QFT Dimensional Analysis In h = c = 1 units, all quantities are measured in units of energy to some power. For example m = p µ = E +1 while x µ = E 1 where m stands for the dimensionality of the mass rather than the mass itself, and ditto for the p µ, x µ, etc. The action S = d 4 xl is dimensionless (in h 1 units, S = h), so the Lagrangian of a 4D field theory has dimensionality L = E +4. Canonical dimensions of quantum fields follow from the free-field Lagrangians. A scalar field Φ(x) has L free = 1 µφ µ Φ 1 m Φ, (1) so L = E +4, m = E +, and µ = E +1 imply Φ = E +1. Likewise, the EM field has L EM free = 1 4 F µνf µν = F µν = E +, () and since F µν = µ A ν ν A µ, the A ν (x) field has dimension Aν = Fµν / µ = E +1. (3) The massive vector fields also have A ν = E +1 so that both terms in L free = 4 1F µνf µν + 1 m A ν A ν (4) have dimensions F = m A = E +4. In fact, all bosonic fields in 4D spacetime have canonical dimensions E +1 because their kinetic terms are quadratic in µ (field). On the other hand, the fermionic fields like the Dirac field Ψ(x) with free Lagrangian L free = Ψ(iγ µ µ m)ψ (5) have kinetic terms with two fields but only one µ. Consequently, L = E +4 implies ΨΨ = E +3 and hence Ψ = Ψ = E +3/. Similarly, all other types of fermionic fields in 4D have canonical dimension E +3/. 1
In QFTs in other spacetime dimensions d 4, the bosonic fields such as scalars and vectors have canonical dimension Φ = Aν = E +(d )/ (6) while the fermionic fields have canonical dimension Ψ = E +(d 1)/. (7) In perturbation theory, dimensionality of coupling parameters such as λ in λφ 4 theory or e in QED follows from the field s canonical dimensions. For example, in a 4D scalar theory with Lagrangian L = 1 µφ µ Φ 1 m Φ n 3 the coupling C n of the Φ n term has dimensionality C n n! Φn, (8) Cn = L / Φ n = E 4 n. (9) In particular, the cubic coupling C 3 has positive energy dimension E +1, the quartic coupling λ = C 4 is dimensionless, while all the higher-power couplings have negative energy dimensions E negative. Now consider a theory with a single coupling g of dimensionality g = E. The perturbation theory in g amounts to a power series expansion M(momenta,g) = N ( g E ) N FN (momenta) (10) where E is the overall energy scale of the process in question and all the F N functions of momenta have the same dimensionality. The power series (10) is asymptotic rather than convergent, so it makes sense only when the expansion parameter is small, g 1. (11) E For a dimensionless coupling g, this condition is simply g 1, but for 0, the situation is more complicated.
For couplings of positive dimensionality > 0, the expansion parameter (11) is always small for for high-energy processes with E g 1/. But for low energies E < g 1/, the expansion parameter becomes large and the perturbation theory breaks down. This is a major problem for theories with > 0 couplings of massless particles. However, if all the particles participating in a > 0 coupling are massive, then all processes have energies E > M lightest, and this makes couplings with > 0 OK as long as g M lightest. (1) Couplings of negative dimensionality < 0 have an opposite problem: The expansion parameter (11) is small at low energies but becomes large at high energies E > g 1/. Beyond the maximal energy E max g 1/( ), (13) the perturbation theory breaks down and we may no longer compute the S matrix elements M using any finite number of Feynman diagrams. Worse, in Feynman diagrams with loops one must worry not only about momenta k µ of the incoming and outgoing particles but also about momenta q µ of the internal lines. Basically, an L loop diagram contributing to N th term in the expansion (10) produces something like g N d 4L qf N (q,k,m) where FN = E N 4L+C, C = const. (14) For very large loop momenta q k,m, dimensionality implies F N q N 4L+C, so for N( ) +C 0, the integral (14) diverges as q. Moreover, the degree of divergence increases with the order N of the perturbation theory, so any scattering amplitude becomes divergent at high orders. Therefore, field theories with < 0 couplings do not work as complete theories. However, theories with < 0 may be used as approximate effective theories(without the divergent loop graphs) for low-energy processes, E < Λ for some Λ < g 1/. For example, 3
the Fermi theory of weak interactions L int = G F appropriate fermions Ψγ µ (1 γ 5 )Ψ Ψγ µ (1 γ 5 )Ψ (15) has coupling G F of dimension G G = E ; its value is G F 1.17 10 5 GeV. This is a good effective theory for low-energy weak interactions, but it cannot be used for energies E > 1/ G F 300 GeV, not even theoretically. In real life, the Fermi theory works only for E M W 80 GeV; at higher energies, one should use the complete SU() U(1) electroweak theory instead of the Fermi theory. Similar to the Fermi theory, most effective theories with < 0 couplings are low-energy limits of more complicated theories with extra heavy particles of masses M < g 1/ but no < 0 couplings. In QFTs which are valid for all energies, all coupling must have zero or positive energy dimensions. In 4D, a coupling involving b bosonic fields (scalar or vector), f fermionic fields, and δ derivatives µ has dimensionality = 4 b 3 f δ. (16) Thus, only the boson 3 couplings have > 0 while the = 0 couplings comprise boson 4, boson fermion, and boson boson. All other coupling types have < 0 and are not allowed (except in effective theories). Here is the complete list of the allowed couplings in 4D. 1. Scalar couplings µ 3! Φ3 and λ 4! Φ4. (17) Note: the higher powers Φ 5, Φ 6, etc., are not allowed because the couplings would have < 0. 4
. Gauge couplings of vectors to charged scalars iqa µ (Φ µ Φ Φ µ Φ ) + q Φ ΦA µ A µ D µ Φ D µ Φ. (18) 3. Non-abelian gauge couplings between the vector fields gf abc ( µ A a ν )Aµb A νc g 4 fabc f ade A b µ Ac ν Aµd A νe 1 4 Fa µν Fµνa. (19) 4. Gauge couplings of vectors to charged fermions, qa µ Ψγ µ Ψ Ψ(iγ µ D µ )Ψ. (0) If the fermions are massless and chiral, we may also have or in the Weyl fermion language qa µ Ψγ µ 1 γ5 Ψ, (1) qa µ χ σ µ χ. 5. Yukawa couplings of scalars to fermions, yφ ΨΨ or iyφ Ψγ 5 Ψ. () If parity is conserved, in the first terms Φ should be a true scalar, and in the second term a pseudo-scalar. In other spacetime dimensions d 3+1, a coupling involving b bosonic fields, f fermionic fields, and δ derivatives has energy dimension = d b d f d 1 δ = b + 1 b+f f δ d. (3) Since all interactions involve three or more fields, thus b+f 3, the dimensionality of any particular coupling always decreases with d. Consequently, there are more perturbativelyallowed couplings with 0 in lower dimensions d = +1 or d = 1+1 but fewer allowed couplings in higher dimensions d > 3+1. In particular, 5
In d 6 + 1 dimensions all couplings have < 0 and there are no UV-complete quantum field theories, or at least no perturbative UV-complete quantum field theories. In d = 5+1 dimensions there is a unique = 0 coupling (µ/6)φ 3, while all the other couplings have < 0. Consequently, the only perturbative UV-complete theories are scalar theories with cubic potentials, L = a ( 1 ( µ Φ a ) 1m a a) Φ 1 6 µ abc Φ a Φ b Φ c. (4) a,b,c However, while such theories are perturbatively OK, they do not have stable vacua. Indeed, a cubic potential is un-bounded from below it goes to along half of the directions in the field space so even if it has a local minimum at Φ a = 0, it s not the global minimum. Consequently, in the quantum theory, the naive vacuum with Φ a = 0 would decay by tunneling to a run-away state with Φ a ±. In d = 4+1 dimensions, the (µ/6)φ 3 coupling has positive = + 1 while all the other couplings have negative energy dimensions. Again, the only perturbative UV-complete theories are scalar theories with cubic potentials, but they do not have stable vacua. The bottom line is, in d > 3 + 1 dimensions, all quantum field theories are effective theories for low-enough energies. At higher energies, a different kind of theory must take over perhaps a theory in a discrete space, perhaps a string theory, or maybe something more exotic. On the other hand, in lower dimensions d = +1 or d = 1+1 there are a lot of allowed couplings with 0. In particular, in d = +1 dimensions the allowed couplings include: Scalar couplings (C n /n!)φ n up to n = 6; gauge and Yukawa couplings like in 4D; Yukawa-like couplings ỹφ ΨΨ involving scalars; gauge-like couplings with g gauge linearly dependent on a neutral scalar field: D µ Ψ = µ Ψ + i(g 0 +cφ)a µ Ψ = Ψ(iγ µ D µ )Ψ cφa µ Ψγ µ Ψ, (5) 6
and likewise D µ Φ D µ Φ icφa µ (Φ µ Φ Φ µ Φ ) + c φ Φ ΦA µ A µ, (6) or non-abelian 1 4 Fa µν Fµνa cφ f abc ( µ A a ν )Aµb A νc c 4 φ f abc f ade A b µ Ac ν Aµd A νe. (7) There are also combinations of g = g 0 +cφ with Chern Simons couplings (like the 3D photon mass term in the mid-term exam and its non-abelian generalizations), but I don t want to get into their details here. There might be some other allowed couplings in 3D, but never mind for now. Finally, in d = 1+1 dimensions there is an infinite number of allowed 0 couplings. Indeed, for d = 1+1the bosonicfields have energy dimension E 0, so ofacoupling doesnot depend on the number b of bosonic fields it involves but only on the numbers of derivatives and fermionic fields, = δ 1 f. (8) Consequently, all scalar potentials V(Φ) including C n Φ n terms for any n, and even the non-polynomial potentials have = +, so any V(Φ) is allowed in D. Likewise, all Yukawa-like couplings Φ n ΨΨ have = +1, so we may have terms like y IJ (Φ) Ψ I Ψ J for any functions y IJ (Φ). At the = 0 level, we are allowed generic Riemannian metrics g ij (Φ) of the scalar field space, hence field-dependent kinetic terms L kin = 1 g ij(φ) µ φ i µ φ j, (9) as well as a whole bunch of fermionic terms with arbitrary scalar-dependent coefficients, L Ψ 1 4 g IJ (Φ) ΨI γ µ ( i µ i µ ) Ψ J + Γ IJk (Φ) µ Φ k Ψ I γ µ Ψ J + 1 R IJKL(Φ) Ψ I γ µ Ψ J Ψ K γ µ Ψ L. In addition, there are gauge couplings with arbitrary scalar dependent g gauge (Φ), chiral couplings to Weyl or Majorana-Weyl fermions, etc., etc.. (30) 7