arXiv:1506.06227v1 [hep-th] 20 Jun 2015 ‡ † ∗ eomlzblt ttesm iei n pctm di- spacetime any in time mension same the and at unitarity achieve re- renormalizability to second possible the makes (or which in quirement, lies theory electrodynamics) Yang-Mills quantum Abelian standard with (or difference perturbative main nonlocality quantum The weak finiteness. or super- (ii) quantum renormalizability (iv) invariance; unitarity; (iii) gauge prop- frame- quasi-polynomiality); following (i) the theory satisfy field theories erties: desired quantum The the [1]. pertur- work in of “validity expansion” “super- and bative to interactions: finiteness” fundamental principles or the guiding renormalizability all two to following common the be require also We t,te h omrhv opsestesm quantum same theories the gauge scale-invariant in possess Furthermore, to have grav- properties. former gauge finite the study or then to super-renormalizable ity, want to we coupled super- whether theories the Moreover, Higgs the in stabilizing unifi- constants potential. 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The theory above consists of a weakly nonlocal kinetic any dimension as we are going to explicitly show in the operator and a local curvature potential Vg crucial to following subsections. achieve finiteness of the theory as we will show later. Moreover, at classical level many evidences endorse In (1) the Lorentz indices and tensorial structures have that we are dealing with “gauge theories possessing been neglected. The notation on the flat spacetime singularity-free exact solutions”. The discussion here is reads as follows: we use the gauge-covariant box op- closely analogous to the gravitational case [5–11]. In par- 2 µ erator defined via D = DµD , where Dµ is a gauge- ticular the static gauge potential for the exponential form covariant derivative (in the adjoint representation) act- factor exp(−/Λ2) is for weak fields given approximately F a a ing on gauge-covariant field strength ρσ = Fρσ T of by: a the gauge potential Aµ (where T are the generators Λr of the gauge group in the adjoint representation.) The Erf( 2 ) Φgauge(r)= A0(r)= g . (4) metric tensor gµν has signature (− + ··· +). We em- r ploy the following definition, D2 ≡ D2/Λ2, where Λ is Λ 2 an invariant mass scale in our fundamental theory. Fi- We used the form factor exp(−/Λ ) and D = 4 to end −1 2 up with a simple analytic solution. However, the result is nally, the entire function V (z) ≡ exp H(z) (z ≡ DΛ) in (1) satisfies the following general conditions [3], [4]: (i) qualitatively the same for the asymptotically polynomial V −1(z) is real and positive on the real axis and it has form factor (3), and Φgauge(r) = const for r = 0. no zeros on the whole complex plane |z| < +∞. This requirement implies, that there are no gauge-invariant poles other than for the transverse and massless gluons. A. Propagator, unitarity and divergences (ii) |V −1(z)| has the same asymptotic behaviour along the real axis at ±∞. (iii) There exists Θ ∈ (0,π/2) Splitting the gauge field into a background field (with D such that asymptotically |V −1(z)| → |z|γ+ 2 −2, when flat gauge connection) plus a fluctuation, fixing the gauge |z| → +∞ with γ > D/2 (D is even and γ natural) for freedom and computing the quadratic action for the fluc- complex values of z in the conical regions C defined by: tuations, we can invert the kinetic operator to get finally C = {z | −Θ < argz < +Θ , π−Θ < argz < π+Θ}. This the two-point function. This quantity, also known as the condition is necessary to achieve the maximum conver- propagator in the Fourier space reads, up to gauge de- gence of the theory in the UV regime. (iv) The difference pendent components, −1 −1 V (z) − V∞ (z) is such that on the real axis 2 2 −1 −iV (k /Λ ) kµkν O (k)= ηµν − , (5) −1 −1 µν k2 + iǫ k2 V (z) − V∞ (z) m N   lim −1 z =0, for all m ∈ , (2) |z|→∞ V∞ (z) where we used the Feynman prescription (for dealing

−1 with poles). The tensorial structure in (5) is the same where V∞ (z) is the asymptotic behaviour of the form −1 of the local Yang-Mills theory, but we see the presence of factor V (z). Property (iv) is crucial for the locality of a new element – multiplicative form factor V (z). If the counterterms. The entire function H(z) must be chosen function V −1(z) does not have any zeros on the whole in such a way that exp H(z) tends to a polynomial p(z) complex plane, then the structure of poles in the spec- in UV hence leading to the same divergences as in higher- trum is the same as in original two-derivative theory. derivative theories. This can be easily proved in the Coulomb gauge, which is An explicit example of weakly nonlocal form factor H(z) manifestly unitary. Therefore, in the spectrum we have e , that has the properties (i)-(iv) can be easily con- exactly the same modes as in two-derivative theories. In structed following [4], this way we have achieved unitarity, but the dynamics

− 2 2 is modified from the simple two-derivative to a super- 1 γE eH(z) = e 2 [Γ(0,e p(z) )+log(p(z) )] renormalizable one with higher-derivatives. Despite that − 2 −e γE p(z) in the UV regime we recover polynomial higher-derivative 2 e = p(z) 1+ −γ 2 + ... , (3) theory, the analysis of tree-level spectrum still gives us a z∈R 2e E p(z) ! p unitary theory without ghosts, because the renormaliz- ability is due to the behaviour of the theory in the very where γE ≈ 0.577216 is the Euler-Mascheroni constant UV limit, while unitarity is influenced by the behaviour +∞ −t and Γ(0, x) = x dt e /t is the incomplete gamma at any energy scale. function with its first argument vanishing. The poly- R In the high energy regime (UV), the propagator in mo- nomial p(z) of degree γ +(D −4)/2 is such that p(0) = 0, mentum space schematically scales as which gives the correct low energy limit of our theory coinciding with the standard two-derivative Yang-Mills O−1(k) ∼ k−(2γ+D−2) . (6) theory. In this case the Θ-angle defining cones C turns out to be π/(4γ + 2(D − 4)). The vertices of the theory can be collected in different The theories described by the action in (1) are unitary sets, that may involve or not the entire function exp H(z). and perturbatively renormalizable at quantum level in However, to find a bound on quantum divergences it 3 is sufficient to concentrate on the polynomial operators because in dimensional regularization scheme (DIMREG) with the high energy leading behaviour in the momenta there are no divergences at one-loop and the theory is k [3, 4]. These operators scale as the propagator, they automatically finite. The reason is of dimensional na- cannot have higher power of momentum k in the scal- ture. In odd dimension the energy dimension of possible ing, in order not to break the renormalizability of the one-loop counterterms needed to absorb logarithmic di- theory. The consideration of them gives the following vergences can be only odd. However, at one-loop such upper bound on the superficial degree of divergence of counterterms cannot be constructed in DIMREG scheme any graph [4, 12, 13], and having at our disposal only Lorentz invariant (and gauge-covariant) building blocks that always have energy ω(G) ≤ DL + (V − I)(2γ + D) − E. (7) dimension two. By elementary building blocks we mean here field strengths or gauge-covariant box operators, or This bound holds in any spacetime, of even or odd di- even number of covariant derivatives (even number is nec- mensionality. In (7) V is the number of vertices, I the essary here to be able to contract all indices). For details number of internal lines, L the number of loops, and E we refer the reader to original papers [12]. is the number of external legs for the graph G. After plugging the topological relation I − V = L − 1 in (7) we In even dimensions we for simplicity consider the poly- γ+ D −2 get the following simplification: nomial p(z) to be a monomial, pγ(z)= ωz 2 (ω is a positive real parameter). In this minimal setup the mono- ω(G) ≤ D − 2γ(L − 1) − E. (8) mial in UV gives precisely the highest derivative term of F 2 γ F the form tr (DΛ) (in D = 4). There is only one We comment on the situation in odd dimensions in the possible way how to take trace over group indices here, next section. Thus, if in even dimensions γ > (D −E)/2, and terms with derivatives
can be reduced to those with in the theory only 1-loop divergences survive. There- gauge-covariant boxes only by exploiting Bianchi identi- fore, the theory is one-loop super-renormalizable [4, 15– ties in gauge theory. These latter terms take the explicit 18] and only a finite number of operators of energy di- a 2 γ µν form Fµν (DΛ) Fa . In four dimensions there is an RG mensions up to M D has to be included in the action to running of only one coupling constant. The contribu- absorb all perturbative divergences. In a D−dimensional tion to the beta function of the YM coupling constant spacetime the renormalizable gauge theory includes all from this quadratic term is actually a dimensionless con- the operators up to energy dimension M D, and schemat- stant (independent of the frontal coefficient of the highest ically reads derivative term), which has been computed in [19] using Feynman diagrams. This number can be cancelled by a 1 L = − tr F2 + F3 + F D2 F + ··· + FD/2 . (9) contribution coming from a quartic (in field strengths) D 4g2 h i gauge killer of the form In gauge theory the scaling of vertices originating from kinetic terms of the type F(D2)γ+(D−4)/2F is lower than sg F2 2 γ−2F2 − tr (DΛ) (10) the one seen in the inverse propagator k2γ+D−2. This is 4g2 because when computing variational derivatives with re-
spect to the dimensionful gauge potentials (to get higher (here there are several possibilities of taking traces). The point functions) we decrease the energy dimension of the contribution to the beta function is linear in the param- result. Hence the number of remaining partial deriva- eter sg and hence the latter one can be adjusted to make tives, when we put the variational derivative on the the total beta function vanish. flat connection background, must be necessarily smaller. The action of the finite quantum theory may take the This means that we have a smaller power of momen- following compact form (for the choice γ = 3 the general tum, when the 3-leg (or higher leg) vertex is written in derivative structure is explicit in D = 4): momentum space. We get the maximal scaling for the gluons’ 3-vertex, and it is with the exponent 2γ + D − 3. 1 2 L = − tr FeH(DΛ)F + s F2(D2 )F2 In this way we can put an upper bound on the degree fin, gauge 4g2 g Λ of divergence for higher-derivative gauge theories even h minimal finite theory with a little excess. Again, for higher-derivative gauge 5 5−j theories and γ > (D − E)/2 we have one-loop super- | {z } + c(j,k) (D2 )kFj , (11) renormalizability. For the minimal choice E = 2 (because i Λ i i j>2 k=0 the tadpole diagram vanishes) we have γ > (D − 2)/2. X X X
i

(j,k) where ci are some constant coefficients. The beta B. Finite gauge theories in odd and even function can succesfully killed by the last operator in the dimensions first line above. The last terms in the formula (11) have been written in a compact index-less notation and the In odd number of dimensions we can easily show that index i counts all possible contractions of Lorentz and the theory is finite without need of gauge potential Vg group indices. 4

II. THE FINITE THEORY IN D = 4 To fix our conventions, we can read the beta function from the counterterm operator, namely As extensively motivated in the previous section the α 1 minimal nonlocal gauge theory in D = 4 candidate to be L := − (Z − 1) F µν F a = −L = − β F µν F a . ct 4 α a µν div ǫ α a µν scale-invariant (finite) at quantum level is: By using the Batalin-Vilkovisky formalism [21] it is pos- α 2 F H(DΛ)F F2 2 γ−2F2 Lfin, gauge = − tr e + sg (DΛ) , (12) sible to prove that for the theory (12) there is no wave- 4 function renormalization for the gauge field Aa . We have h i µ only renormalization of the gauge coupling constant. The where the function H(z) is given in (3). We here evaluate (γ) (sg ) contribution to the beta function β due to the nonlocal the contribution to the beta function β from the two α α kinetic term was obtained in [19], namely following independent killer operators quartic in the field 1 strength 2 (γ) (5+3γ + 12γ ) β = − C2(G) , γ ≥ 2 , (20) s α 192π2 g a µν γ−2 b ρσ 1. − 2 Fµν Fa Λ Fρσ Fb , (15) 4g where C (G) is the quadratic Casimir of the gauge group s 2 g a µν 2 γ−2 b ρσ G. By imposing the following condition for scale invari- 2. − 2 Fµν Fb (DΛ) FρσFa . (16) 4g ance

All details of the computation are not included in this β(γ) + β(sg ) =0, (21) letter because they are very cumbersome, but the results α α ∗ are: we can find the special value of the coefficient sg that kills the beta function. Using for example the first killer (sg ) sg 1. βα = , (17) (15) we get 2π2ω

(sg ) sg 2. β = (1 + NG), (18) ∗ 2 (γ) α 4π2ω sg = −2π ωβα , (22) where NG is the number of generators of the Lie group. and the Lagrangian for a finite nonlocal gauge theory in These results have been checked using two different four dimensions can be explicitely written techniques: the method of Feynman diagrams and the 2 α a H(DΛ) µν Barvinsky-Vilkovisky trace technology [20]. Lfin, gauge = − F e F (23) 4 µν a The computation has been done for the nonlocal the- h 2 (5+3γ + 12γ ) a µν 2 γ−2 b ρσ ory with general polynomial asymptotic behaviour pγ (z) +ω C2(G)Fµν Fa (DΛ) FρσFb of degree γ. By choosing the monomial p (z) = ωzγ 96 γ i the prototype kinetic term used to evaluated the beta where we assumed γ ≥ 2. function reads It is possible to kill the beta function also in nonlo- cal theories, where we have Abelian symmetry groups. 1 a 2 γ µν Lfin, kin.gauge = − 2 Fµν 1+ ω (DΛ) Fa . (19) For concreteness we can study the one-loop beta func- 4g 3 2 tion of QED βe = e /12π for electric charge e. In terms
of the inverse coupling α this function is expressed as As already explained all the other contributions of the β = −1/6π2, which is a constant and gives logarith- form factor fall off exponentially in the UV and do not α mic scaling with the energy for the coupling constant α. contribute to the divergent part of the quantum action. Since pure two-derivative QED is a free theory, then the running comes entirely from quantum effects of charged matter. Here we assume one species of charged fermions 1 coupled minimally to photon field. If we extend QED to It is worth noting that if we choose the gauge group G = SU(N) and in the adjoint representation, it holds the nonlocal version (1) with killer operator (15) and we replace tr(T aT bT cT d)= δabδcd + δadδbc , (13) ω Therefore, the killers we have considered exhaust all the possible ∗ 2 (γ) sg = −2π ωβα = (24) operators we can construct, regarding the structure in the inter- 3 nal indices. On top of this we have the freedom of using different contractions of Lorentz indices and covariant derivatives in the in (12), then the theory is completely finite regardless expressions for quartic killers. Indeed, if we plug the formula of the parameter γ. It is important to notice that even above (13) in the following general Lagrangian in the Abelian case the killer operator has crucial im- sg µν 2 γ−2 ρσ pact on the beta function because it contains photon Lkiller = − tr Fµν F (DΛ) FρσF , (14) 4g2 h i self-interactions. In this way we solve the problem of we get the sum of the two killers (15) and (16) with the same Landau pole for the running of the electric charge in the front coefficient. UV regime of QED. The same can be repeated for any 5 gauge theory coupled to matter, provided that in the no possibility for appearance of a new real pole in it. On matter sector we do not have self-interactions and the the other hand, in the IR the analytic form factor does coupling to gauge fields is minimal [19]. not play any role and there is no pole because the beta We want to comment on what we can achieve if we stick function is negative. The outcome is a theory pertur- to one-loop super-renormalizable gauge theories without bative in both the UV and in the IR regime. Therefore attempts to make them finite. The final result (20) high- we are left with two possible options. We can choose lights a universal Landau pole issue in the UV regime completely UV finite (no divergences) nonlocal theories for the running coupling constant g(µ) (where µ is the or super-renormalizable nonlocal theories with negative renormalization scale). This is true for any value of the beta functions (βα) and hence without any singularities integer γ ≥ 2, when we do not introduce any potential in asymptotic behaviours of the couplings. The second Vg with killer operators. The sign of the beta function is option seems to be very appealing in models that attempt negative because the discriminant ∆ < 0 of the quadratic to realize a unification of all coupling constants. polynomial in γ in (20). For the particular choice (22) the theory (12) is one-loop finite, but if the front coeffi- cient sg has a bigger value than in (22) then we enter the III. CONCLUSIONS regime in which the UV asymptotic freedom is achieved. We here summarize the three possible scenarios for the We have explicitly evaluated the one-loop exact beta value of the sg function for the weakly nonlocal gauge theory recently 2 proposed in [2]. The higher-derivative structure or quasi- <ω (5+3γ+12γ ) C (G) , Landau pole, 96 2 polynomiality of the action implies that the theory is  2 super-renormalizable, and in particular only one-loop di- (5+3γ+12γ ) ∗ sg  = ω C (G) ≡ s , finiteness,  96 2 g vergences survive in any dimension. Once a potential,   2 at least cubic in the field strengths, is switched on, it (5+3γ+12γ ) >ω C2(G) , asymptotic freedom. is always possible to make the theory finite. We evalu-  96  ated the beta function for the special case of D = 4, but However, in weakly nonlocal higher-derivative theories the result can be generalized to any dimension where a we must read out the poles from the quantum effective careful selection of the killer operators should be done. action and not only from the beta functions of the cou- In short, in this paper we have explicitly shown how plings in the theory. In particular, in the case of the to construct a finite theory for gauge bosons in D = theory (1) the one-loop dressed propagator is devoid of 4 (23). We have considered both cases of Abelian any pole because its UV asymptotic behaviour is entirely and non-Abelian gauge symmetry groups. The super- due to the form factor exp H(z) [4], namely, up to the renormalizable structure does not change if we add a tensorial structure, general extra matter sector that does not exhibit self- 2 e−H(k ) interactions. − i 2 . (25) The minimal nonlocal theory without any killer oper- k2 1+ β e−H(k ) log(k2/µ2) α 0 ator shows a Landau pole for the running coupling con- Moreover, as a particular feature of
the super- stant, regardless of the special asymptotic polynomial ∗ renormalizable theory, when sg =0or sg

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