UV/IR mixing in Noncommutative SU(N) Yang-Mills theory

We show that there are one-loop IR singularities arising from UV/IR mixing in noncommutative SU(N) Yang-Mills theory defined by means of the $\theta$-exact Seiberg-Witten map. This is in spite of the fact that there are no ordinary U(1) gauge fields in the theory and this is at variance with the noncommutative U(N) case, where the two-point part of the effective action involving the ordinary SU(N) fields do not suffer from those one-loop IR singularities.

itself as a form of hierarchical UV/IR mixing and is tied to the interaction between the weak gravity conjecture and nonlocal (possibly noncommutative) gauge operators. Namely in the scalar field theory, the mass of the scalar is far from ultra-violet scale. Thus the form of hierarchical UV/IR mixing restricts the mass of the scalar field to a IR scale deep below the UV scale which is associated to quantum gravity. Additionally one can think about naturalness of the scalar mass in a way that the UV physics knows nothing about the theory in the deep IR. So instead of introducing the higher dimensions and new symmetries one might hope that connecting the deep-low IR and far-high UV could solve the naturalness problem [48,49].
In the literature, all above notions with UV/IR mixing connections are considered as possible windows to Quantum Gravity.
A chief characteristic of noncommutative U(N) theory is that only the U(1) part of the two-point function for the gauge field exhibits UV/IR mixing at the one-loop level. Indeed, the SU(N) part of that two-point function does not get any contribution from one-loop nonplanar diagrams. This raises the question of whether noncommutative SU(N) gauge theories experience this UV/IR mixing, for there is by construction no fundamental U(1) degrees of freedom in them. The purpose of this paper is to answer this question, which as we shall see demands to carry out lengthy computations. Let us recall that to formulate Yang-Mills theories on noncommutative spacetime for arbitrary gauge groups in arbitrary representations one has to use the noncommutative framework put forward in references [2,4,50,51]. In this formalism the noncommutative gauge fields and noncommutative gauge transformations are defined solely in terms of the ordinary counterparts by using the Seiberg-Witten map, which in this case take values in the universal enveloping algebra of the Lie algebra of the gauge group. Thus, any -unitary-representation of the Lie algebra is admissible; but, then, one needs to address the problem of too many degrees of freedom, since all coefficient functions of the monomials in the generators could a priori be physical fields. The solution furnished in [2,4,15] is that those coefficient fields are not all independent: they are functions of the correct number of ordinary gauge fields via SW maps. Thus, for semisimple Lie algebras, no fundamental U(1) degrees of freedom occur in the noncommutative gauge theory. That the Seiberg-Witten map in question exists for any compact gauge group -for SU(N), in particular-in any representation has been shown in reference [15] by using BRST techniques, thus proving that the Seiberg-Witten exists not only for the U(N) group in the fundamental, antifundamental or bifundamental representations. The noncommutative Yang-Mills action for arbitrary compact gauge groups has been introduced in references [2,4,50,51] and fully studied in reference [15]. It is by using so-called enveloping algebra formalism that we have just mentioned that the noncommutative Yang-Mills theory for SU(N) in the fundamental representation is formulated -see reference [50]-without the need of introducing the interacting U(1) fundamental degrees of freedom which occur in the noncommutative U(N) case.
Using the enveloping algebra formalism we have just made discussed, the background-field and path integral methods, from the classical Yang-Mills (YM) action we shall first construct the BRST exact noncommutative effective action S BRST exact , with the NC fields spanned on the Moyal manifold and being expressed in terms of ordinary YM fields by means of θ-exact SW maps. Let stress that the quantization of the theory will be carried out by integrating in the path integral over the ordinary SU(N) gauge fields and ghosts -this is the SU(N) analogue of quantization method succesfully used in [35,36,52] for the U(1) case. Then, we investigate the UV/IR mixing phenomena in the gauge sector and for a Dirac fermions in the fundamental representation of SU(N). Finally we have to state clearly that this theory is not really the pure Moyal NCYM, but a class of Moyal deformed gauge theory which is different, and not yet completely understood.
This article is on the line of our previous works [52][53][54][55], and it is organized as follows. In the next section we introduce classical and the background-field effective actions. We compute one-loop two-point functions by using DeWitt method [56][57][58][59] and in Section three prove existence of the gauge independent UV/IR mixing phenomena. Section four is devoted to the fundamental Dirac fermions in the framework of the noncommutative QCD. Conclusion is given in fifth Section, while details of θ-exact SW maps for all relevant fields and details of divergent integral solutions are given in Appendices A and B, respectively.

II. THE CLASSICAL ACTION AND THE BACKGROUND-FIELD EFFECTIVE ACTION
Let a µ = a a µ T a , in terms of component fields a a µ , be an ordinary gauge field, with T a being the generators of SU(N) in the fundamental representation, normalized so that Tr(T a T b ) = δ ab . The symbol A µ will denote the noncommutative gauge field defined in terms of the previous SU(N) ordinary gauge field a µ by means of the Seiberg-Witten map [10].
The classical action, S cl [A], of our noncommutative SU(N) YM theory reads , with star(⋆)-product being the Moyal-Weyl one, and self-evident notations for ⋆-commutator. Important to note is that the dynamical field variable in S cl [A(a λ )] is a λ . For details of SW maps see Appendix A.
To quantize the classical theory whose action is, S cl [A(a λ )], in (1), we shall use the background field method [29] and, thus, we shall split a µ as follows where b µ = b a µ T a denotes the background field and q µ = q a µ T a stands for the quantum field -the field to be integrated over in the path integral.
Since the noncommutative gauge field, A µ , is a function of a µ -A µ = A µ (a λ )-, the splitting in (2) gives rise to the following splitting of A µ Notice is given by the Seiberg-Witten map applied to the ordinary SU(N) gauge field b µ . Mark that Q µ (b, q) is a function of both b µ and q µ . The background field effective action Γ[b] is given by the following equation with Classical action, S cl [A(b + q)], is defined in (1) and S BRST exact stands for the gauge-fixing terms including the ghost's contributions. To render the sum S cl [A(b + q)] + S gf +gh as simple as possible we shall choose the following Feynman type of gauge fixing (S gf ) and ghost (S gh ) actions sum S gf +gh where s denotes de ordinary BRST operator, andC =C a T a is the antighost, while F = F a T a Lautrup-Nakanishi auxiliary field. As noted the S BRST exact is BRST-exact. The BRST operator s acts on the ordinary and noncommutative fields, respectively, with c = c a T a denoting ordinary ghost field. The notion of Seiberg-Witten map implies the action of s on the noncommutative fields B µ (b), Q µ (b, q) and C(b + q, c) is the following where s N C is the noncommutative BRST operator whose action on the noncommutative field thus runs as follows By construction, we also have s N CC = sC = F and s N C F = sF = 0. Using the definitions above, one gets III. THE TWO-POINT CONTRIBUTION TO THE EFFECTIVE ACTION AT ONE-LOOP The final purpose of this section is the computation of one-loop contribution to the effective action Γ[b], which has been defined in (4). We shall begin by expanding in powers of q µ and b µ the action, S, in (5) and, then, dropping the terms with more than two b µ 's and more than two g's -g being the coupling constant.
A. Removing O(g 3 ) terms from the path integral Let us first integrate out the field F a in the path integral in (4) by taking advantage of (10). Thus, one obtains where S is the sum while actions S gf and S gh being given by To obtain above S gf we have to use the following SU(N) generators identity We start with definitionsq a µ = Tr (T a Q µ (b, q)), andc a = Tr (T a C(b, q)), where the Seiberg-Witten maps Q µ (b, q) and C(b, q) are fully discussed in Appendix A -see (A4) and (A6). Next, in the path integral (11), we shall make the following variable change q a µ →q a µ , c a →c a , for thus we shall remove from the path integral the lengthy interaction terms due to contributions to the classical action coming from the SU(N) part -see (A7)-of the Seiberg-Witten maps Q µ (b, q) and C(b, q) when expressed as functions of q µ . This is a much welcome simplification since, as we shall see below, we will still have to deal with the interaction terms the U(1) part of the former Seiberg-Witten maps introduce. Notice that the Jacobian of this transformation is trivial in Dimensional Regularization, since all the momentum integrals it involves vanish -for details see [54]. (11). The change of variables (16) and the use of (A13) and (A15) from Appendix A, after some laborious algebra, leads to where and with self-evident notations for equations of motions (eom), and ⋆-anticommutator. Notice that S 2 and S 2gh only involve the U(1) part of the Seiberg-Witten maps Q µ (b, q) and C(b, c) as defined in (A4) and (A6): the SU(N) parts of those maps have been disposed of by introducing the SU(N) fieldsq andc-see (A13) and (A15) in Appendix A. Now, let p µ and q µ be two arbitrary vectors. We shall use the following notation Then,Â , are given by the following expressions: B. The one-loop contribution to the two-point function By evaluating the order g 2 contribution to the right hand side of eq. (17), we shall obtain the one-loop contribution to Γ 2 [b], the two-point bit of the effective action. The result that we obtained, with help of integral basis given in Appendix B and [52,60], runs thus where Γ 2 [b] (eom0) , which obviously vanishes upon imposing the equations of motion, contains all the contributions on which integral in (4), is involved. That is, is the sum of all the contributions which involve either S 1eom or S 2oem from (17), and it is given by the following long expression Note that f µν (−p) = i(p µ b ν (−p) − p ν b µ (−p)). B 1 , C 1 , C 2 , C 4 ,Ã,B 1 ,C 1 ,C 2 andC 4 in (25) read thus: C. Gauge independent UV/IR mixing (23) develops, as a result of UV/IR mixing, IR divergences in the region wherep µ = 0 and that these divergences do not survive the large N limit. And yet, Γ 2 [b] is a gauge-fixing dependent quantity so one may ask whether all these IR singularities are gauge-fixing artifacts. We shall answer this question by putting the background field b µ on shell so that our Γ 2 [b] will boil down to the 2-point on-shell DeWitt effective action [56][57][58][59], for we are computing radiative corrections at order g 2 . It is known that on-shell DeWitt effective action is a gauge-fixing independent object.
Since we are working at order g 2 , we only demand the b µ be a solution, b µ , to the free equation of motion to put b µ on-shell: Now, any solution to (27) is of the form where α(x) is an arbitrary function taking values in the Lie algebra of SU(N) and b ⊥ µ (x) is such that Let us replace b µ in (23) with b (0) µ (x) in (28) to obtain the 2-point on-shell effective action [56][57][58][59]: Notice that b ⊥ µ (p) above is the Fourier transform of the b ⊥ µ (x) satisfying the conditions in (29) and that is related to b ⊥ µ (p) by a gauge transformation. Notice that the right-hand side of equation (30) is invariant under arbitrary on-shell gauge transformations. The latter as defined thus Finally, Γ 2 [b (0) ] above develops quadratic IR singularity, in the limitp µ → 0, that is given by which goes away in the large N limit. Note that obtained (32) has completely different Lie algebra structure with respect to the famous IR singularity in the one-loop two-point function of the ordinary gauge field of noncommutative U(N) case [26],

D. The UV/IR mixing phenomenon
Let us show now that the IR singularity in (32) which occur whenp µ = 0 is a consequence of the UV/IR mixing phenomenon and not the result of the Seiberg-Witten maps in (22) having denominators which vanish for specific values of the momenta. We shall display below each and everyone of the contributions to the path integral in (17) which yield (30).
Let us begin with the following definition O(q a µ ,c a ,C a ) 0 = Dq a µ Dc a DC a e i(S0+S 0gh ) O(q a µ ,c a ,C a ), where S 0 and S 0gh are given in (18) and (20), respectively. Then, gives rise to the following contribution tends to a constant as kp goes to zero. Hence, the singularity atp 2 = 0 of the right hand side of (36) is a consequence of the fact that, whenp µ = 0 in the integral over k exponentials e ±ikp kills the UV divergent behaviour of the rest of the integrand, rendering a finite result. Of course, these UV divergent behaviour resurfaces in the form of a IR divergence asp 2 → 0. This is precisely celebrated UV/IR mixing of the U(N) noncommutative gauge field theories, discovered first in [21][22][23][24][25]27].
The contribution to (30) coming from In the above equation (39) we meet the very same type of loop integral -the integral over k-, so the IR divergent behaviour asp µ goes to zero that occurs on the right hand side of (39) has the UV/IR mixing origin that we discuss in the paragraph below (36).
Let us now deal with It can be shown that the previous expression is equal to approaches zero asp µ goes to zero. Hence, the vanishing, whenp µ → 0, of the denominator of the integral over k in (41) has no bearing on the IR divergence atp µ = 0 that occurs on the right hand side of (41). Indeed, again, this IR divergence atp µ = 0 occurs as a consequence of the fact that e ±ikp cuts-off the UV divergent behaviour of the integral over k we have just mentioned: clearly herep acts as a cut-off giving rise to the UV/IR mixing phenomenon. The loop integral -the integral over k-which occurs in (41) is also the only responsible for the IR singularity at p µ = 0 of the following contributions to Γ 2 [b (0) ] in (30): There remains to discuss the origin of the singularity atp µ = 0 of the following contributions to (30): The singular behaviour atp µ = 0 in the expression above comes uniquely from various types of integrals over k, all being given in Appendix B: The UV/IR mixing phenomenon that the latter integrals bring about has been amply discussed in the literature already [21,22]. Notice that one obtains (30) by adding the right hand sides of (36), (39), (41), (43) and (44). Some final comments are in order. The U(1) part of the Seiberg-Witten map Q µ (b, q) as defined in (A7), contributes to the singularity atp µ = 0 due to UV/IR mixing -see (36)(37)(38)(39)(40)(41)(42)(43). The UV/IR mixing effect the SU(N) bit of Q µ (b, q) which is involved in, is obtained by using the fieldq µ , -see (44).

IV. ADDING DIRAC FERMIONS IN THE FUNDAMENTAL: NONCOMMUTATIVE QCD
We shall show in this section that the inclusion of a Dirac fermion transforming under the fundamental representation of SU(N) does not change the one-loop IR singular behaviour of the two point function of b µ (x), Γ 2 [b], that we have unveiled in the previous section.
Let ψ(x) = ψ j (x), j = 1, ..., N -the SU(N) index, be an ordinary Dirac fermion transforming under the fundamental representation of SU(N). Let Ψ(x) = Ψ(a, ψ) j (x), with a µ (x) = b µ (x) + q µ (x), be the noncommutative fermion field obtained from ordinary fields ψ j (x) by using the Seiberg-Witten map. Then the classical action S Dirac , of Ψ(a, ψ) j (x) coupled to the noncommutative gauge field whereΨ(a, ψ) is the Dirac conjugate of Ψ(x), and / D [A] = i(/ ∂ − g / A⋆). The one-loop contribution, Γ Dirac [b], to the background field effective action Γ[b] due to the noncommutative Dirac field Ψ(a, ψ) is given by the path integral with respect to the ordinary fields ψ,ψ Now, since Ψ(a, ψ) j is a four component spinor, one can make the following change of variables in the path integral (47) and obtain: where J F (b) andJ F (b) are the appropriate Jacobians: As with the Seiberg-Witten map for the gauge field, a SW map defining the noncommutative field Ψ(b, ψ) can be obtained by solving the following "evolution" problem: where B µ (b, t) is given by the "evolution" equation discussed in Appendix A.
It can be seen that F µ1µ2·µn−1 (p 1 , p 2 , ..., p n−1 ; p n ; θ) in (52) is a linear combination of functions of the type where P(p 1 , ...., p n ) is a polynomial of the momenta p i 's and Q(p i ∧ p j ) only depends on p i ∧ p j = p µ i θ µν p ν j , i, j = 1, ..., n. This fact leads to the conclusion that all the loop integrals involved in the evaluation of vanish in dimensional regularization, and, hence, The detailed derivation of (55) is analogous to the one carried out in Appendices B and C of Ref. [54]. Clearly, in dimensional regularization we also havē Let us introduce the following notation: · · · f 0 = DΨDΨ · · · e iΨ(i/ ∂−m)Ψ.
Substituting (55) and (56) in (49), one readily obtains, in dimensional regularization, the following one-loop result . From (59) we draw the conclusion that the full one-loop Γ Dirac [b] in (47) lacks any singular behaviour when any of the momentap µ i = θ µνp ν i vanishes ∀i. Finally, it is plain that (58) and (59) hold whatever the gauge group and representation provided the noncommutative Dirac field transforms as follows under infinitesimal noncommutative gauge transformations defined by Ω(b, ω), while ω(x) defines the infinitesimal ordinary gauge transformations.

V. CONCLUSIONS AND OUTLOOK
The main conclusion of this paper is that noncommutative SU(N) defined by means of the θ-exact Seiberg-Witten map has a two-point function for the gauge field that exhibits UV/IR mixing at the one-loop, in spite of the fact that there are no fundamental U(1) degrees of freedom. This is at odds with the noncommutative U(N) case where it is only the U(1) part of the one-loop two-point function for the ordinary gauge field the one which is affected by the famous UV/IR mixing. Indeed, in the noncommutative U(N) case the famous noncommutative IR singularity in the one-loop two-point function of the ordinary gauge field reads [26] In Appendix C, we re-derive the above formula using the enveloping algebra approach together with the background field and the path integral methods employed in our recent works [52][53][54][55], as well as in the previous sections of this paper, confirming eq. (61) and showing this way implicitly the correctness of our computations. The noncommutative IR singularity of the noncommutative SU(N) theory reads -see (32)-which has a novel, as yet unknown, UV/IR mixing tensor and Lie algebra structure with respect to that in (61). Indeed, in (61), we have the product of two traces over the generators of the Lie algebra so that only the U(1) part of the contribution gets affected by the noncommutative IR divergence. However, in (62), we have only one trace over the product of two SU(N) generator and, hence, the full SU(N) part of the contribution receives the noncommutative IR divergence. No η µν occurs in (61). In subsection 3.4 -see (36), (39), (41), etc-we have shown that the noncommutative IR divergence in (62) occurs because the noncommutative phase e iθ µν kµpν -k being the loop momenta and p the external momenta-regularizes, provided θ µν p ν = 0, otherwise UV divergent integrals. Thus, the UV divergence is turned into an IR divergence at θ µν p ν = 0. This the UV/IR mixing phenomenon and its origin has nothing to do with the use of the Seiberg-Witten map and its vanishing denominators at θ µν k µ p ν = 0.
We have also shown that this conclusion also holds in noncommutative QCD, since the addition of Dirac fermions in the fundamental representation to the noncommutative SU(N) theory does not modify the UV/IR mixing behaviour of the two-point function at hand.
The phenomenological implications of the UV/IR mixing unveiled in this paper are worth studying. Notice that this UV/IR mixing affects the noncommutative Standard Model of [50] and the noncommuative GUTs models of [51,61] when defined by means of the θ-exact Seiberg-Witten map.

Appendix A: Seiberg-Witten maps
Let a µ and c be, respectively, an ordinary gauge field and the corresponding ghost field taking values in a Lie algebra in a given irreducible representation. A Seiberg-Witten map which defines a noncommutative gauge field A µ (a) and the corresponding noncommutative ghost field C(a, c) in terms of ordinary a µ and c is obtained [13][14][15] by solving, between t = 0 and t = 1, the following "evolution" problem The product ⋆ t has been defined in (21) and The "evolution" problem in (A1) can be solved [17] by expanding in power of the coupling constant g as befits the definition a field theory in perturbation theory: A µ (a) = a µ + g A (2) µ (a, a; t = 1) + g 2 A C(a, c) = c + g C (2) (a, c; t = 1) + g 2 C (3) (a, c, a; t = 1) + O(g 3 ), j (a, a; s)} ⋆s + {A (2) i (a, a; s), 2∂ j a µ − ∂ µ a j } ⋆s − i{a i , [a j , a µ ] ⋆s } ⋆s , j (a, a; s)} ⋆s . (A3) To obtain B µ (b) and Q µ (b, q) in (3), we replace a µ with b µ + q µ in (A2) and (A3), and then, expand in powers of a µ and b µ the resulting expressions: µ (a = q, a = q, a = q; t = 1).

(A5)
Since our purpose is to compute Γ 2 [b] at one-loop, we shall not need the q-dependent bits of C(b + q, c): Assume that q µ and b µ take values in the SU(N) Lie algebra in the fundamental representation. Then Q µ (b, q) in (A4) can be expressed as the following linear combination: where I N is the N × N identity matrix and T a are the SU(N) generators. We shall call Q (0) µ (b, q) and Q a µ (b, q) the U(1) component and SU(N) components of the noncommutative field Q µ (b, q).
Integral I 3 is a standard nonplanar integral, on which we apply the decomposition I 3 = η µν p 2 · I 31 + (p µ p ν − η µν p 2 ) · I 32 +p µpν · I 33 , and after integration over parameter α obtain I 31 = 2 p 2 I 1 , x(1 − x)p 2p2 , Here I 33 can be re-expressed via I 31 and K D 2 −2 x(1 − x)p 2p2 as follows Integral I 4 is a standard planar integral and we have the standard tensor reduction result: Trivially one can see that I 6 = I 13pν , I 9 = I 16pν , and integrals I 13 and I 16 can be solved by an NC tensor reduction.
Since the only relevant momentum in these two integrals isp µ , we can prescribe the following simple tensor structures: I 13 = η µνp2 A 13 +p µpν B 13 , which after contraction with η µν andp µpν gives: We are now left with I 11 and I 14 , and they are not as easy as they seem to be. Priorly we evaluated them in [52] as the following sum denoted as T −2 : Two methods were used to calculate T −2 explicitly, one is based on Grozin parametrization [60] and the other contour integral parametrization. We encounter immediately problem with the second method because the poles at x = 0 are second order in I 11 and I 14 while first order in T −2 , which means that we can not define the principle value around this pole for I 11 or I 14 . This leaves only the Grozin parametrization producing: