with inclusion of light quarks in the limit of their masses vanishing. If the equation 9^ = 0 were true also on the quantum level, then QCD could not have any relation to the real world since all the hadrons should have been massless as well.

"Fortunately" there exists the quantum conformal anomaly which makes 9nonvanishing. Indeed, to be defined on the quantum level, QCD needs regularization. Moreover, renormalizability of QCD implies that all the dependence on the UV cutoff is hidden in the coupling which is running. The Lagrangian, in particular, is defined in terms of the fields n

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