ANOMALIES IN QUANTUM FIELD THEORIES Текст научной статьи по специальности «Физика»

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ПРЕГЛЕДНИ РАДОВИ ОБЗОРНЫЕ СТАТЬИ REVIEW PAPERS

° Anomalies in quantum field theories

cn Nicola Fabiano

o

™ University of Belgrade, "Vinca" Institute of Nuclear Sciences - National

uu Institute of the Republic of Serbia, Belgrade, Republic of Serbia,

ct e-mail: nicola.fabiano@gmail.com,

O ORCID iD: ©https://orcid.org/0000-0003-1645-2071

C

< DOI: 10.5937/vojtehg71-38164;https://doi.org/10.5937/vojtehg71-38164

IC

FIELD: mathematics o ARTICLE TYPE: review paper

E

Abstract:

Introduction:purpose: Noether's theorem connects symmetry of the La-grangian to conserved quantities. Quantum effects cancel the conserved quantities.

Methods: Triangle diagram, Path integral, Pauli-Villars regularisation. to Results: Quantum effects that spoil conserved quantities of local gauge

LA symmetries endager renormalisability.

Conclusion: A careful treatment of anomalies is needed in order to ob-

y tain correct results. The n0 ^ 77 decay is perhaps the most notable

i "impossible" effect allowed by anomalies.

LU

Key words: symmetry, quantum anomalies.

Noether's theorem

What happens when a Lagrangian is invariant under certain symmetry? It happens that there is a conserved quantity, as stated by Noether's theorem (Noether, 1918), which is perhaps the most important theorem in theoretical physics. We have a generic Lagrangian with fields 0a that undergo an infinitesimal transformation 60a after applying the symmetry. The Lagrangian is invariant by the hypothesis, so we have (summation is understood on repeated indices a)

6L = 0 = ^60a + jd^5(9^a) = 064a + jd^Wa) , (1) 50a 6(0^a) 60a 6 a)

where the order swapping of 5 and dM is possible because < variation. There are also equations of motion that read

SC

= d»

S

is afunctional

(2)

and by combining the two, we obtain

" SC

d»

0 .

Defining a current

J » =

S

(3)

(4)

S(dd^a)

from eq. (3), we see that is conserved, i.e., ddJV = 0.

In QED, for example, the wave function ^ is invariant under the phase transformations ^ ^ eia^, for which we have the conserved current

JH _ ^ such that dJV = 0 . (5)

There is also the global chiral transformation ^ ^ eifil&^ which has a conserved current only in the massless limit,

J5» = ^ such that d»J5» = 2im^j

(6)

Quantum symmetries

In the previous section we have discussed the invariance of the La-grangian under symmetry transformations and its consequences. There was no mention of quantum effects in Noether's theorem, which are not relevant for its proof. In fact, it was tacitly assumed that classical and quantum symmetries are the same thing. This belief was shattered in the late 60-ies when it was discovered that quantum effects could indeed spoil classical symmetries. Such symmetries that are broken by quantum effects are called anomalies.

A posteriori, this belief had actually no grounds. While classically there is a transformation that implies SC = 0, at the quantum level, by means of a path integral, the things are different. One computes

D$ e

(i/K)S(t)

(7)

and, while under the transformation 5$ the action is also invariant, 5S = 0, there remains the Jacobian of the transformation in which in general will

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to

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CO (M o (M

yy

0£ ZD O

O -j

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LU

I—

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not equal unity. Therefore, classical symmetry has in general no reasons to survive quantisation. For a brief review of path integrals, see (Fabiano, 2022a).

Anomalies are bad when they afflict symmetries necessary to renormalise the theory: their presence disallows the possibility of obtaining finite predictions (Schwinger, 1959). This case happens when local gauge symmetry is anomalous. Anomalies of global symmetries, on the other hand, are considered relatively harmless because they contribute finitely to physical processes.

U(1) anomaly

Consider a theory with massless fermions, where

C

(8)

whose conserved currents we have already encountered in the section entitled Noether's theorem. We already know that U(1) symmetry is conserved and for massless fermions both vector current and axial current are con-

served, duJu

0 and duJ5

h1

0.

For this theory, we will now calculate the three point function

(X1,X2) = (0 TJ5A(0)J(xi)Jv(X2) 0) •

(9)

In plain language, the vector current causes a fermion-antifermion pair creation at the point x\ and another such creation at the point x2, then a fermion from one pair and an antifermion from another pair annihilate, while at the point 0 the chiral current annihilates the remaining fermion -antifermion pair.

Two relevant Feynman diagrams for this process are two triangles represented in Figures 1 and 2.

The Fourier transform of the three point function (9) is written as

AXuv(kl,k2) = (-l)i3 JTr(

,5 1 1 1 1

Y Y -zY -¡tt "7+

,5 1 „,u 1 Y Y -Y -Y ~

102

1

2

YXY 5

5

p - q

Y v

p - ki

Y v

p - q

P - k2

Y v

Figure 1 - Triangle diagram, 1 Рис. 1 - Треугольная диаграмма, 1 Слика 1 - Ди]аграм троугла, 1

Figure 2 - Triangle diagram, 2 Рис. 2 - Треугольная диаграмма, 2 Слика 2 - Ди]аграм троугла, 2

the first part belonging to the first diagram, the other part on the last line describing the second diagram, with q = k\ + k2. The overall minus sign comes from the closed fermion loop. Both diagrams are needed in order to obtain Bose statistics.

An immediate observation is that the integral of eq. (10) is linearly divergent because it contains three fermionic propagators. This linear divergence is at the origin of the breaking of U(1) symmetry, i.e. some current will not be conserved anymore at the quantum level.

Consider an integral of a function over the whole real line

r+ro

dx f (x)

(11)

and then shift the variable, x ^ x + a. The possible consequences of this action will be evidenced by this integral

dx [f (x + a) - f (x)] ,

(12)

this action is usually harmless and eq. (12) would be zero. Expanding this expression with the Taylor series, we obtain

dx

af '(x) + -f'' (x) + O(a3)

= a[f (+«>) - f (-rc)] +

y[f'- f'(-«)]+ O(a3) .

(13)

If the integral converges, the variable shift has no consequences, but if the integral is linearly divergent, the result is given from eq. (13), and equals a[/(+rc>) - f (-ro)j.

103

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This ambiguity can be generalised to an arbitrary (Euclidean) dimension. Define the function

A(a) = JdDx [f (x + a) - f (x)] = 1

§ dD x

a^ f + (x) + O(a3)

2

c£ R

£ /(R)Sd(R) , (14)

o applying the Gauss theorem. All terms except the first vanish when inte-

< grating over the surface R — SD(R) = 2nD/2R(D-1)/r(D/2) is the surface of the D-dimensional sphere. In the four dimensional Minkowskian

X

o case, we have

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A(a) = lim (2n2i)a^RR2/(R) . (15)

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Triangle diagram

The conservation of two classical currents = 0 and d^J5^ = 0 translates respectively to the equations

S

CD >o

k\uAx^v = 0 and k2vAVv = 0 , (16)

i for the vector current, and

qxAx»v = 0 (17)

g for the pseudovector one. The first of eq. (16) is given by the expression

k^A^ (ki,k2) =

d * Tr (tVA /1 +

(2*)

yVA:Yv1 ) , (18)

1 — m m —

by substituting the first occurrence of /1 as /1 = p- (p-/1) and the second occurrence of k/1 as k1 = (p - k2) - (/ - /), we obtain

k^A^ (k1,k2) =

^ (W 7Vk-qY^[k - (k - ^+

i

2

YXY5l</ - *-) - - YVp

(2n)

— i a in — tht m —

and an analogous expression is obtained for k2vAX^v(k1, k2) by exchanging k1 o k2 and ¡i o v.

When observing the last line of eq. (19), we see that the second term is obtained by the first term by shifting the integration variable p ^ p -k1, so one would infer that the net result is zero and the vector current is conserved. However, by virtue of eq. (13), this deduction is wrong.

Define the integrand function present in eq. (19)

2 / j (P - k2)2p2

4ieTvaXk2Tpa

(p - k2)2p2

(20)

where eTvaX is the totally antisymmetric Levi-Civita tensor, with e0123 = +1, and from eqs. (12) and (15), we obtain

ki„AX^v(ki,k2) = -^ lim i(-ki)»PR 4ieTVaX4k2TP° 2n2p3 = (2n)4 p p4

^kiTk2a = 0 . (21)

Above, we have used the expression p^pa/p2 = /4: by contracting both sides with the inverse g^a, we have

pp g^a = p2 = g^ g^a = 1 ^

p2 p2 4

We will now verify the behaviour of eq. (10) with respect to a different choice of the shift in the integrand. Define the function of an arbitrary vector

AX-(a fc, fe) = {-1)i3 JTV (YXY5 j+O-qY' *

-1-Y^—1--h YXY5-1-Y^-1-Yv 1 I (23)

.+/ - /i' p+a p+/ - / p+/ - /2 p+a)'

105

1

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dp Tr(YX Y5—1— Yv—- YXY5^tYv ^ , (19) *

up

U1

<u

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ro

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f ( ) X 5 1 v A TrY 5(p - h)Yv/YX ] Q

f (p) = Tr Y Y —nTY

<2 ro

2

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and compute Ax»v(a, k-\_, k2) - Ax»v(k1,k2) with the aid of eq. (15) applied to the function

o >

g Tr[7A75(k - /)YV(/ - /1)7^/]

/(p) = Tr ( jV-^-Y»- 1 =

a:

yy

ct

o o

-J

< J ^ —

° p6 p6

x

o

LU

I—

>-

CC

^ so we obtain

(p2 - q2)(p2 - kf)p2 We have the following property:

Tr (7xj5/yvf^f) 2p»Tr (7A75pYv/

(24)

/(p) lim

p2Tr (YXY5pYv= -4ippeav»x p6 p6

(25)

Ax»v(a,k1,k2) - Ax»v(k1,k2) = ^i2 lim a"^^eav»x+

8n2 p2

<5 {(», k1) o (v, k2)} = 8-2e™»xaa + {(», k1) o (v, k2)} . (26)

CD n

>0 We can parametrise the shift vector a by the two independent momenta

k1 and k2 in the following manner:

a = a(k! + k2) + ^(k1 - k2) , (27)

and by inserting back this expression into eq. (26), we obtain

Ax»v(a, k1,k2) = Ax»v(k1,k2) + eav»x(k1 - k2)a . (28)

We notice that the dependence from a drops out and the result depends only on the difference k1 - k2.

We will now impose the conservation of the vector current in eq. (16). Not doing so would in fact lead to the non conservation of electric charge Q: fermions would be created out of nowhere. As this violation has never been observed in Nature, this constraint on the J» current is of paramount

1

importance and can not be avoided. Recalling from eq. (21) that

i

o

kMA X^v(ki,k2) = 8^2eXVTakiTk2a , (29) 5

we have .§

o

kMA X^v(a, ki, k2) = ^eX^^^kiTk2a + £2e^(ki - k2)a , (30) "

and by choosing 3 = -1/2, we obtain the vector current conservation.

A possible way of understanding this phenomenon is that the Feynman rules as such are not enough to determine the three point function of eq. (9). Because of its ambiguity, one has also to impose the constraint of the vector current conservation. 1

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Chiral current

So far, we have discussed the conservation of the vector current. Is it possible to impose also the chiral current conservation in the massless limit? We have already encountered all the necessary machinery necessary to compute the expression d^J5^:

qxA X»v(a, ki,k2) = qXA X»v(kiM) + e^vX°kixk2a . (31)

We have

qixA X^v(ki ,k2) = i J(04 Tr (V— Yv-r-irY'

4r„ /1 1

5 1 Yv 1

(2n)4

Y5-^-Yv 1Y^ + {(P,ki) o (v,k2)} =

^e»vX°kiXk2a , (32)

in a fashion analogous to eq. (18). Eventually, for the chiral current, we obtain:

QX A X^v (a, ki, k2) = n e^V X a ki x k2a = 0 , (33)

i.e., the chiral current is not conserved even in the massless limit. This phenomenon is known as the chiral anomaly, the axial anomaly or the Adler-Bell-Jackiw (ABJ) U(1) anomaly (Adler, 1969; Bell & Jackiw, 1969). For the path integral formulation, see (Fujikawa, 1979).

33

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Consequences of the chiral anomaly

We have just seen how the triangle diagrams are dependent upon a variable shift, and that it is impossible to impose both constraints of the conservation of a vector and chiral current at the same time. This leads to the breaking of U(1) symmetry and has many consequences. Some of them will be illustrated briefly.

a:

qE Add photons - We could add photons to our naive theory of eq. (8), i.e. 8 L = - ieA»)^. It is equivalent to adding two external photon

lines attached to the vertices n and v of Figs. 1-2. Of course, clas-o sically, the J5 current is still conserved, as we did not add the mass

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CO CM o CM

X

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LU

<

CO

LU I—

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o

term. At the quantum level, eq. (33) becomes

e2

k dJl = ^ e»vx° Fxa , (34)

which is an operator that produces two photons. This term is very often written as

e2

d^Jl = j~a—\ *F»VF»v , (35)

3 " (4n) lv

CD

g where *Flv = el"xaFxa is the dual electromagnetic tensor .

x n0 ^ yy decay - With the argument shown in the Triangle diagram section, attaching an external line of a pseudoscalar n0 at the two vertices of Figs. 1-2 and two photons as described above, one can calculate the decay rate of n0 ^ 2j. Historically, people used the erroneous quantum conservation of the chiral current to prove that this decay cannot occur at all! It is interesting to note a posteriori that r(n0 ^ 2y) ^ 7.82 eV, and that its branching ratio is B = r(n0 ^ 2y)/r ^ 99%, for a "non existing" decay channel.

Add a mass term - OurLagrangian becomes L = -ieAl)-m}^,

and explicitly spoils the conservation of J5 as illustrated in eq. (6). So we have

— e2

dlJl = 2im^ + ^elvx°F»vFx„ , (36)

i.e. the classical explicit mass term that violates the chiral current conservation and the quantum term with the analogous effect add up.

Régularisation - Since the triangular diagrams in Figs. 1-2 are linearly divergent, one could wonder whether some sort of regularisation would be able to cancel the anomaly. Dimensional regularisation (Bollini & Giambiagi, 1972; 't Hooft & Veltman, 1972) cannot be used this time, because the Y5 matrix in D dimensions defined as y5 = ij0j1. ..jD-1, -in the odd dimensional spacetime still obeys the Clifford algebra } = and Y, y5} = 0, but is inconsistent with the trace properties, i.e. does not obey the relation

1

o o

£± CO

CO Œ

CO CD

Tr(YMYvYpYaY5) = 0 . (37)

One could use the Pauli-Villars regularisation (Pauli & Villars, 1949) discussed in (Fabiano, 2022b). Keeping to 0 the electron mass and | introducing a regulator mass M, the behaviour of the integrand in the three point function of eq. (10) is unchanged for p < M and is superficially logarithmically divergent, so one is allowed to shift the integrand variable. Yet the chiral current is again not conserved after the intro- § duction of M, as precisely this mass term violates the chiral symmetry. w This breaking still persists even after the regulator M ^

Yang-Mills theory - Consider the massless version of the La-grangian (Yang & Mills, 1954) of eq. (20) of (Fabiano, 2022c), i.e. L = - igA^Ta)^. The difference with the Abelian case

is that we insert a factor Ta at the vertex labelled by n and a factor Tb at the vertex labelled with v. For a non Abelian gauge theory, one obtains

dJl = e^uXa G x- (38)

where G^v = Ga^vTa. Because the field strength defined in eq. (21) of (Fabiano, 2022c) contains also the terms cubic and quartic in A, beyond the triangle anomaly, we also have square and pentagon anomalies.

References

Adler, S.L. 1969. Axial-Vector Vertex in Spinor Electrodynamics. Physical Review, 177(5), pp.2426-2438. Available at: https://doi.org/10.1103/PhysRev.177.2426

Bell, J.S. & Jackiw, R. 1969. A PCAC puzzle: n0 ^ 77 in the a-model". II Nuovo Cimento A (1965-1970), 60(1), pp.47-61. Available at: https://doi.org/10.1007%2FBF02823296

Bollini, C.C. & Giambiagi, J.J. 1972. Dimensional renormalization : The number > of dimensions as a regularizing parameter. II Nuovo Cimento B (1971-1996), 12(1), co pp.20-26. Available at: https://doi.org/10.1007/BF02895558 ° Fabiano, N. 2022a. Path integral in quantum field theories. Vojnotehnicki glas-

& nik/Military Technical Courier, 70(4), pp.993-1016. Available at: ¡2 https://doi.org/10.5937/vojtehg70-35882

o Fabiano, N. 2022b. Regularization in quantum field theories. Vojnotehnicki

" glasnik/Military Technical Courier, 70(3), pp.720-733. Available at: < https://doi.org/10.5937/vojtehg70-34284

x

Fabiano, N. 2022c. Beta functions in the quantum field theory. Vojnotehnicki ô glasnik/Military Technical Courier, 70(1), pp.157-168. Available at: m https://doi.org/10.5937/vojtehg70-32131

Fujikawa K. 1979. Path Integral Measure For Gauge Invariant Fermion Theories. Physical Review Letters, 42(18), pp.1195-1197. Available at: https://doi.org/10.1103/PhysRevLett.42.1195

't 'Hooft, G. & Veltman, M. 1972. Regularization and renormalization of gauge fields. Nuclear Physics B, 44(1), pp.189-213. Available at: <5 https://doi.org/10.1016/0550-3213(72)90279-9

^ Noether, E. 1918. Invariante Variationsprobleme. Nachrichten von der

,q Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1918, pp.235-257 [online]. Available at: https://eudml.org/doc/59024 w [Accessed: 25 April 2022].

o Pauli, W. & Villars F. 1949. On the Invariant Regularization in Relativistic

g Quantum Theory. Reviews of Modern Physics, 21(3), pp.434-444. Available at: https://doi.org/10.1103/RevModPhys.21.434

Schwinger, J. 1959. Field Theory Commutators. Physical Review Letters, 3(6), p.296. Available at: https://doi.org/10.1103/PhysRevLett3.296

Yang, C.N. & Mills, R. 1954. Conservation of Isotopic Spin and Isotopic Gauge Invariance. Physical Review, 96(1), pp.191-195. Available at: https://doi.org/10.1103/PhysRev.96.191

Аномалии в квантовых теориях поля

Никола Фабиано

Белградский университет, Институт ядерных исследований «Винча» - Национальный институт Республики Сербия, г. Белград, Республика Сербия

РУБРИКА ГРНТИ: 29.05.03 Математические методы

теоретической физики, 29.05.23 Релятивистская квантовая теория.

Квантовая теория поля 29.05.33 Электромагнитное взаимодействие ВИД СТАТЬИ: обзорная статья

Резюме:

Введение/цель: Теорема Нётер (Ш1Ьег) устанавливает соответствие между обобщёнными симметриями Лагран-жа и сохраняемыми величинами. Квантовые эффекты отменяют законсервированные величины.

Методы: Треугольная диаграмма, интеграл по траекториям, регуляризация Паули-Вилларса.

Результаты: Квантовые эффекты, влияющие на законсервированные величины локальной калибровочной симметрии ставят под угрозу перенормируемость.

Выводы: Для получения точных результатов необходимо провести тщательные исследования. Распад п0 ^ 77 вероятно является самым уникальным "невозможным" эффектом, допускаемым аномалиями правильные результаты.

Ключевые слова: симметрия, квантовые аномалии.

Аномали]е у квантним теори]ама по^а Никола Фабиано

Универзитет у Београду, Институт за нуклеарне науке „Винча" -Национални институт Републике Срби]е, Београд, Република Срби]а

ОБЛАСТ: математика

КАТЕГОРША (ТИП) ЧЛАНКА: прегледни рад Сажетак:

Увод/цил: Нетерина (Ш1Ьег) теорема повезуjе симетри-jе Лагранжиана са конзервираним величинама. Квантни ефекти поништаваjу конзервиране величине.

Методе: Д^аграм троугла, интеграл пута, Паули-Виларсова регуларизац^а.

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Резултати: Квантни ефекти ко\и кваре конзервиране величине симетри]а локалне калибраци}е угрожава]у ренор-мализабилност.

Закъучак: Аномали}е jе потребно пажъиво третирати ка> ко би се добили тачни резултати. Распад п0 ^ yy jе можда на]наглашени]и „немогуЪи" ефекат ко\и дозвоъава}у анома-° ли]е.

ш Къучне речи: симетри]а, квантне аномали]е.

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Paper received on / Дата получения работы / Датум приема чланка: 30.05.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы / о Датум достав^а^а исправки рукописа: 28.01.2023.

ш Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 29.01.2023.

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ОТ © 2023 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military

^ Technical Courier" (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Данная статья в открытом доступе

О и распространяется в соответствии с лицензией "Creative Commons"

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