Massive Self-Duality Solution Associated with Invariant One-Forms

Chinese Physics C Vol. 42, No. 1 (2018) 013107

Massive self-duality solution associated with invariant one-forms

Ibrahim˙ S¸ener S¸eyh S¸amil Mahallesi 137. Cadde No:19 D:9, P. B. 06824 Eryaman, Etimesgut - Ankara / TURKEY

Abstract: A massive self-duality solution associated with invariant 1-forms is presented. At the zero mass limit the massive self-dual theory of the SO(3) gauge group on 4 dimensions cannot be reduced to that of massless self-duality. In such a case the self-dual connection turns to the ﬂat connection and one cannot obtain a massless theory in such an approach.

Keywords: massive gauge ﬁelds, self-duality, invariant one-forms PACS: 02.40.Ma, 11.15.-q DOI: 10.1088/1674-1137/42/1/013107

1 Introduction a connection 1-form. The curvature of this connection is F A =dAA+AA AC =(F A) dxµ dxν Λ2(g). (1) Massive [1] and massless [2] Euclidean Yang-Mills B B C ∧ B B µν ∧ ∈ fields [3] have been handled by some authors in the sense In the sense of Hodge duality we write the self-duality of Feynman diagrams [4] in the quantum field theories. equations for the curvature of a SO(3)-connection on 4 In their results, a massless gauge theory cannot be ob- dimensions as follows: tained as a limit case of a finite massive theory [5]. How- 1 1 (F2 )12 =(F2 )34 =q1, (2) ever, the self-duality of massive Yang-Mills gauge fields is 2 2 not extensively discussed today. General opinion about (F3 )13 = (F3 )24 =q2, (3) 1 − A self-dual Yang-Mills fields [6], [7] is that they are non- (F3 )14 =(FB )23 =q3, (4) trivial solutions to the vacuum Yang-Mills equation. In ∞ where q1,q2,q3 (M). The components of the curva- such a case, the masses of the gauge fields mostly seem ture matrix become∈C unimportant. 1 µ ν 1 2 3 4 However, together with the investigation of the sym- F = q1Iµν dx dx =q1 (dx dx +dx dx ), (5) 2 ∧ ∧ ∧ metry breaking mechanism [8], [9], [10], in which the F 2 = q J dxµ dxν =q (dx1 dx3 dx2 dx4), (6) 3 2 µν ∧ 2 ∧ − ∧ gauge bosons gain mass, the gauge group SU(2) of the F 1 = q K dxµ dxν =q (dx1 dx4+dx2 dx3). (7) weak nuclear force becomes important for massive gauge 3 3 µν ∧ 3 ∧ ∧ fields. We know that this group has covering groups where such as SO(4)=SU(2) SU(2) and has an isomorphism × 0 1 0 0 0 0 1 0 SU(2) = SO(3). Therefore, since there exist two iden- ∼ SO(4) SO(4)  1 0 0 0   0 0 0 1  tical quotients SU(2) = SO(3) , we can handle a self-dual I = − ,J = − , 1 R4  0 0 0 1   1 0 0 0  connection A Λ (so(3), ) together with its curvature     F Λ2(so(3),R∈4). Thus, in this paper we purpose a self-  0 0 1 0   0 1 0 0  ∈  −   −  duality solution to a massive gauge theory of the gauge 0 0 0 1 group SO(3) on 4 dimensions.  0 0 1 0  K = . (8)  0 1 0 0  2 Self-duality  −   1 0 0 0   −  Let P be a principal G bundle of a Lie group G with It is easily seen that Lie algebra g on a 4-dimensional Euclidean manifold M L2 = I , L L = L L , i,j=1,2,3, (9) with local coordinates xµ R4. We show by Λr(g) i − 4×4 i j − j i { } ∈ 0 1 the bundle of g-valued r-forms. Let : Λ (P ) Λ (P ) where Li = I,J,K. This means that the triplet (I,J,K) be a connection on this bundle together∇ with co→variant presents a quaternionic structure on R4. We know al- derivative =d+A, where A=(AA ) dxµ =AA Λ1(g) is ready that some solutions to the SU(2) Yang-Mills the- ∇ B µ B ∈

Received 31 August 2017, Revised 7 November 2017, Published online 6 December 2017 1) E-mail: sener [email protected] ©2018 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

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2 2 ory have quaternionic structures [11] and that the in- 2 q2 (r +r0 ) ν µ A3 = 2 x Jµν dx stanton equation on 4 dimensions presents a quaternion 2 r0 valued connection in the sense of division algebras [12]. q (r2+r2) = 2 0 (x3dx1 x4dx2 x1dx3+x2dx4), (18) The most classical self-dual potential (or connection 2 2 r0 − − in the geometrical sense) concepts in the Yang-Mills the- 2 2 1 q3 (r +r0 ) ν µ ories are the BPST [6] and ’t Hooft [7] solutions on 4 A3 = x Kµν dx 2 r2 dimensions of the Euclidean case. We choose an so(N)- 0 q (r2+r2) valued connection 1-form as follows 3 0 4 1 3 2 2 3 1 4 = 2 (x dx +x dx x dx x dx ), (19) 2 r0 − − α A x A µ A µ AB =h(r) (εB )µαdx =h(r)(ωB )µdx , (10) Therefore, from Eq. (15) we have the following gauge r invariant quadratic terms where h ∞(M) is a smooth scalar, ωA = (g−1dg)A ∈ C B B tr F A 2 = 2(q2+q2+q2)dVol (20) is the Maurer-Cartan 1-form for g SO(N), and we k B k − 1 2 3 A∈ A 2 1 A 2 choose an auxiliary constant tensor (εB )µα which is skew- tr tr ωB = 4 2 FB . (21) symmetric with respect to each of the indices A and B k k 4r0 h k k of the group and µ and ν of coordinates. The tensorial properties of the coordinates with respect to the metric 3 Massive theory tensor ηµν on the base manifold are It is well known that to add a mass term into a gauge 2 µ ν ν µσ µ invariant Lagrangian is not easy, because the mass term r =ηµν x x , xµ =ηµν x , η ησν =δν . (11) tries to break the gauge invariance. However, the first at- Therefore, the curvature of this connection is written as tempt was the Proca action [13], which can be considered as a massive Abelian action. For a massive non-Abelian 2 1 gauge theory, the mass term µ of the gauge potential, or (F A) = (h h2)(εA ) + (h +2r(h h2)) B µν − r − B µν r2 − connection 1-form A is added into the Lagrangian by an (εA) x (εA) x xα, (12) invariant term such that µ2A A, similar to the Higgs × B να µ− B µα ν ∧∗ mechanism [8]. Due to this mass term the gauge fields are where h = ∂h/∂r. In order to preserve the tensorial acquired by a (spontaneous) symmetry breaking mecha- structure of the curvature it must be nism [14]. The Euclidean massive Yang-Mills theory is non- h+2r(h h2)=0. (13) renormalizable, so it cannot be interpreted as a correct − quantum field model to describe physical interactions, This indicates that the connection is self-dual like the but the spontaneous symmetry breaking mechanism pro- BPST instanton [6]. The solution to this equation is as vides an adequate way to reach a renormalizable massive follows: Yang-Mills theory. In contrast to this method, we intend to find a non-Abelian self-duality solution to this theory r2 without symmetry breaking by adding a mass term to h(r)= , 06r 6r. (14) 2 2 0 the Yang-Mills action after modifying the connection. r +r0 Any connection on a fiber bundle transforms as in- homogeneous under a local gauge transformation g G, Then the self-dual connection is obtained such that so that A0 =g−1Ag+g−1dg, where the term violating∈the homogeneity is the Maurer-Cartan form 1 (r2+r2)2 (ωA) = 0 (F A) xν , (15) B µ 2 r2r2 B µν ω=g−1dg Λ1(g) (22) 0 ∈ 1 (r2+r2) A A 0 A ν and it satisfies the Maurer-Cartan equation (AB )µ =h(r)(ωB )µ = 2 (FB )µν x . (16) 2 r0 dω+ω ω=0. (23) Hence, considering the components of the curvature in ∧ Eqs. (5), (6) and (7), the components of our self dual In contrast to the connection, its curvature transforms − connection are obtained as as homogenous, that is F g 1F g, and it presents a gauge invariance quadratic →term, i.e. F F . The differ- 2 2 ∧∗ 1 q1 (r +r0 ) ν µ ence of two connections on a bundle is a tensor. There- A2 = 2 x Iµν dx 2 r0 fore, if we choose two connections on the same principal q (r2+r2) G-bundle, for example if A is any connection and the = 1 0 (x2dx1 x1dx2+x4dx3 x3dx4), (17) 2 Maurer-Cartan 1-form ω of the group G is another, then 2 r0 − −

013107-2 Chinese Physics C Vol. 42, No. 1 (2018) 013107 we write Therefore, the dynamical variables for the gauge invari- $=A ω, (24) ant action integral (30) of the massive self-dual gauge − and so this 1-form has a homogeneous transformation field become (Q,Q ). rule under the local group transformation In order to compare the massive self-dual gauge field with the massless one, we present an action integral for $ g−1$g, g G. (25) → ∈ the massless self-dual gauge field as follows Consequently the quadratic term tr $ 2 becomes a k k A 2 2 2 2 gauge invariance. Therefore the action integral for a Sµ=0 = gtr FB =2g (q1 +q2 +q3 )dV ol. (34) massive Yang-Mills theory without symmetry breaking | | ZM k k ZM can be written as follows Therefore the integrals of massive and massless gauge A 2 2 A 2 fields are rewritten with respect to the new dynamical Sµ=06 = gtr F +µ tr $ , (26) | | Z k B k k Bk variables (Q,Q) such that M where µ is the mass of the gauge field, g is the coupling µ2 constant, tr is the trace operator and F A 2 =F A F C . S(Q,Q) = (Q2 2λQ) g+ fdr, (35) k B k C ∧∗ B µ=06 2 Also, means that the metric signature of the base man- | | ZM − 4h || ifold is not considered and the action integral is positively 2 S(Q,Q )µ=0 = (Q 2λQ )fdr, (36) defined in every case. This action gives the following field | | ZM − equation ∞ A 2 A where f (M) and dV ol = f(r)dr. The field equa- FB =µ $B . (27) ∈ C ∇∗ tions for the massive and massless self-dual gauge fields, When the connection is self-dual, i.e F = F , then we respectively, are A ∗ A have FB =0 because of the Bianchi identity FB =0. ∇∗ ∇ This case tells us that the gauge field is massless, µ=0, or µ2 f f Qf+ (Q+gλ) +λ = 0, (37) a pure gauge (flat connection) A=ω =g−1dg. However, 4g h2 h2 we consider a gauge field which is massive and self-dual. Qf+(λf) = 0. (38) On the other hand one can write 1 Then the term Q for the massive case is tr $A 2 =(h 1)2tr ωA 2 = tr F A 2. (28) k Bk − k Bk 4h2 k B k µ2 (gλ+ln(f/h2))˙ Thus the action integral (26) for the massive self dual Q = (39) massive µ2 gauge field becomes − 4 g+ 2 2 4h µ A 2 Sµ=06 = g+ tr F . (29) 2 B From Eq. (31), q1,q2,q3 are found as follows | | ZM 4h k k This action integral presents the field equation q2 =√a bq1, q3 =√bq1, Q=αq1, (40) (h F A)=dh F A =0 with respect to the self-dual con- − ∇ ∗ B ∧ B nection. Of course this equation cannot be interpreted where a and b are constants and as a field equation of a massive gauge field. α=1+√b+√a b. (41) 4 Conclusion − On the other hand, from Eq. (32) one gets In order to get a field equation for the massive self 3β dual gauge field, the action integral (29) is rewritten as q = , (42) 1 2λ 2 2 2 2 µ Sµ=06 =2 (q1 +q2 +q3 ) g+ 2 dV ol. (30) where | | ZM 4h From the Bianchi identity dF +A F F A = 0 for β=(1+a b) 2 √b+√a b+ b(a b . (43) ∧ − ∧ the connection A Λ1(so(3)R4) and its curvature F − − − p − 2 R4 ∈ ∈ Λ (so(3), ), one gets the following equations: Considering Eq. (39) together with Eq. (14), one finds q2q3 q1q3 q1q2 the following solution = = =λ, (31) q q q 1 2 3 2 2 2 3β g r0 (r +r0 )f 2 2 2 1 2 q = 4 +1 r +C +ln , q +q +q = (Q 2λQ ), (32) 1 2 2 1 2 3 2 − − 2g µ − r r (44) where λ ∞(M) and ∈C where C is a constant. Hence, the components of the Q=q1+q2+q3. (33) connection given as the self-duality solution in Eqs. (17,

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18, 19) are re-obtained as the massive self-duality solu- the field equation (38) of the massless self-duality case. tion with respect to Eq. (44) such that Thus, if µ 0, then Q 0, that is the curvature tends → → 2 2 to vanish. This means that at the limit µ 0 a massive 1 q1 (r +r0 ) ν µ → A = x Iµν dx , (45) self-dual SO(3)-connection on 4 dimensions turns into 2 2 r2 0 the flat connection. However, the opposite of this case q (r2+r2) 2 √ 1 0 ν µ is not necessarily true. In addition, the zero limit of the A3 = b 2 x Jµν dx , (46) 2 r0 mass term in Eq. (44) does not give a massless case. q (r2+r2) 1 √ 1 0 ν µ Therefore, we cannot obtain a massless theory of self- A3 = a b 2 x Kµν dx . (47) − 2 r0 dual gauge fields at such a limit. Our solution is a non- It is easily seen that at the limit µ 0 the field equa- trivial self-duality solution to massive Euclidean Yang- tion (37) of the massive self-duality case→is not reduced to Mills theory without spontaneous symmetry breaking.

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