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ПРЕГЛЕДНИ РАДОВИ ОБЗОРНЫЕ СТАТЬИ
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BETA FUNCTIONS IN THE QUANTUM FIELD THEORY
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Nicola Fabiano
University of Belgrade, "Vinca" Institute of Nuclear Sciences -Institute of National Importance for the Republic of Serbia, Belgrade, Republic of Serbia, e-mail: [email protected], °
ORCID iD: https://orcid.org/0000-0003-1645-2071
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DOI: 10.5937/vojtehg70-32131;https7/doi.org/10.5937/vojtehg70-32131
FIELD: Mathematics ARTICLE TYPE: Review paper
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Abstract: "S
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Introduction/purpose: The running of the coupling constant in various Quantum Field Theories and a possible behaviour of the beta function are illustrated. ¡5
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Methods: The Callan-Symanzik equation is used for the study of the beta function evolution.
Results: Different behaviours of the coupling constant for high energies are observed for different theories. The phenomenon of asymptotic freedom is of particular interest.
Conclusions: Quantum Electrodynamics (QED) and Quantum Chromo-dinamics (QCD) coupling constants have completely different behaviours in the regime of high energies. While the first one diverges for finite energies, the latter one tends to zero as energy increases. This QCD phenomenon is called asymptotic freedom.
Key words: Quantum Electrodynamics, Quantum Chromodynamics, Quantum Field Theory, renormalization group, beta function.
Fixed points
In (Fabiano, 2021) we have seen how a generic coupling constant behaves at different renormalisation scales. It should be remarked that this result is valid also for different renormalisation schemes, not only for dimensional regularisation. In this sense the coupling constant is a func-
tion depending on the energy scale and is often regarded to as running coupling constant. Just for the sake of simplicity define the new variable p~ t = log^ (the t variable could be also thought of as a "time" parameter). With this position, the Callan-Symanzik equation (Callan, 1970; Symanzik, 1970) could be rewritten in a nicer form as:
* §=«»> • <1)
UJ
which is a differential equation governing the behaviour of the coupling cono stant g upon the energy scale considered. As such it also needs some initial ^ conditions in order to be solved - a Cauchy problem. The points g for which
m = o (2)
o
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o
are called fixed points (Symanzik, 1971), and once the coupling g reaches £ one of these points, it does not evolve anymore. In Fig. (1) a possible scenario for the fi function is shown. The origin 0, the points gi and g2 are fixed points. If for the initial scale t = 0 the coupling constant g is at one of these points, then it will remain there for any energy scale considered (or "forever", depending on the language one prefers).
There are different kinds of fixed points. Consider the point g1 and its neighbourhood. From the Figure, for 0 <g < g1, fi(g) > 0 then the coupling y constant increases with the scale because of eq. (2) (i.e. dg/dt > 0), moving towards g1 for t ^ On the contrary, in the interval g1 < g < g2 the ¡5 fi function is negative, so the coupling constant decreases and approaches again g1 as t ^ We conclude that g1 is a stable fixed point, as g tends to it from either side. It is called the ultraviolet stable fixed point - the term "ultraviolet" is present because g ^ g1 as t ^
On the other hand, for the points 0 and g2 it is clear that the inverse of the previous argument holds true: the coupling g "escapes" from them as t ^ and approaches them as energy decreases, for t ^ 0. Such points are named the infrared stable fixed points.
It is important to know that the fixed points of the fi function are difficult to calculate because they are usually determined by nonperturbative effects, apart from the trivial zero at the origin, for g = 0.
Behaviour of p function
We shall consider some possible asymptotic behaviours of the fi function for energy scale ^ ^ The exact problem we consider is given
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Figure 1 - The Beta function with fixed points 0, g1 and g2. The arrows indicate
the direction of the flow of g with increasing scale / Рис. 1 - Бета-функция с фиксированными точками 0, gi и g2. Стрелки указывают направление потока g с увеличением шкалы / Слика 1 - Бета функц^а са фиксним тачкама 0, g1 и g2. Стрелице показ^у правац тока g са повеЪаъем скале /
<5
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log (. (4)
by
=
g(po) = go, (3)
whose formal solution is written as
J9o P(g) W
Different behaviours of the " function are shown in Fig. 2. For such functions, the running coupling constant g will, for different cases:
(a) approach infinity for a finite value of g, with "(g) > 0
— b .... d
Figure 2 - Beta functions with different asymptotic behaviour Рис. 2 - Бета-функции с различным асимптотическим поведением Слика 2 - Бета функци]е с различитим асимптотским понашаъем
(b) approach infinity as g ^
(c) have a finite fixed point in gi, ^(gi) = 0
(d) approach -ro for increasing g, with ^(g) < 0
Case (a)
Suppose that ^(g) grows sufficiently rapidly in such a manner that the integral of eq. (4) converges (for instance, any power of g larger than 1), namely
fdg
в(д)
< ,
(5)
then it is clear that the scale ^ has a finite upper bound corresponding to the coupling g = given by the relation
= ^q exp
'go
в(g)J '
(6)
We have already encountered such behaviour, for the QED coupling § case as discussed in (Fabiano, 2021), where ^(g) = g3/12n2 and is 1 given by the Landau pole (Landau et al, 1954; Landau & Pomeranchuk, 1955) of eq. (40) in (Fabiano, 2021).
Another example is the scalar field theory with the interaction term ° $^4/4! given by the Lagrangian
l=2^)2 - mm2^ - 4^, (7)
3g2 18 g3
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for which the ^ function is
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The Lagrangian (7) is almost the same of the Higgs field in the Standard Model (Glashow, 1959; Salam & Ward, 1959; Weinberg, 1967). The ^ only difference, yet an essential one, is that in the latter case the scalar m field is coupled to fermion fields ^ via a term A^^^, so called Yukawa coupling (Yukawa, 1935), where A is another coupling constant different from g.
Using only the first term of eq. (8), we arrive at the expression
g =-3 g0 ^ , (9)
1 - 132go log(£)
which has the same form of eq. (39) in (Fabiano, 2021) as anticipated; it has also a pole for ^ = exp(16n2/(3g0)).
Case (b)
The integral of eq. (5) diverges. It means that the coupling constant g becomes infinite only at an infinite energy scale, ^ = For instance, assume that ^(g) = agk, with a > 0 and k < 1 but k = -1 - then one obtains for eq. (3) the solution
i/(i-fc)
. (10)
The growth of g in ^ is very slow, but in the very high energy limit the coupling becomes independent from the initial condition g0.
g=
g1-k + a(1 - k) log (
Case (c)
We encounter a fixed point like previously discussed in the section Fixed points for the ultraviolet fixed point gi, that is "(gi) = 0. The " function stays positive for 0 < g < g1 and turns negative afterwards. Either if the initial condition g0 is such that g0 < g1 or g0 > g1 the coupling constant g will evolve towards the fixed point, g ^ g1 as ^ ^ +rc>. Assuming that the root of" in g1 is simple, then
"(g) = a(g1 - g) for g ^ g1 (11)
with a > 0. The solution to eq. (3) is then
g1 - g - n~a (12)
with the assumption that g0 < g, g0 < g1 and g < g1.
It is worth noticing that we have already discussed a case in which, apparently, an ultraviolet fixed point is obtained. The scalar theory presents such a point. From eq. (8) one computes the fixed point g1 as
g1 = 8n2, (13)
which, however, has a huge value of g1 ^ 80 thus spoiling the perturbation theory as g » 1. As the " functions that have been encountered so far have been computed using only the perturbation theory, it is clear that the result obtained above is invalid. The discussion regarding eq. (8) proves the statement of the section Fixed points, for which a fixed point could be basically only computed by means of nonperturbative techniques.
Case (d)
So far, all " functions discussed were positive at least for small positive g, so the renormalisation group flow drives away g(^) from the origin g = 0. Now suppose that "(g) < 0 for small positive g, like
" (g) = —agn , (14)
where a > 0, n > 1 and an integer. The solution to eq. (3) is then written as
- g0 (15)
1+ gn-1 (n — 1)a log( £)
1/(n 1)
A dramatic difference between this and the previous cases is that, for § large energy scales, the coupling constant vanishes, i.e.
g = 0 for ^ ^ . (16)
C = ^ Cngn (17)
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This phenomenon is called asymptotic freedom (Gross & Wilczek, 1973; Politzer, 1973). With growing energy, the theory has a weaker coupling constant, approximating a free theory, i.e. one without interactions. So at larger energy scales, the perturbation theory gives better results. Remember actually that corrections C of any kind (propagator, coupling, etc.) are w computed as series of powers of g, °
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c
CO c o
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and this formal series is supposed to converge for small g.
A toy model that exhibits asymptotic freedom could be obtained from
the Lagrangian (7) with a negative potential1 -g^4/41. Its " function has m the form
№) = -16-2 g2 (18) |
for which
g =[dh log( £)] 1 for ^ ". (19)
i.e. g goes to zero at logarithmic speed.
A very important class of theories that have the property of asymptotic freedom is the Yang-Mills theory (Yang & Mills, 1954), with the gauge group SU(N). Of particular relevance is one of them, quantum chromodynamics - QCD - that is the theory of strong interactions embedded in the Standard Model, whose gauge group is SU(3). The QCD Lagrangian is written as
L = ^ [¿(7^)jfc - mj] ^k - 4Gif (20)
where ^ (x) is the j-th quark field, indexed by j, k; Ai are the gluon fields, a = 1... 8. y^ are the usual Dirac matrices, the covariant derivative is given
1 We neglect the fact that this theory is ill-defined and that the perturbation theory cannot be applied.
by D = <9^ - igA^T". Gis the gluon field strength tensor, similar to the electromagnetic tensor, defined by
G% = ^A" - dvA" + gfabcAbA (21)
where fabc are the structure constants of SU(3), [Ta, Tb] = if abcTc, with T" being generators of the group.
For a generic SU(N) the Yang-Mills theory coupled to fermions the £ function at one-loop level is given by
£(g) = - A (11N - 1+ O(g5) , (22)
4n2 V12 3
and for the QCD case C2 = nf /2,
g3 A1 f , /n^
£ <»> =- T - if J + O(g5) ■ (23)
where nf is the number of quark flavours with masses much lower than the energy scale considered which can be considered massless.
Defining the QCD strong coupling constant as = g2/4n2 in an analogous fashion to QED, where a = e2/4n, we obtain from eq. (49) in (Fabiano, 2021)
asW = (33 - fkgWA) (24)
which exhibits asymptotic freedom as far as the number of quark flavours is nf < 17. Another property due to the presence of (approximately) massless particles is that a dimensionless coupling g0 is exchanged for a dimension-ful parameter A, which is an integration constant with dimensions of energy. This phenomenon is referred to as dimensional transmutation (Coleman & Weinberg, 1973; Weinberg, 1973). The £ function eq. (24) is known today to four-loop order O(a4), with three and four-loop coefficients being renormalisation scheme dependent. The measured value of a strong coupling constant at the Z peak is
as(mz) = 0.1197 ± 0.0016 , (25)
while the corresponding value of A is about 0.2 GeV.
A few remarks are in order. In the 1950s, Landau argued that in QED the increasing powers of logarithmic terms, that we already encountered at
one-loop level in (Fabiano, 2021), of the form log(E/M), would coalesce § and give raise to singularities for finite values of the energy E. This is the (a) case, with the Landau poles, also known as the Landau ghosts or the Moscow zero (because e0/e(u) = 0), discovered by himself (Landau et al, 1954; Landau & Pomeranchuk, 1955). This argument does not rule out the ° cases (b) or (c), though. This possible inconsistency in the renormalisation procedure has not yet been proved but it is believed to actually exist.
Today, there is a broad agreement on the fact that the interacting field theories like QED or scalar we have discussed (which are not asymptotically free) are not mathematically consistent. About QED, there is some evidence against the case (c) with a finite fixed point that would be only possible in the presence of yet unknown nonperturbative effects. However, even if (c) is ruled out, there still remains the possibility (b) with a fixed point at infinity. £
There is an electromagnetic analogy for different behaviours of QED and QCD couplings. In QED, the charge is stronger at shorter distances, i.e. is the vacuum acts like a dielectric medium with a dielectric constant
e > 1, (26)
shielding the charge. Remembering the relation of the relative magnetic permeability ^ to the dielectric constant to the speed of light, which in our units is 1,
e^ = 1 , (27)
we have a duality relation. The QED case corresponds to ^ < 1, also known as Landau diamagnetism, where charged particles in the medium in response to an external magnetic field generate an opposed magnetic field, a phenomenon seen in superconductors, water, copper, and gold. In QCD, the opposite behaviour is observed: the chromoelectric charge is weaker at shorter distances, so its vacuum is anti screening, with a dielectric constant
e < 1 . (28)
The equivalent magnetic permeability is ^ > 1, known as Pauli para-magnetism, where the particles tend to align with the external field, as in tungsten, aluminium, or lithium. It has to be stressed that the electromagnetic terminology used for QCD is just an analogy to the QED case: by "the charge" we mean the colour charge, by "the magnetic moment" the colour magnetic moment.
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References
Callan, C.G. 1970. Broken Scale Invariance in Scalar Field Theory. Physical Review D, 2(8), pp.1541-1547. Available at: https://doi.Org/10.1103/PhysRevD.2.1541.
Coleman, S. & Weinberg, E. 1973. Radiative Corrections as the Origin of Spontaneous Symmetry Breaking. Physical Review D, 7(6), pp.1888-1910. Available at: https://doi.org/10.1103/PhysRevD7.1888.
Fabiano, N. 2021. Quantum electrodynamics divergencies. Vojnotehnicki glas-nik/Military Technical Courier, 69(3), pp.656-675. Available at: https://doi.org/10.5937/vojtehg69-30366.
Glashow, S. 1959. The renormalizability of vector meson interactions. Nuclear Physics, 10(February-May), pp.107-117. Available at: https://doi.org/10.1016/0029-5582(59)90196-8.
Gross, D.J. & Wilczek F. 1973. Asymptotically Free Gauge Theories. I. Physical Review D, 8(10), pp.3633-3652. Available at: https://doi.org/10.1103/PhysRevD.8.3633.
Landau, L.D., Abrikosov, A.A. & Khalatnikov, I.M. 1954. Dokl. Akad. Nauk SSSR, 95, 497, 773, 1177 (in Russian).
Landau, L.D. & Pomeranchuk, I.Ya. 1955. Dokl. Akad. Nauk SSSR, 102, 489 (in Russian).
Politzer, H.D. 1973. Reliable Perturbative Results for Strong Interactions? Physical Review Letters, 30(26) pp.1346-1349. Available at: https://doi.org/10.1103/PhysRevLett.30.1346.
Salam, A. & Ward, J.C. 1959. Weak and electromagnetic interactions. Il Nuovo Cimento, 11(4), pp.568-577. Available at: https://doi.org/10.1007/BF02726525.
Symanzik, K. 1970. Small distance behaviour in field theory and power counting. Communications in Mathematical Physics, 18(3), pp.227-246. Available at: https://doi.org/10.1007/bf01649434.
Symanzik, K. 1971. Small-distance-behaviouranalysis and Wilson expansions. Communications in Mathematical Physics, 23(1), pp.49-86. Available at: https://doi.org/10.1007/BF01877596.
Weinberg, S. 1967. A Model of Leptons. Physical Review Letters, 19(21), pp.1264-1266. Available at: https://doi.org/10.1103/PhysRevLett.19.1264.
Weinberg, S. 1973. New Approach to the Renormalization Group. Physical Review D, 8(10), pp.3497-3509. Available at: https://doi.org/10.1103/PhysRevD.8.3497.
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.
Yukawa, H. 1935. On the interaction of elementary particles. Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, 17, pp.48-75. Available at: https://doi.org/10.11429/ppmsj1919.17.0_48 .
БЕТА-ФУНКЦИИ В КВАНТОВОЙ ТЕОРИИ ПОЛЯ
Никола Фабиано
Белградский университет, Институт ядерных исследований «Винча» - Институт государственного значения для 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: В данной статье представлено, как работает константа связи в различных квантовых теориях поля и возможные модели поведения бета-функции.
Методы: Уравнение Каллана-Симанзика используется для изучения эволюции бета-функции.
Результаты: Наблюдается различное поведение константы связи при высоких энергиях в различных теориях. Особый интерес представляет явление асимптотической свободы.
Выводы: Константы связи квантовой электродинамики (КЭД) и квантовой хромодинамики (КХД) ведут себя совершенно по-разному в режиме высоких энергий. Первая отличается конечной энергией, в то время как вторая стремится к нулю, когда энергия увеличивается. Данное явление КХД называется асимптотической свободой.
Ключевые слова: квантовая электродинамика, квантовая хромодинамика, квантовая теория поля, ренормализаци-онная группа, бета-функция.
БЕТА ФУНКЦШЕ У КВАНТНОJ ТЕОРШИ ПО^А
Никола Фабиано
Универзитету Београду, Институт за нуклеарне науке "Винча"-Институт од националног знача]а за Републику Срби]у, Београд, Република Срби]а
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ОБЛАСТ: математика ВРСТА ЧЛАНКА: прегледни рад
Сажетак:
Увод/цил: Илустровани су рад константе спреге у разним квантним теори}ама пола као и могуче понашак>е бета функци]е.
Методе: Калан-Шиманзикова ¡едначина користи се за про-учаваъе еволуци]е бета функци]е.
Резултати: Применено jе различито понашак>е константе спреге за високе енерги]е за различите теорбе. Од по-себног интереса }е феномен асимптотске слободе.
Заклучак: Константе спреге квантне електродинамике (QED) и квантне хромодинамике (QCD) има]у потпуно ра зличито понашак>е у режиму високих енергща. Док се прва разилази за коначне енерги]е, друга тежи нули како се енер-ги}а повеЬава. Ова] феномен QCD назива се асимптотска слобода.
Клучне речи: квантна електродинамика, квантна хромо-динамика, квантна теори]а пола, ренормализациона гру-па, бета функци]а.
Paper received on / Дата получения работы / Датум приема чланка: 05.03.2021. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 30.12.2021.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 31.12.2021.
© 2022 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
© 2022 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 и распространяется в соответствии с лицензией "Creative Commons" (http://creativecommons.org/licenses/by/3.0/rs/).
© 2022 Аутори. Об]авио Во]нотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Ово ]е чланак отвореног приступа и дистрибуира се у складу са Creative Commons лиценцом (http://creativecommons.org/licenses/by/3.0/rs/).
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