Ganguly, Avijit K

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Indian J. Phys. 75A (3). 277-278 (2001)U Pan internutionul )ournalAn exact solution in SU(2) Yang Mills theoryAvijii K G anglySaha Institute of Nuc%ar Physics, l/AF Bidhannagar. Calcutt|i 700 064, IndiaR e c e iv e d  15  N o v e m b e r  2 0 0 0 , a c c e p te d  7 F e b r u a r y  2 0 0 1Abstract : An exact solution of Yang Mills equation in (l + l) dimension his been found rhis solution shows a chaiacterisiic oscillation in time, vkhosc frequency depends on amplitude and it vanes linearly with spatial coordiihatesKeywords . Yang Mills equation, exact solutionPACS No. : 12.10 -gThe solution of classical sourcelcss Yang-Mills (YM) equation has attracted the attention of many workers [1,2] for a long time ioi obtaining more information about the structure of the QCD ground slate. It is also important from the point of view of studying Quantum Chromo Dynamics (QCD) vacuum, in the back ground of such a solution and there from getting information about the stability of the system. Here, we will motivate the study of such a solution from the point of view of QCD flux tube model, introduced originally by Low [3] and Nussinov [4] to account for the observed scaling behaviour of scattering cross sections in hadron hadron collisions as well as in the nucleus nucleus collisions.In this model, it is assumed that at high energy, when the two highly Lorentz contracted nuclei pass through each other, the partons of one nucleus interact with the partons of the other nucleus by the exchange of soft (color octet) gluons. If the fly by lime of the nuclei is less than the time scale of interaction of the partons, the receding nuclei get randomly color-charged by exchange of soft gluons. These color octet partons in the receding nuclei get connected to each other by means of color Hux lubes with color electric fields inside them. The purpose of ihis note is to show that due to the presence of the nonlinear terms in the Lagrangian, the fields interact among each other, generating an electric field that undergoes a characteristic non­linear, non-abelian oscillation in time. For this purpose, we will make certain assumptions based on the geometry of the prolem. These assumptions however do not change the qualitative nature of our observation. The first assumption is that each of ihe color-charged nucleus has a uniform distribution of charge in the plane transverse to the direction of motion so that thereexists no gradient of the field m this direction. Our second assumption is that these color charges produce a chromoelcclric field such that and are the only non-zero potentials.With these simplifying assumptions, the dynamics of the gluon fields can essentially be described in (l + I) dimensions rather than (3+1) dimensions. Therefore in order to gel information about the nature of the classical gluon fields, one needs to solve the classical Yang-Mills field equations in (1 I ) dimensions.We next show that in (l + I) dimensions, the Yang-Mills equations have a solution with a sinusoidally time varying component whose frequency depends on the amplitude. Wc first write the sourcelcss Yang Mills equation in (1 +1) dimensions( 1)where the Greek indices ^ and v take values 0 and 1 only. The covarianl derivative is defined aswith g as the coupling constant and K as the commutator bracket. Since we are working with an SU(2) color symmetry, is defined as A  ^ = where are the generators obeyingthe commutation rules(3)The indices a,bx  takes values from one to three. Further, the only non-zero components of the vector field in this case are Aq©2001 lACS278 Avijit K Gangulyand A and wc have chosen the axial gauge A, = 0. With this choice of the gauge wc get from cq. ( I ) for v = 0,whose solution is(4)(5)Here and are arbitrary integration constants. In order to find out an exact solution of this equation, wc take todepend on time and to be a constant. Eq. (1) for v = 1 gives= 0 . (6)Substituting the solution (5) in cq. (6), we arrive at+ (7)One can derive a conservation law from this equation, namely= constant. (8)A summation over repeated indices is implied and a  is .scaledby g-We solve this set of coupled first order linear differential equations by Euler's method; i.e. we choose a solution of the forma„(t) = a_e (9)A l= [p ,  +P,p ,[e “^ ' -e - '“']]z + ^ , .(13)= [ ^ 2  p^2P,[e““' +(14)^  =  [/3, +  Pi [e'“ ' +  e-"** ] -  +  P ,. (15)Tlius from the solution, it is clear that the electric field inside a chrom o-electric flux tube oscillates with frequci^v(0 = [ { P^f +  {P2 )  ^+ ( P i f ] \w h ic h  depends on the amphiudc of oscillation. It may be worth, pointing out several features n.i this solution, namely the time independent piece corresponds to the covariantly constant field case. The time dependeru oscillating piece is due to the self interaction of the fields, thai shows dependence of amplitude on the frequency. It may nKi, be pointed out that there exists [2] an exact time-dependem vacuum solution of (SU(2)) Yang-Mills equations of the typeA " = { 0 , HS ? , H5 ^ , HS ^ ) ,whereW = y i g B i t  -  to)]fl6i( 1 7 )Substituting eq. (9) in eq. (7), we obtain a set of coupled algebraic equations whose solution is of the forma , = p , + p , P , [ e ‘^ ‘ + e-'‘“’] - im p 2 [e'“ ' (10)«2 = ^2 + PiPy y '“‘ + P '“ ' -  (11)«3 = ^3 + P i [e '" ' + [e““‘ + e - '“ ']- (12)Here, 0) = [(/J|)^ + i p 2)^  +03,)^]^ and the constants of integrations are set equal to one.Once the or's are known, one gets the solution for the Aq's by substituting eqs. (10), (11) and (12) in cq. (5). Without giving the unnecessary mathematical details, the final expression iswith frequency, dependent on am plitude. In cq. (16), (= 0,1,2,3) is the Lorentz index, a  (= 1,2,3) is the color indexand 5 “ is the Kronecker delta. In cq. (17), cn represents the Jacobi elliptic function and B is a constant determining the amplitude of the oscillating field. Physically, these solutions represent a non-linear collective oscillation of gluons with acharacteristic amplitude dependent time period -*-(7^^)  ^which is a manifestation of the intrinsic nonlinearity present in the system.This is worth mentioning that earlier there had been attempts to study the break down of QCD vacuum, in the context ot RHIC, in the background of a covariantly  constant, homogeneous Chromo Electric field, as well as a sinusoidally oscillating Chromo Electric field [5]. In the light of those results, we feel, it will be interesting to look for the same effect in the presence of the type of background obtained in this note. Some work along this direction is currently under progress.AduiowledgmentThe author would like to thank Prof. P. K. Kaw for s u g g e s t io n s  and advices provided during the course of this work.References[IJ S Coleman P h y s , Utt, 70B 59 (1977) ; A Actor Rev, M od . P fm  51 461 (1979)[2] S G Matinyan, G K Savidy and N G Tcr-Arutyunyan-Savvidy P h y s .  J E T P  53 421 (1981)13] F E Low P h y s  R ev . D12 163 (1975)(4] S Nussinov P h y s . R ev . U t t  34 1286 (1975) ; H Nueberger Phys R ev. D20 2936 (1979) ; C B Cliiv and S Nussinov P h y s . Rev. 1>20 945 (1979)[5] Avijii Kr Ganguly, Predhiman K Kaw and Jitendra C Parikh Ph}^  R e v  D46 R2983 (1993)

An Exact Solution in SU(2) Yang Mills Theory

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An Exact Solution in SU(2) Yang Mills Theory

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