научная статья по теме ON DECAY AND CANCELLATION OF AXIAL ANOMALY IN TRANSITION AMPLITUDE FOR MASSIVE FERMIONS Физика

Текст научной статьи на тему «ON DECAY AND CANCELLATION OF AXIAL ANOMALY IN TRANSITION AMPLITUDE FOR MASSIVE FERMIONS»

ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 11, с. 1455-1464

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

ON Z ^ YY DECAY AND CANCELLATION OF AXIAL ANOMALY IN Z ^ yy TRANSITION AMPLITUDE FOR MASSIVE FERMIONS

©2014 E. V. Zhemchugov*

Institute for Theoretical and Experimental Physics; NRC "Kurchatov Institute", Moscow, Russia

Received April 25, 2014

Z ^ yy decay amplitude is considered and proven to be zero due to properties of polarization vectors and Bose statistics. Triangular diagrams for a pseudoscalar ^ yy and Z ^ yy processes with massive fermions in the loop are explicitely calculated. In the Standard Model axial anomaly vanishes in the sum of these diagrams as Z boson is mixed with one of the Goldstone bosons.

DOI: 10.7868/S0044002714100158

1. INTRODUCTION

A couple of articles has appeared recently concerning decay of a spin-1 particle into two photons in spite of the fact that such a decay is prohibited by Landau—Yang theorem [1, 2]. Paper [3], released in two versions, one in 2011, and the other in 2013, considers Z — yy decay. The first version claimed that the decay width is not zero, albeit less than the currently established experimental boundary. The decay width was calculated from an amplitude which does not satisfy Ward identities for photons, apparently by applying photon polarization matrix proportional to

. Since actual photon polarization matrix contains an ambiguous term proportional to photon momentum which gives zero only as long as Ward identities are satisfied, this calculation produced an incorrect result. It has been corrected in the second version of paper [3], by noticing that the amplitude equals zero. However, the expression for the amplitude has not been changed, so Ward identities remained invalid. A proper expression for the amplitude both on and off shell is provided in Section 3.2 of this work.

Another paper, [4], alerts us that we should keep considering the 126-GeV Higgs boson candidate as a particle with spin 1 in spite of the clearly observed two-photon decay of it. The paper claims that Landau—Yang theorem is inapplicable in this case. While not addressing this statement, one can demonstrate that decay of a spin-1 particle into two photons is impossible considering only the tensor structure of the amplitude, Bose statistics, and properties of polarization vectors of photons and the decaying particle. This demonstration is provided in Section 2.

E-mail: jini.zh@gmail.com

Z — yy transition amplitude is a textbook example of an axial anomaly appearing in a triangle diagram. A self-consistent theory has to be free of anomalies. In the Glashow—Weinberg—Salam theory of weak interactions with massless fermions, due to U(1) hypercharge values, anomalies cancel out when a sum of all possible fermions running in the fermion loop is accounted for [5]. However, in the case of massive fermions, straightforward calculation of Z - yy amplitude shows that its derivative is proportional to a term dependent on fermion mass. Since fermion masses are free parameters of the theory, contributions of different fermions in the loop can no longer cancel out. Nevertheless, the Standard Model features a mechanism to keep anomalies being zero which stems from the way fermion masses are generated — spontaneous symmetry breaking. In Section 4.2 it is shown that Z boson is mixed with one of the Goldstone bosons, and the latter provides the exact value to cancel out the mass-dependent term of the derivative of Z - yy amplitude. Since mass-independent terms keep cancelling out in the same way as in the massless theory, it is concluded that in the Standard Model Z — yy transition amplitude is free of anomalies.

2. Z - yy DECAY Amplitude for Z — yy decay MZis proportional to the following expression:

Mz- e^eVexT^x(k,k'), (1)

where e and e' are polarization vectors of photons, e is the polarization vector of the Z boson, k and k' are photons momenta, and t^vx(k,k') is the tensor corresponding to the sum of diagrams in Fig. 1:

T^uX(k, k') = T»vX(k, k') + TVtlX(k', k), (2)

k '

Fig. 1. Lowest-order Feynman diagrams of Z ^ 77 transition.

where T^vX (k,k') is the tensor corresponding to diagram in Fig. 1a.

Z boson couples with fermions via both vector and axial currents of the form

(gv ^ ^ + qa № Y5 ^ Z, (3)

where gv and gA are coupling constants. It follows from the Furry's theorem [6, § 79], that proportional to the vector coupling part of the amplitude equals zero. Thus, we need to consider only axial coupling. The most general representation of the latter is [7]1)

T^vX(k, kf) = (Aka + A'k'a)s^vXa + (4)

+ (Mk» + M' k)svXap k'a kp +

+ (NkV + N' k'V )s^Xap kfa kp,

where A, A', M, Mr, N, Nr are some functions depending on k and k'. Substitution of (4) into (2) results in the following expression:

T^vX (k,k') = (5)

= ((A - A'*)ka + (A' - A*)k'a)e^vXa +

+ ((N - M'*)kV + (N' - M*)k'V)s^Xapkfakp +

+ ((M - N'* k + (M' - N * )k'» )svXap k'a kp,

where A*, A'*, ... are A, A', ... with k and k' interchanged.

The expression for the amplitude should satisfy Ward identities

k^T^vX(k, k') = 0, kVT^vX(k, k') = 0. (6)

The third Ward identity,

qX T»VX (k,k' ) = 0, (7)

where q = k + k' is Z-boson momentum, is violated, and this fact is referred to as axial anomaly. We will consider it in Section 4.1. Equations (5), (6) provide the following relation for the coefficients:

A' - A* = (M - N'* )k2 + (M' - N * )kk', (8)

and another one with k and k1 interchanged. Hence, T^vX (k, kk) has the following structure:

T^vX (k,k1 ) = (9)

= (N1 - M * )(k12 s^Xa ka + k1v s^Xaß kfa kß ) + + (N - Mf*)(kkfS^Xaka + kVS^Xaßk'akß) + + (M - N1*)(k2£^vXakfa + k^£vXaßkfakß) + + (M1 - N * )(kk1 s^Xa kfa + k1^ evXaß kfa kß ).

On the mass shell k2 = k12 = 0, and there is only one Lorentz-invariant scalar which depends on either k or k! : kk!. It does not change under k ^ k! interchange, consequently, on the mass shell M = M *, M 1 = = M1*, ...Therefore,

TVvX(k,k1) = (N - M1 ) x (10)

x (kk1 s^vXa (k - k1 )a +

+ (kve^Xaß - k*£vXaß)k'akß} +

+ (N1 - M)(k1Vs^Xaß - k^evXaß)k'akß.

Let us now work in a system of coordinates where Z boson is at rest and consider relations between vectors appearing in the problem. Let the z axis be parallel to the spatial part of the photon momentum k. Then photon momenta can be written as follows:

k = (ko, 0,0, ko), k1 = (ko, 0,0, -ko), (11)

where 4k2 = 2kk1 = (k + k1 )2 = q2 = m2Z is Z boson mass squared. Photon polarization vectors are orthogonal to photon momenta, and can be chosen as follows:

= (0,ei ,62, 0), e'v = (0,6[ ,ef2, 0).

(12)

Finally, let us take the physical polarization of Z boson:

(13)

1)Note the absence of the kxk'akp term. This is due to it being linearly dependent on other terms as follows from (A.7) (see Appendix).

With these equations in mind it becomes clear that substitution of (10) into (1) gives zero, because:

1) e^k^ = eVk'V = 0,

k

ON Z — YY DECAY AND CANCELLATION

1457

k

Fig. 2. Lowest-order Feynman diagrams of y ^ yy transition.

2) e^k'^ = eVkV = 0,

3) e^e'Ve\s^vXa (k — k')a = 0 since the only a when e^e'Vex= 0 is a = 0, but (k — k')0 = k0 — k0 = = 0.

Hence, amplitude of Z — yy decay is equal to zero in the Z-boson rest frame. Since the amplitude is Lorentz invariant, it equals zero in any other coordinate system as well, and Z — yy decay amplitude vanishes.

It should be stressed that the amplitude vanishes on the mass shell; it does not if one of the photons is off-shell. For example, amplitude of the Z — yy* — — Ye+ e- transition gives nonzero contribution to the

Z — Ye+e

decay amplitude.

3. TRIANGLE DIAGRAMS

Further, in Section 4.2, we will need explicit expressions of two transition amplitudes: Z — yy and p — yy, where p is a pseudoscalar particle. Calculation of these amplitudes is relatively simple but tedious, and most of the necessary information is provided in any general quantum field theory course,

so we will omit some intermediate steps and provide only the final results.

3.1. p — yy

p — YY Transition amplitude is equal to the sum of Feynman diagrams in Figs. 2a and 2b:

M p—

(14)

= -tr

— k' — m

— i — г[ф^i/j]^--г[ф7^] x

x /

^[фрф]л Y5

— m d4 p

+

+ (e

+ k — m"1^rjA/ J (2n)4 e>,k ^ k) = ]2[ФрФ]ae^eVT»v(k,kf),

where yP] is the coupling constant between fermion and photon (fermion electric charge), pip]a is the pseudoscalar coupling constant between p and fermion, e and e' are photon polarization vectors, m is the fermion mass,

T^V(k, k') = T^V(k, k') + TV^(k', k), (15)

k

i

= f tr^-ff + mfr^ + mfr^ + ft + mfr5) dAp (16)

' J ((P — k!)2 — m2) (p2 — m2) ((p + k)2 — m2) (2n)4

The trace is equal to2) 1 1-X dz dx

x

tr((p — k + m)YV(p + m) x (17) J J 2xzkk — m2 + x(l — x)k2 + z(1 — z)k'2

0 0

x Y^(p + k + m)Y5) = 4ime^af3k'ake. On the mass shell

Calculating the integral and substituting the result M^^ = x (19)

into (14), we obtain:

" 2тг2

1 1-х

dz dx

М^ = $г1>?[Ф<РФ]Ае^,/Т'»'(к,к>)= (18) xe^kpj j 2xzkk,_m2-

2)We use the following conventions: y5 = —y0Y1Y2Y3,

£°123 = +1. p \mp

VV=(— (20)

where mv is the mass of y. Then

= Л _ I+^TO . (21,

This expression can be rewritten as follows:

^ = x (22)

—2iarccot-

X

( 71 1

arctan2^^ „ >1 arctan 2^-1' 2'

3.2. Z ^ YY

Z ^ YY transition amplitude is equal to the sum of Feynman diagrams in Figs. 1a and 1b:

i

X £

Mz^YY = J -tr i

— fc' — m

i[tpYФ] x (23)

i[ipZip ]a)y f

— m '' ' " p + k — m

+ (e ^ e',k ^ k') =

dAp (2^

+

= [^f^ZiPUev e'^€XT^vX(k,k'),

where [tpY^] is the coupling constant between fermion and photon, [tpZtp]A is the axial coupling constant between Z boson and fermion, e, e', e are polarization vectors of photons and Z boson, m is the fermion mass,

T»VX(k, k') = T^uX(k, k') + TV^X(k', k), (24)

T^x{k k') = [ ~ ^ + + + ^ + d4P (25)

' J {('P — k')2 — m2) (p2 — m2) ((p + k)2 — m2) (2n)4

The trace is evaluated with the help of (A.5). Unlike the case of (16), now the trace depends on integration variable p. The resulting expression is carried through F

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