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Induced theta vacua in heavy ion collisions Buckley, Kirk B. W. 2000

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I N D U C E D T H E T A V A C U A IN H E A V Y I O N COLLISIONS By Kirk B.W. Buckley B. Sc., Bishop's University, 1998  A THESIS S U B M I T T E D  IN P A R T I A L F U L F I L L M E N T O F  T H E REQUIREMENTS - -  FOR T H EDEGREE OF  M A S T E R OF SCIENCE  in T H E FACULTY OF GRADUATE  STUDIES  D E P A R T M E N T O F PHYSICS A N D A S T R O N O M Y  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y O F BRITISH  COLUMBIA  June 2000 © Kirk B.W. Buckley, 2000  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h C o l u m b i a , I agree that the L i b r a r y shall make it freely available for reference and study.  I further agree that permission for extensive copying of this  thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Department of Physics and A s t r o n o m y The University of B r i t i s h C o l u m b i a 6224 A g r i c u l t u r a l Road Vancouver, B . C . , Canada V 6 T 1Z1  Date:  "3W 1-2, 2-000  Abstract In the following we demonstrate that it is possible to create non-zero ^-states during heavy ion collisions. Using the effective Lagrangian for low energy Q C D derived in [1, 2], we w i l l show numerically that i n the period immediately following a quench, the chiral condensate phases fa and fa w i l l relax to the value ~ Q/Nf.  T h i s is a true condensate  in the sense that if the volume of the system is changed, the zero mode still remains. If such a state can be created, it would decay by various mechanisms to the 0f  und  = 0 state  which exists i n our world. We w i l l discuss the experimental signature for the produced non-trivial 0-state. In particular, we w i l l provide evidence that the creation of a non-zero 0-state would result in an excess of low momentum particles (7r°, rj, and Ty'-mesons) , in the (10 — 100) MeV  range. T h i s phenomena could possibly account for the excess of low  momentum dileptons observed at C E R N [3, 4].  ii  Table of Contents  Abstract  ii  Table of Contents  iii  List of Figures  v  Acknowledgements  1  2  3  4  vii  Introduction  1  1.1  Review of Q u a n t u m Chromodynamics and the 6 Term  2  1.2  Overview  8  Effective Lagrangians and Q C D  10  2.1  A Simple Example: the Linear Sigma M o d e l  10  2.2  D i Vecchia-Veneziano-Witten Effective C h i r a l Lagrangian  11  2.3  Halperin-Zhitnitsky Anamolous Effective C h i r a l Lagrangian  13  Induced Theta Vacua  18  3.1  W h a t is the difference between 6  3.2  Disoriented C h i r a l Condensate  20  3.3  #-State i n Heavy Ion Collisions  21  3.4  E v o l u t i o n of the Equations of M o t i o n  24  ind  S i g n a t u r e s o f a n I n d u c e d 9-Vacuum  4.1  and the fundamental 0 parameter?  State  Properties of the pseudo-Goldstone Bosons i n the non-zero ^-Background  iii  .  19  29  30  4.2  Signature of the Creation of 9-Vacua  31  4.3  Results  34  5 Conclusions and Future Considerations  42  Bibliography  44  iv  List of Figures  3.1  \<f>k\ is shown for various \k\ as a function of time. In a time ~ 1 0 ~ the zero mode relaxes to <£j ~ 9/Nf  23  sec,  and a l l other modes decay to zero.  T h i s is very similar to the idea of a disoriented chiral condensate 3.2  26  Above we demonstrate that the system exhibits the coarsening phenomenon (amplification of the zero mode as time increases). T h e data was sampled at three times w i t h i n the first 1000 time steps of the evolution  3.3  26  The zero mode and a non-zero mode are shown as a function of time for three different volumes. The heavily dashed line represents the smallest volume (8 fm) , 3  (16 fm) , 3  the medium dashed line represents the middle volume  and the solid line represents the largest volume (32 fm) . 3  The  volume independence of the zero mode reinforces the claim that a true nonperturbative condensate has been formed 4.1  27  \4>i(k = 0)| is plotted as a function of time for the up, down, and strange quark. Notice that the zero momentum modes of the fa fields settle to a non-zero value in a time on the order of 1 0 ~ t  2  23  s.  The times t\ and  represent the value we chose for T h ii, the the time when the shell S  separating the two regions disappears  v  e  35  4.2  We plot the number of pions produced, N o(k) v  as a function of the mag-  nitude of the wave vector, \k\. Above we show that the momentum distribution of the 7r°-mesons produced is primarily < 25 MeV  for two different  values of r h u- T h e solid line represents the earlier time ti and the dotted s  e  line represents the later time £2 (see F i g . 4.1 for the positions of ti and t  2  in the evolution of the fa fields). This is expected as the formation of the E s t a t e can be attributed to the enhancement of the low momentum modes. 36 4.3  N (K) V  4.2.  is plotted as a function \k\, at two different times t and t as in F i g . x  2  The higher peak represents the later time w i t h the m a i n difference  being that now a larger percentage of the produced particles lie in the k < 25 MeV 4.4  N i(k) v  range  37  vs. \k\ is shown above. T h i s graph shows that the 7/ clearly domi-  nates the spectrum compared to the other neutral particles 4.5  T h e momentum distribution (N i(k)vs.\k\) n  38  is now shown for the case where  the geometry of the ^-region is considered as a rectangle.  We consider  the case where the size of the ^-region is larger along the longitudinal directions compared to the transverse directions. T h e solid and dotted lines once again represent times T h ii = t\ and £2 respectively. A l t h o u g h S  e  the majority of the spectrum still exists in the low momentum range, the higher momentum modes now become more evident 4.6  40  For the scenario shown in F i g . 4.5, we consider the angular dependence on the azimuthal angle for the ir° instead of the 77'. T h e azimuthal angle is defined so that 6 = 0 coincides with the beam axis  vi  41  Acknowledgements I would like to thank several people for their help along the way. I would like to thank my supervisor Ariel Zhitnitsky for providing a sense of enthusiasm, ideas, and for his patience in answering a seemingly infinite number of questions.  I would also like  to extend my gratitude to Nathan Weiss for useful discussions, as well as Konstantin Zarembo for reading over this thesis. My friends that I have met while at U . B . C . , Todd , Matt , and Mark, have made my time here quite enjoyable as well as always being available for discussions and input. I would also like to thank my collaborator Todd Fugleberg for working with me over the last year. Finally, I am grateful to my family and Lauren for their constant support and encouragement throughout the course of this work.  vn  Chapter 1 Introduction  Over the past 30 years the theory of quantum chromodynamics ( Q C D ) has been universally accepted as the description of the strong force. T h e theory has been tested experimentally w i t h great precision i n accelerator experiments. A l t h o u g h the Lagrangian of the theory is well known, there are still many mysteries which surround Q C D . W i t h the advent of modern heavy ion colliders, such as the Relativistic Heavy Ion Collider ( R H I C ) at Brookhaven National Laboratory, it may be possible to answer many of these pending questions. T h e hope is that i n these colliders it may be possible to indirectly observe a plasma of quarks and gluons, which is thought to have existed at the beginning of the universe. T h i s could signify the creation of a new form of quark matter at high temperature and large chemical potential. T h e study of Q C D under extreme conditions has been the subject of extensive research i n the last few years. One of the most exciting ideas is that at high densities and low temperatures such as the interior of neutron stars, quark matter could take on a superconducting phase where a condensate of quark-quark pairs would form. In the following we provide evidence that an additional induced restate could be observed i n heavy ion collisions. T h i s could be very exciting as it has been verified experimentally that \Qf  \  undamental  < icr . 9  We will also provide suggestions  of experimental signatures that can be observed i n order to verify that such a non-trivial vacuum state has been created.  1  Chapter 1.  1.1  Introduction  2  Review of Quantum Chromodynamics and the 9 Term  T h e spectrum of Q C D consists of six flavors of fermions (quarks) and eight vector bosons (gluons) which are the mediators of the strong force.  In addition to the character-  istic quantum numbers of a spin 1/2 fermion, the quarks carry an additional quant u m number, color charge.  There are six different flavors of quarks w i t h each one  having a distinct charge and mass. up, down, strange,  T h e quarks are represented by ip , ak  where a =  charm, bottom, top is the flavor index and k = 1, 2, 3 is the color index.  The gluons also carry color charge and are represented by the  field  w i t h a = 1,  2 , 8  being the color index and JJL the usual Lorentz index. T h e Lagrangian of the theory is given by the following equation:  CQCD  =  -\G%,G%  + £  where g is the coupling constant.  ^ ' ( » 7 ^ * + rn°V )V* k  k  +  B  ^  G  %  G  (1.1)  ^  T h e first term of E q . ( l . l ) is the kinetic term of the  gluon, otherwise known the gluon field strength tensor, the second term represents the kinetic term of the quark and the quark-gluon interaction, with the covariant derivative Dfj, = <9 + igA^/j, M  and the t h i r d term represents an anomaly which does not appear  at any order i n perturbation theory (this w i l l be discussed in more detail shortly). The gauge group for the color charge is SU(3) w i t h the generators represented by A and a set a  of corresponding structure constants given by f . abc  T h e generators of the gauge group  obey the following commutation relations: if j. abc  The gauge field can be written as  = A ^-  Eq.(1.2), the field strength tensor G  a  C  (1.2)  Using the commutation relations given in  is given by the following:  Chapter  1.  Introduction  3  =  (drAl-drAl  + gf^AlAl)-,  (1.3)  which implies that Gl = d»Al-d„Al  + gf AlAl  (1.4)  abc  The dual field strength tensor is given by  = e^p^G^,  with e  apixl/  the totally anti-  symmetric four index tensor. Due to the n o n - A b e l i a n nature of the 577(3) gauge group of Q C D , the commutator i n the first line of Eq.(1.3) does not vanish and therefore leads to a self-coupling between gluons. T h i s complicates matters greatly by allowing the creation of v i r t u a l gluon pairs. Consequently, if one wishes to calculate a diagram containing a virtual fermion-antifermion pair, the corresponding Feynman diagram w i t h a gluon pair included must also be calculated. A l t h o u g h the Q C D Lagrangian given in E q . ( l . l ) at first glance looks similar to the equations of other well known theories such as the electroweak theory, it contains many unique features. T h e first of these is the property of asymptotic freedom. T h i s means that contrary to quantum electrodynamics or classical gravity, the coupling constant g grows larger when we go to large distances or conversely, small momenta. renormalization group equation, a  s  = ^  Using the  is given as a function of Q according to the  following equation: 2TV  a (Q) S  = . ,  ,  b where  AQCD  n  /  r,  A  (1.5)  \og(Q/A cD)  a  Q  is the momentum scale at which a becomes strong as Q is decreased and the s  constant b = 11 — §Ay, w i t h Nf the number of quarks flavors being considered. Current 0  experimental measurements indicate that  AQCD  ~ 200MeV.  T h i s is what essentially  what sets the characteristic energy scale of the strong force. One of the consequences of this is that quarks are essentially free inside baryons. T h i s is somewhat counterintuitive as the strength of the gravitational and electromagnetic forces are inversely proportional to the distance.  Due to the spontaneous creation of charged fermion-antifermion pairs  Chapter 1.  Introduction  4  in the Q E D vacuum, we know that electric charge is screened.  In Q C D , we have the  opposite situation where color charge is antiscreened. T h e Q C D vacuum receives similar effects from quark-antiquark pairs, but these are suppressed by contributions from v i r t u a l gluons. In addition to asymptotic freedom, Q C D also exhibits confinement of color. T h a t is, only finite-energy asymptotic states of the theory exist as color neutral combinations of quarks.  The consequence of confinement is that the experimental observation of a  solitary quark is forbidden. For example, if we tried to p u l l an up quark out of a proton (2 up quarks, 1 down quark), a pair of quarks would be created leaving us w i t h a pion and a neutron. A l t h o u g h there is no current theoretical proof for confinement derived from ordinary Q C D , all experiment evidence obtained so far points to this. The  unique features of Q C D definitely do not end w i t h confinement. We know that  the Q C D vacuum is filled w i t h a condensate of quark-antiquark pairs. In other words, the following quantity has a non-zero vacuum expectation value:  <o#v|o> = ( O l V v M O ) + ( 0 | V ^ V L | 0 ) ,  (1.6)  which means that we lose the freedom to freely rotate left handed quarks independent of right handed quarks. T h i s phenomenon is referred to as dynamical chiral symmetry breaking. The Lagrangian possesses this chiral symmetry (with m  q  uum  = 0) but the vac-  does not share a similar property. The chiral condensate given in Eq.(1.6) can be  calculated by using so called soft-pion techniques to give (iptp) = — ( 2 4 0 M e V ) . 3  The  Q C D Lagrangian w i t h all quark masses set to zero is invariant under the large  symmetry group SU(3)  x SU(3)  color  right  is broken down to SU{3)i f + ight e t  metry group, SU(3) i  co or  T  x SU(3)  left  x U(l)  b  x U(l)  a  x R  s c a i e  , but this  x U(l)b by various mechanisms. In the unbroken sym-  is the gauge group for the gluons, SU(3) t(SU(3)i f ) righ  e t  is the  Chapter  1.  5  Introduction  freedom to freely rotate right-handed (left-handed)  among one another, U(l)b  sponds to baryon number (common phases for all quark fields), U(l)  a  corre-  is axion baryon  number (equal and opposite phases for all left-handed and right-handed quark fields), and Rscaie corresponds to scale invariance. T h e color symmetry is hidden by the confinement phenomenon. The SU(3) i ht r g  x SU(S)i f  e t  is broken down to SU(3)i f + ight e t  r  by dynamical  chiral symmetry breaking. Conservation of axial baryon number is violated through the triangle anamoly or chiral anamoly.  Finally, the running of the coupling constant or  asymptotic freedom breaks scale invariance. A l t h o u g h there does not exist conclusive evidence, lattice calculations indicate that a chiral symmetry phase transition occurs at a temperature of about T ~ 150 c  In E q . ( l . l ) , the last term O-^^G^G^  restoration/deconfinement MeV.  is connected w i t h the non-trivial vacuum  structure of Q C D as well as the chiral anomaly.  T h i s term is very important to this  thesis, so we will give a brief explanation of its origin (for a more detailed explanation, see [5]). It can be shown that there is a conserved quantity, called the winding number n, which is characteristic of all gauge potentials A . It is defined as: M  ,3  (1.7) If we consider the integral: (1.8) we see that it is actually the difference between winding numbers n*. The quantity Q is referred to as the topological charge. T h i s is surprising as the integrand can be rewritten as a total derivative: (1.9)  Chapter  1.  Introduction  6  with K, = 2e^Al{G%  -  -r A ^)  9  bc  b  (1.10)  Naively, we should assume that since this is a total derivative we can integrate the action by parts and discard this term as it is merely a surface term evaluated at  = ± o o . This  is simply not true as the different w i n d i n g numbers actually belong to different homotopy classes. In other words, one gauge field configuration cannot be continuously deformed into another if their respective w i n d i n g numbers are different. Gauge invariance implies that the Q C D vacuum must receive contributions from a l l homotopy classes and is a coherent superposition of all w i n d i n g states:  |0> = X>" '|n>.  (1-11)  iB  where 9 is the Q C D vacuum label. To see how the Lagrangian must be modified in order to take into account the non-trivial vacuum structure, we consider the following m a t r i x element of an arbitrary operator  X:  (0\X\Q) = J2 e - (n\X\m). m,n l{m  (1.12)  n)e  Notice that the path integral acquires an extra phase ( m — n is just the difference in winding states). In order to properly take into account this phase when doing calculations we must add to the Q C D Lagrangian the #-term: cf CQCD  a  ~  a  CQCD + O-^^G^G^.  (1.13)  Now we w i l l discuss the connection between 9 vacua and the axial anomaly. If we consider E q . ( l . l ) as a classical field theory, then Noether's theorem tells us that the axial current is a conserved current: i S ? = U7f.75« + dy^ d 5  + 57^755,  (1.14)  Chapter  1.  Introduction  7  ie. dpjfy = 0 i n the chiral l i m i t where a l l quark masses vanish. U p o n quantization, this current is no longer a conserved current of the theory as it receives quantum corrections. In order to see that the axial current is not divergenceless, the famous triangle diagram must be calculated. W h e n this calculation is done, we see that the chiral current has an anomaly given by: dJS  = ^ G - G "  (1.15)  T h e only way to ensure that the axial current is conserved is to add to  an extra term  given by E q . ( l . l O ) : & - r r $ = & - ^ K  (1-16)  B y performing a gauge transformation of the conserved charge Q$ = f d xj^°Q, 3  we see  that the different \9) vacua are related by a chiral U(1)A transformation: e Q*\d) = \d-2N a) ia  f  (1.17)  where a is an arbitrary constant. T h e actual 9 parameter that appears i n the Q C D Lagrangian receives contributions from two different sources: 0 = 0Qd? + a r g ( d e t M ) ,  (1.18)  where the first term is the Q C D vacuum label and the second term is introduced upon diagonalization of the quark mass matrix, M.  If 9 is non-zero i n Nature, then one  should observe a magnetic dipole moment for the neutron.  T h i s has been measured  experimentally and results indicate that the two contributions in Eq.(1.18) cancel to precision better than 10~ . T h e strong CP problem asks why these two terms should 9  cancel w i t h such high precision (see [6] for a recent review). One possible solution to the strong CP problem is to promote the 9 parameter to a dynamical field to represent a theoretical particle called the axion (for further details regarding axion physics see  Chapter 1.  8  Introduction  [7, 8, 9]). If this proves to be true, it provides a solution for the strong CP problem as well as accounting for some of the unexplained dark matter i n the universe. We would also like to note that if the ^-parameter possesses a non-zero value, the P and CP symmetries are strongly violated. T h i s comes from the fact that the ^-parameter comes into the equations with the anti-symmetric four index tensor e  apilv  which breaks  parity invariance. The vacuum state w i t h 9 ^ 0 is a stable ground state when one works in the thermodynamic limit (V —» oo). If one considers only gauge-invariant observables O in Q C D , it is well known that there are no transitions between two distinct 9-vacua as (9\0\9') ~5(9-9')  1.2  [10, 11, 12].  Overview  T h i s thesis is organized in the following manner. In Chapter 2, we introduce the effective Lagrangian approach. We w i l l present the well known linear sigma model as a simple example that is used frequently to model low energy Q C D . Next, we w i l l briefly review the large i V D i Vecchia-Veneziano-Witten effective Lagrangian for Q C D as given in c  [13, 14, 15, 16]. T h e anamolous effective chiral Lagrangian ( E C L ) constructed by Halperin and Zhitnitsky [1, 2] represents an improvement to the potential given in [13, 14, 15, 16], as it is valid for finite VV . We will concentrate on this as all calculations that follow will C  rely upon it. In Chapter 3, the motivation for studying an induced #-state is presented. It is crucial to clarify the difference between a non-zero # -state and a non-zero  gf  md  L t te.  undamenta  We will also present a strong analogy between the creation of a non-zero 9 -state ind  s a  in  heavy ion collisions and the possibility of observing disoriented chiral condensate(DCC). The  idea of a D C C in heavy ion collisions has been studied extensively over the past  ten years [17, 18, 19, 20, 21]. Chapter 3 w i l l be concluded with results from a numerical  Chapter 1.  9  Introduction  simulation that support the possibility of the creation of a 0 -state in heavy ion collisions md  [22]. As we will see in Chapter 4, the physics of this world is very different from our world with 9 = 0. The properties of the Goldstones Bosons are altered [23, 24] and CP (charge conjugation times parity) is no a longer valid symmetry of the system. Furthermore, there will be an excess of low momentum particles (in the (10-100) MeV range) as our calculations demonstrate [25]. We conclude this thesis in Chapter 5 with a summary of our results and future considerations.  Chapter 2  Effective Lagrangians and Q C D  In certain circumstances, the full field theory which incorporates all degrees of freedom may be unnecessary. W h e n these situations arise, it is convenient to define an effective Lagrangian which represents the dynamics of the low energy degrees of freedom.  The  effective Lagrangian is obtained by integrating the heavy particles out of the full action and incorporating their effects into a few simple constants.  Effective Lagrangian tech-  niques have proven to be extremely useful in practically a l l areas of physics, from particle physics to condensed matter physics. In order to study the vacuum structure of Q C D , we must consider the low energy regime  (E  < AQCD  ~ 200 MeV).  T h i s is a theory that is highly nonperturbative and we  must formulate a new plan of attack. We are now presented with an ideal setting to use effective Lagrangian techniques.  In the following few sections we will describe various  effective Lagrangians which are relevant to the calculations presented in this thesis.  2.1  A Simple Example: the Linear Sigma M o d e l  We will begin w i t h a well known field theory model, the linear sigma model. T h i s theory was an early candidate for Q C D which described the dynamics of a doublet of nucleons (proton and neutron), a triplet of pions  (7r ,7r°,  Lagrangian is constructed so that SU(2)i  x SU(2)  and  +  R  10  7r~),  and a scalar field a.  is a symmetry of the theory:  The  Chapter  2. Effective  Lagrangians  and  QCD  11  (2.1) with (2.2)  7T =  and r =  ( T \ T  2  , T  3  )  ( T T  1  ^  2  the three P a u l i matrices.  ^  3  )  (2.3)  T h e scalar a field represents the chiral  condensate, Eq.(1.6). U p o n m i n i m i z i n g the potential in Eq.(2.1), we see that the vacuum expectation value of the 7r and a fields are: (2.4) and the chiral condensate has a non-zero vacuum expectation value, as desired in order to model Q C D . Considering only low energy physics, below the scale of the m^, all the matrix elements that can be calculated using Eq.(2.1) can be incorporated into the following effective Lagrangian: (2.5) w i t h F = v from Eq.(2.4) to first order i n the coupling constant. In order to calculate any pion scattering amplitudes, we can expand U in powers of pion field, U = exp(if-Tv/F) 1+  2.2  ~  if-if/F.  D i Vecchia-Veneziano-Witten Effective C h i r a l Lagrangian  Next, we will briefly review the D i Vecchia-Veneziano-Witten effective chiral Lagrangian as given in [13, 14, 15, 16]. In these papers, they begin by constructing the most general  Chapter 2.  Effective  Lagrangians  and  12  QCD  equation due to symmetry arguments alone. In the low energy l i m i t , we know that the theory must possess the following properties: 1. In the large JV limit, Q C D exhibits confinement and chiral symmetry breaking. C  = 0, we have a £7(3) x U(3)  2. In the case of three flavors of light quarks w i t h m  q  chiral symmetry in the l i m i t JV —> oo. Because of chiral symmetry breaking, this C  is broken down to £7(3) spontaneously and there is a nonet of pseudo-Goldstone bosons which consists of the 7r-mesons, if-mesons, ry, and the rj'. 1  3. The axial anomaly is proportional to 1/N  and is therefore absent i n the large N  C  c  limit. In order to describe the lowest energy particles of the theory in a world w i t h 3 flavors of quarks, we parameterize the pseudo-Goldstone bosons by a 3 x 3 unitary m a t r i x U = £7 (1 + iJ2t°"K I'F A  0  n  +  0(IY )), 2  where U is the vacuum expectation value of U, t  a  0  are the generators of the group U(3), constant F  n  and 7 r represents the Goldstone bosons.  is the pion decay constant, F  a  n  = 93 MeV.  The  In order for the Lagrangian to  correctly transform under a £7(3) x U(3) transformation U —> AUB^,  w i t h A and  arbitrary unitary 3 x 3 matrices, there are only a limited number of terms which can be introduced. The kinetic term must assume the form Tr{d Ud U~ ) 1  ll  the form Tr(A4U  tl  and the mass term  + A f "•"£/*), w i t h M. a 3 x 3 m a t r i x which is proportional to the quark  mass m a t r i x . In principle, higher order derivative terms could be included, but these can all be neglected in the low energy l i m i t . In [13, 14, 15, 16], it was also pointed out that the axial anomaly breaks £7(3) x £7(3) but preserves SU(3)  x S'£7(3)(x£7(l)). O b e y i n g  the above constraints, the effective Lagrangian is given as follows: C = 3-  (Trd^Ud^U-  1  +Tr(MU  + MU) ]  ]  - ^(-ilndet  £7) ) , 2  (2.6)  I n the c h i r a l l i m i t where a l l q u a r k masses vanish, there w o u l d be a nonet of true massless G o l d s t o n e bosons. 1  Chapter  2.  Effective  Lagrangians  and  QCD  13  where a is a constant of order one and the constant F^ is included to ensure that the action is a dimensionless quantity. We can diagonalize the matrix M by an SU(3) x SU(3) transformation M. = e ^M.  We are also free to perform the following transformation,  l9  U —y e~ l U l6 z  and therefore lndet U —>• lndet U — i6. Once this is implemented i n Eq.(2.6),  we are left with: C = S  (rrd^Ud^U-  1  + Tr(MU  + MU ) ]  - ^(-ilndet  U - 0) ) . 2  (2.7)  The W W effective Lagrangian describes the light matter fields of Q C D and incorporates the Q C D vacuum angle 9, as well as the axial anomaly. T h i s can be used to calculate various meson amplitudes to lowest order as well as allowing the vacuum structure of Q C D to be studied.  2.3  Halperin-Zhitnitsky A n a m o l o u s Effective Chiral  Lagrangian  The D i Vecchia-Veneziano-Witten effective Lagrangian has proven to be very useful in the past in explaining such phenomena as the U(l)  problem. The U(l)  problem asks  why the ?7'-meson so much heavier than the octet of pseudo-Goldstone bosons. A s was demonstrated by Halperin and Zhitnitsky (HZ) in [1, 2], the W W effective potential could be improved upon.  T h e improvements included the potential being valid for a  finite number of colors, N , and the quantization of the topological charge being explicitly c  incorporated from the beginning. T h e H Z anomalous effective chiral Lagrangian can be further tested by showing that it reproduces the anomalous conformal and chiral W a r d identities of Q C D and can be expanded to reproduce the large N D i Vecchia-Venezianoc  W i t t e n effective chiral Lagrangian plus corrections proportional to  l/N . c  Usually there are two definitions of an effective Lagrangian. One is the Wilsonian effective Lagrangian which describes the low energy dynamics of the lightest particles i n the theory and the other is defined as the Legendre transform of the generating functional  Chapter 2. Effective Lagrangians and QCD  for connected Green functions. The W  14  W effective potential is of the former type i n that  it describes only the lightest particles in the theory (the Goldstone bosons), but it also incoporates the effects of the chiral anomaly. In deriving their Lagrangian, H Z take the generating functional approach and arrive at the W i l s o n i a n effective Lagrangian upon integrating out the massive "glueball" fields. We w i l l now briefly summarize the steps that were taken i n order to arrive at the improved effective potential for low energy Q C D (see [1] and [2] for a more complete description). The construction of the Lagrangian starts off w i t h imposing that the anomalous conformal and chiral W a r d identities of Q C D hold at the tree level. Once consistency with the anomalous W a r d Identities is ensured, the light matter fields of the theory, which are parameterized by a Nf x Nf unitary m a t r i x U, are introduced by the substitution: 9 -)• 9 - iTr log U.  (2.8)  The chiral anomaly dictates the exact form of the above substitution.  If we consider  Nf = 3 flavors of quarks, the light matter fields consist of an octet of pseudo-Goldstone bosons  (IT'S,  ICS,  and the rj) and the rj singlet field. The mass term for the U field remains  unchanged from the arguments given i n the previous section for the W  W potential.  T h e unitary m a t r i x Uij corresponds to the 75 phases of the chiral condensate, ( ^ ^ i ) — \{^R^L)\Uij,  =  and takes the following form: U = exp  U  ^Nf  (2.9)  In'  where U is the vacuum expectation value of the U field, 7r represents the pseudoscalar a  0  octet, Nf is the number of flavors, A are the generators for SU(Nf), a  U  =  F-KV?  = 133 MeV  and fa = 86 MeV.  and the constants  U p o n integrating out the heavy "glueball"  fields, the potential for the U field is given by the following expression [1, 2]: 1 . ^~ // 11 -\im^ logJ2 MVEcos(—(9-i\ogDetU N c  V(U,6)  =  e  v  1=0  l  ..  „  .„  +  2TT 2vr  \  —l))  Chapter 2. Effective  +  Lagrangians  ]-VTr(MU  where V is the 4-volume, M E = (ba /(32ir)G ) 2  s  and QCD  15  + MW)) =  (2.10)  —diag(rni(^i^i))  w i t h b = 11NJ3  is the diagonal mass matrix, and  - 2N /3.  Once the 9 parameter in Eq.(2.10) is  f  specified, there are no free parameters remaining. E v e r y t h i n g is fixed in terms of the vacuum condensates:  the chiral condensate is given by (ty^i)  = —(240 MeV)  and E  3  is given in terms of the gluon condensate by E = (ba /'(32ir)G ) 2  s  ~ 0.003 GeV . 4  The  summation over a l l branches, parameterized by I, is the main difference between this effective Lagrangian and the W W effective potential. T h i s potential also possesses cusp singularities at certain values of the fields, which is a result of the quantization of the topological charge from the beginning. T h i s type of piecewise potential is also present i n certain supersymmetric theories [26]. Considering only small values of (9 — ilogDetU)  in the thermodynamic limit (V —»  oo), the / = 0 branch dominates and the potential takes the following form: V(U,d)  =  -Ecos ^-(6-ilogDetU)]  - \Tr{MU  + M U ).  (2.11)  with q — u,d,s.  Once  fa  (- )  ]  We are free to choose a diagonal basis where U = diag(expi(f> ) q  ]  this is done, the potential in terms is the fas is:  V(9, fa) = -Ecos  -E > M  ^ ~ ^ )  cos  2  12  where M j are the diagonal entries of the quark mass m a t r i x which was introduced above. Notice that in Eq.(2.12), the 9 parameter appears only in the combination Jlfa — 9. In order to study the 9 dependence on the vacuum energy, we must minimize the potential and solve the following equation: . (9-Zfa\ sin I —  N M~i =-^-smfa. c  j  (2.13)  Chapter 2. Effective Lagrangians  and QCD  In the case where we have an SU(Nf)  16  isospin symmetry and M j - C E, the approximate  solution is given by fa ~ -^j. M a k i n g this substitution, the vacuum energy density as a function of 9 is given as: E (9) vac  = -E  + m N ($ib) q  f  cos (J^J  + 0(m ).  (2.14)  2  q  In addition to Eq.(2.14), one can also calculate the topological charge density Q = ( 0 | § J G G | 0 ) as a function of 9: (9\^GG\0)  =  = -m,|(W>|sin  .  (2.15)  The effective potential shown above in Eq.(2.10) represents the anomalous effective Lagrangian for Q C D realizing broken conformal and chiral symmetries. The following three points reinforce this statement: 1. Eq.(2.11) correctly reproduces the D i Vecchia-Veneziano-Witten effective chiral L a grangian ([13, 14]) i n the large N limit. A s was already mentioned, in the thermoc  < TV, the I = 0 branch dominates.  dynamic limit for small values of (9 — ilogDetU) E x p a n d i n g this branch i n powers of l/N , c  V(U, 9) = -E  + ^-{9  -ilogDetU)  2  we arrive at the following: -  -Tr{MU  l  + MW)  + ....  (2.16)  which is identical to the W W potential (Eq.(2.7)) except for the shift by the constant E. We see that the constant a in Eq.(2.7) just corresponds to  in the ^c  above expansion. It turns out that this can be identified w i t h a quantity called the topological susceptibility in pure Y a n g - M i l l s gauge theory. 2. It reproduces the anomalous conformal and chiral W a r d identities of QCD. consider the limit of SU(Nf)  If we  isospin symmetry w i t h Nf = 3 flavors of light quarks  Chapter 2. Effective Lagrangians and QCD  (m  q  <C AQCD),  17  the anomalous W a r d identity given in [27, 28] can be obtained if we  then differentiate Eq.(2.14) twice w i t h respect to 9:  Bmi/^OIT^GGfxJ^GG^IO)  = c  in the approximation where 9 is small.  ^ ( # )  + O K ) ,  (2.17)  The other W a r d identities can also be  reproduced in a similar fashion from Eq.(2.10). 3. It reproduces the known dependence in 9 {ie. 2TT periodicity of observables) [13, 14]. If one. examines the ^-dependence of the potential given i n Eq.(2.11), the naive claim would be that the 27r periodicity desired for physical observables is absent and instead it is periodic i n 9/N . c  T h i s is not true as Eq.(2.11) represents only one  particular vacuum state, / = 0. If we consider the sum over a l l vacuum states as in Eq.(2.10), we see that 9 and 9 + 2TT are physically equivalent states and there is actually no problem at all. We w i l l now proceed and use Eq.(2.12) as a model for the creation of an induced 9state i n heavy ion collisions. It should be noted that the following analysis is independent of the exact form of the potential and i n principle one could choose the W W potential in place of the H Z potential which was just presented.  Chapter 3  Induced T h e t a V a c u a  T h e long awaited Relativistic Heavy Ion Collider ( R H I C ) at Brookhaven N a t i o n a l Laboratory could yield many fascinating results concerning the accepted theory of the strong force, Q C D . A l t h o u g h the main goal of this experiment is to study the quark-gluon plasma, there exists many other possibilities. One of the most sought after results is the nature of the chiral symmetry restoration and/or deconfining phase transition. Another possible outcome is the so called disoriented chiral condensate ( D C C ) . In the following chapter we w i l l provide evidence that it may be possible to create an induced 9 / 0-state in heavy ion collisions. If this is possible, the consequences could include the observation of a new form of matter where the physics is quite different from what is familiar to us. In the following chapter, we use the low energy effective Lagrangian to show that a non-trivial # -vacua can be created i n heavy ion collisions. O u r calculations will follow mrf  closely the ideas presented i n discussing the formation of a disoriented chiral condensate. Through the use of a simplified numerical model based on the Lagrangian presented i n Section 2.3 we w i l l estimate the time it takes for a non-trivial # -state to form. Despite of mf/  the fact that our calculations are based on a simplified model which differ somewhat from the real physics associated w i t h high energy collisions, there is no reason to believe that a # -state cannot be created at R H I C . If the 9 -state md  ind  can be experimentally detected,  it must form within the time that the central region of the fireball (quark-gluon plasma) is isolated from the true vacuum.  18  Chapter 3. Induced Theta  3.1  Vacua  19  What is the difference between 9  md  and the fundamental 9 parameter?  T h e idea of an induced 0-state is very similar to the creation of the Disoriented C h i r a l Condensate ( D C C ) i n heavy ion collisons [17, 18, 19, 20, 21] (see [29] for a discussion of D C C as an example of an out of equilibrium phase transition). D C C refers to regions of space (interior) i n which the chiral condensate points in a different direction from that of the ground state (exterior), and separated from the latter by a hot shell of debris. For both cases ( D C C and # -state) the difference i n energy between a created state and md  the lowest energy state is proportional to the small parameter m and negligible at high q  temperature (see Eq.(3.3) below). T h i s is necessary as it was previously believed that the creation of a (9 -state would be associated w i t h the larger constant E. md  We would like to stress the point that an induced ^-parameter is very different from Qfund  T  c  i  s z  e  r  0  m  o  u  rW  o r l d and which cannot be changed. Above the temperature  at which the Q C D phase transition occurs, the quark phases fa may be, in general,  arbitrary.  G i v e n this, the singlet term  4> in iet s  g  — Y.fa, which is related to the ?/, is  non-zero. T h i s can be identified with # -state as it can be rotated away by performing mrf  a U(1)A rotation of the fundamental Q C D Lagrangian, at the cost of introducing an induced 9 term i n Eq.(2.12), 9  ind  = —Nf(f> i . sing et  T h i s non-vanishing singlet term could  also be referred to as a zero (spatially constant) mode of the rj field. Since the symbol rj is usually associated w i t h the excitation of the //-field corresponding to the //-meson and not w i t h a classical constant field (condensate) configuration, we w i l l refer to it as a 0 - s t a t e as we are free to perform a U(1)A rotation. A l t h o u g h the fundamental 9md  parameter remains zero, we would expect that an induced ^-parameter w i l l mimic the physics of the world with 9^  und  ^ 0. From now on we will refer to a # -state as simply  a #-state, thus o m i t t i n g the "ind" label.  md  Chapter 3. Induced Theta  3.2  Vacua  20  Disoriented C h i r a l Condensate  We w i l l now describe the disoriented chiral condensate scenario as given i n [17] to emphasize the similarities between the D C C and the 0-vacuum state. Rajagopal and Wilczek [17] use the 0 ( 4 ) linear sigma model (see Sect. 2.1) to describe the low energy dynamics of the pions and the chiral condensate. A l l fields are represented by a 4-vector w i t h components <j> =  fr), where o represents the chiral condensate and TT represents the triplet  of pions. Throughout the exterior region the vacuum expectation value of <f> is (v,0). In the interior region, however, the pion fields can become non-zero and (a, TT) wanders in the four dimensional configuration space. T h e high energy products (shell of debris) of the collision expand outwards at relativistic speeds and separates the misaligned vacuum interior from the exterior region. In [17], all calculations were done under the assumption of a quenched system. In the quenched approximation, the (a, TT) fields are suddenly removed from a heat bath (T > T ~ 150 MeV) c  and evolved according to zero temperature Lagrangian dynamics.  This is done numerically by giving the fields a non-zero vacuum expectation value (0) / 0 and letting this field configuration evolve i n time according to the zero temperature equations of motion. There is no reason to ignore the possibility that regions of misaligned vacuum with an arbitrary isospin direction w i l l be created. In [17], it was shown that if the process is very rapid and the system is out of equilibrium, there is a temporary growth of long wavelength spatial modes of the pion field corresponding to domains where the chiral condensate is approximately correlated. In the case of D C C , the created state will relax to the true vacuum by coherent emission of pions w i t h the same isospin orientation producing clusters of charged or neutral pions. D C C has been suggested as a possible explanation of certain events reported in the cosmic ray literature called Centauro and anti-Centauro events.  Chapter  3. Induced Theta  Vacua  21  Let us consider the case Nf = 2. T h e m a t r i x U is parameterized by the misalignment angle <j> and the unit vector n in the isospin space: U =  , UW = 1 , (¥ ¥ ) L  R  = - K * * H > | Uij. L  The energy density of the D C C is determined by the mass term: E$ = -^Tr(MU  + M^U ) ]  = - 2 m | ( M ) | cos(0),  E q . (3.1) implies that that any <j> ^ 0 is not a stable vacuum state because ^  (3.1) 7^ 0, ie. the  vacuum is misaligned. Since the energy difference between the misaligned state and the true vacuum is proportional to m , the probability to create a state w i t h an arbitrary c/> q  at high temperature T ~ T is proportional to exp[V(Ee c  — E )/T] 0  and depends on (j) only  very weakly. In other words <p is a quasi-flat direction. Right after the phase transition when ( ^ ^ ) becomes non-zero, the pion field oscillates. T h e coherent oscillations of the 7r-meson field would correspond to a zero-momentum condensate of pions. Eventually these classical oscillations produce real 7r-mesons which can hopefully be observed.  3.3  #-State in Heavy Ion Collisions  A s we discussed in Sect. 3.1, the formation of the (9-vacuum state could occur when the U{1)A  phase of the chiral condensate is in general non-zero. T h e difference between the  formation of #-vacua and D C C is twofold: (i) In addition to chiral fields differing from their true vacuum expectation values the ^-parameter of Q C D , which is zero in the real world, becomes effectively non-vanishing in the interior of the shell of debris. (ii) The disoriented chiral condensate involves the amplification of the charged pions while the formation of the ^-vacuum state involves the amplification of the low momentum modes of only the neutral particles. T h i s includes the 77' singlet as we shall see below.  Chapter 3. Induced Theta  Vacua  22  We w i l l choose a diagonal basis for the m a t r i x Uij i n order to take into account the U(1)A  phase associated w i t h 9-vacua.  T h e energy density of the misaligned vacuum is  determined in this case by the potential given by Eq.(2.12) and derived i n [1, 2]:  V(9,fa) where E  = (ba /(32ir)G )  3  ~  2  s  10~ GeV .  = -£cos( 10~  2  9—  GeV  (b  N f  - £  Mi cosh,  is much larger than M  (3.2)  A  t  = -m (M) g  ~  The fact that the ^-parameter appears only in the combination Y fa — 9 is  4  very important for the formation of a 0-state. If we consider the l i m i t where all the quark masses are equal and Mi <C E, the approximate m i n i m u m of this potential is given by fa ~ JJJ.  In this case, the difference i n the vacuum energy when 9 = 0 and 9 j= 0 for  Nf = 2 is given by: V (e)-V {d vac  vac  = 0)~-2m \(W)\(\cos^\-l). q  (3.3)  From this we see that the cost of creating a non-trivial 0-state goes like the much smaller parameter m (#^/) and that a 0-state is degenerate w i t h the 9 = 0-state in the chiral 9  limit where m  q  = 0, as expected. T h e combination Yl fa ~ 9 is a direct consequence of  the transformation properties of the chiral fields under a U(1)A rotations. If we compare Eqs. (3.1) and (3.2), we see that i n E q . (3.2) there is a large term ~ E 3> m \(^f^)\. q  In the chiral limit, this term does not vanish and provides a non-  zero mass for the ?/-meson which is expressed in terms of the parameter, E . Since 9 is associated with the large parameter E , it is often assumed that the creation of non-trivial 0-vacua would involve too large an energy cost. It was pointed out in [30] that this is not necessarily the case. T h e crucial point arises when one considers the scenario suggested for D C C where the chiral phases fa acquire random values, the energy of a misaligned state differs by a large amount proportional to E from the vacuum energy. In order to produce long  Chapter 3. Induced Theta Vacua  23  wavelength oscillations (a macroscopically correlated region) , the potential must have 1  a quasi-flat (~ m ) g  directions along the fa coordinates.  T h i s is not the case i f the  misaligned state differs from the vacuum state by an amount ~ E. However, when the relevant combination (J2ifa  ~ Q) from Eq.(3.2) is close by an amount ~ 0(m ) q  to its  vacuum value, a B o l t z m a n n suppression due to the term ~ E is absent, and an arbitrary misaligned 0-state can be formed. In order to study the time evolution of the chiral phases numerically, several assumptions must be made. First, assume that the 0-parameter acquires a non-zero value on a macroscopically large domain when the temperature T ^> m  q  and the system is out-  of-equilibrium. Once again, ( 9 / 0 follows from the assumption that the singlet phases (YI fa) 7= 0 immediately upon cooling through the phase transition. Performing a U(1)A rotation then gives us 9 /  0. T h e second assumption is that the phases fa, (f> , and d  4> have small random values at these high temperatures. T h i r d , the rapid expansion of s  the high energy shell leaves behind an effectively zero temperature region i n the interior which is isolated from the true vacuum. T h e high temperature non-equilibrium evolution is very suddenly stopped, or "quenched", leaving the interior region in a non-equilibrium initial state that then begins to evolve according to zero temperature Lagrangian dynamics. Starting from an initial non-equilibrium state we can study the behavior of the chiral fields using the zero temperature equations of motion. T h e equations of motions are non-linear and cannot be solved analytically but we can solve them numerically in order to determine the behavior of the fields. Finally, by realizing that the chiral fields relax to a constant, equal, and non-zero values on a time scale over which spatial oscillations of the fields vanish (ie.  only the zero momentum mode remains), we show  that a 0-state has been formed. T h e formation of a nonperturbative condensate is also I n referring to a macroscopically large domain, the region considered to be macroscopic compared to the scale of the system, AQ . 1  CD  Chapter  3. Induced Theta  Vacua  24  supported by observation of the phenomenon of coarsening (see below) and by a test of volume-independence of our results.  3.4  E v o l u t i o n of the Equations of M o t i o n  We will now present the results obtained from the numerical evolution of the chiral fields on a cubic lattice.  The equations of m o t i o n for the phases of the chiral condensate  with two quark flavors consists of two coupled second order nonlinear partial differential equations: fawhere V  2  ^ fa 2  + lfa + ~V{fa,fa,9)  = 0,  i = l,2,  (3.4)  is a three dimensional spatial L a p l a c i a n and the potential is given in E q . (3.2).  There is obviously some sort of damping present in any real system, and this damping can be attributed to the expansion of the region inside the shell (dilution of energy) and/or the radiation of Goldstone bosons. T h e damping constant, 7, was chosen to be of the same order of magnitude as  AQ D C  ~ 200 MeV,  which should be a reasonable value  according to all other scales present in the system. The initial data for each of the chiral fields fa is placed on a 3 — d grid of 16 points, 3  with fa chosen from a uniform distribution and fa = 0. T h e initial data was evolved i n time using a Two-Step Adams-Bashforth-Moulton Predictor-Corrector method for each grid point w i t h the spatial Laplacian approximated at each grid point using a finite difference method. We used periodic spatial boundary conditions. T h e lattice spacing was set according to the the length of the side of the spatial grid, which was varied in order to vary the volume. We evolved the data for 8000 time steps of about 10~ MeV" , 5  1  which was much smaller than the lattice spacing. In order to  examine the momentum dependence, we applied a Fast Fourier Transform [31] to the spatial data at evenly spaced time steps. T h e data was then binned i n small increments  Chapter 3. Induced Theta  Vacua  25  of the magnitude of the wave vector \k\ i n order to obtain the angular averaged power spectrum. T h e b i n w i d t h was chosen to be as small as possible without resulting i n some intervals being left empty. The numerical simulation was carried out for different initial conditions and different volumes, w i t h the results being qualitatively the same. Once the chiral phases (f>i were given random values, they begin to approach the true solution fc « 9/Nf  and of course  overshoot it. We saw an initial growth of long wavelength modes as in [17] and subsequent damped oscillation of all modes.  The \k\ = 0 modes oscillated and approached  equilibrium values of the fields (fc ~ 6/Nj).  A l l of the modes with \k\ /  the  0 decayed to  zero before the \k\ = 0 mode reached the vacuum expectation value (fc ~ 6/Nj).  The  fact that this behaviour manifests itself for different total volumes and grid sizes suggests that our results are not due to finite size effects. Since we are working i n a finite box w i t h periodic boundary conditions, our \k\ = 0 mode is really only a quasizero mode. However, our quasizero mode approaches the same value irrespective of the total spatial volume indicating that this really is a condensate.  If it were not we would expect the value of  the coefficient to decrease when the volume of the system increases (as Ak ~  1/Ax).  The evolution of the Fourier modes of the fc fields as a function of time is shown in F i g . 3.1 for the specific case of 0 = 27r/16 and a a spatial volume of (10 fm) . 3  assigned each of the fields an equal mass of about 5 MeV. phases were chosen randomly, with \fc\ < 7n/16  We  T h e initial values of the chiral  and fc = 0. In F i g . 3.1, we show the  zero mode and three of the higher momentum modes. It is obvious that the zero mode settles down to a non-zero value ~ 0/Nj  while a l l higher modes vanish extremely rapidly  and are negligible long before the zero mode settles down to its equilibrium value.  —> We have also examined fc as a function of \k\ for different times. T h e instantaneous distribution of Fourier modes for the evolution above is shown i n F i g . 3.2 at a few  Chapter 3. Induced Theta  Vacua  —i , , , , i 0  100  ,  ,  26  ,  ,  i  200  ,,,,i ,,,,i 300  Time (1.6x1(Hlv1eV-i)  ,  ,  400  ,  ,  i_i  500  Figure 3.1: \(f>k\ is shown for various \k\ as a function of time. In a time ~ 1 0 ~ sec, the zero mode relaxes to fa ~ 0/Nf and all other modes decay to zero. T h i s is very similar to the idea of a disoriented chiral condensate. 23  0 .15  1  t -5t 3  0 . 05  0  -  1  V  V2T,  v L -  ti =1.6x10- M e V "  • 3  5  —- 10  15  1  20  •—  25  Magnitude of k (arbitrary units) Figure 3.2: Above we demonstrate that the system exhibits the coarsening phenomenon (amplification of the zero mode as time increases). T h e data was sampled at three times within the first 1000 time steps of the evolution.  Chapter 3. Induced Theta Vacua  27  0 .1 0.08  4>k  0.06 0.04 0 . 02 0 0  100  200  300  400  500  Time (1.6x1(HMeV-i) Figure 3.3: The zero mode and a non-zero mode are shown as a function of time for three different volumes. T h e heavily dashed line represents the smallest volume (8 fm) , the medium dashed line represents the middle volume (16 fm) , and the solid line represents the largest volume (32 fm) . T h e volume independence of the zero mode reinforces the claim that a true nonperturbative condensate has been formed. 3  3  3  different times. T h i s graph clearly demonstrates the phenomenon of coarsening , which is the amplification of the zero mode with increasing time. Coarsening and the formation of a nonperturbative condensate is discussed in greater detail in [29]. In F i g . 3.3 we plot \(/>k\ as a function of time for three different volumes. We chose 9 = 7r/16 and the volumes (8 fm)  3  , (16 fm) , 3  and (32 fm) . 3  We plotted the zero  mode and one of the higher momentum modes for each volume. Notice that the zero mode is independent of the volume of the system, while the magnitude of the non-zero mode decreases with increasing volume. T h i s is the signature that a real nonperturbative condensate has been formed. For a total volume of (10 fm)  3  and 9 = 27r/16 the time for relaxation from the initial  nonequilibrium state following the quench to the non-trivial ^-vacuum is approximately  Chapter 3. Induced Theta  0.064 MeV^  1  % 4 x 10~  23  28  Vacua  s. T h e central region of the fireball is estimated to be isolated  from the true vacuum (9 = 0) for a time Tfi t, u ~ 10 fm/c re  a  ~ 10  - 2 3  s. Given this estimate  for Tfireball, we see that the formation of a 0-state is of the same order of magnitude. T h e volume we have used is just at the upper limit of what we would expect at R H I C for the transverse direction. In spite of the fact that we have not taken into account the correct physical collision geometry, our simplified calculation suggests the possibility of producing non-trivial 0-vacua. In the preceding sections, we have shown that the realization of an induced 0-state may be possible in heavy ion collisions. Through the use of a numerical simulation of the zero temperature equations of motion, we have shown that the chiral fields fa go to a spatially constant non-zero value related to the ^-parameter (fa ~ 9/Nf) of about 1 0 ~  23  s.  in a time  A l l simulations relied upon the quenched approximation where we  assume that the system is suddenly removed from a high temperature heat bath ~ T  c  and subsequently evolved according to the zero temperature equations of motion. The following two facts confirm that the a true nonperturbative condensate (9-vacuum state) has been formed: (i) A l l non-zero modes decay to negligible values long before the zero mode reaches its equilibrium value, (ii) The zero mode is independent of the volume of the system while the non-zero modes decrease w i t h increasing volume.  Chapter 4  S i g n a t u r e s o f a n I n d u c e d 9-Vacuum  State  One might ask, "How could such a (9-state be detected if it is created in heavy ion collisions?" If the correlation length of the created region is large and the system is in thermal equilibrium, axions would be the obvious thing to look for. If we assume the existence of axions, they should i n principle be produced upon relaxation of the 9  md  ^ 0  to the lower energy state w i t h 9 = 0. However, since it is unlikely that the system w i l l be in thermal equilibrium and the size of the created region is not expected to be very large (~ (10 fm) ) 3  axion production does not look very promising if we compare to the  l i m i t already achieved from astrophysical and cosmological considerations [7, 8, 9]. T h i s forces us to look to must other possibilities. Another possibility is that the 0-state could be observed through Goldstone bosons w i t h specific P , C P - o d d correlations [32, 33]. T h i s possibility must be considered due to the non-zero value for the topological charge density w i t h 9 ^ 0 in Eq.(2.15). It was recently pointed out [24] that this effect would probably be washed out due to the rescattering of pions and their interactions in the final states, which mimic true C P - o d d effects. A s was shown in Chapter 3, the creation of the 9-vacuum state involves the enhancement of the low momentum modes of the chiral fields. Considering this, we should look for enhanced production of particles such as the 7r°, 77, and //-mesons w i t h low moment u m , on the (10 — 100) MeV  scale (depending on the size of the domain L). These should  decay by various processes to photons and dileptons, and if it is possible to detect a large number of these low momentum particles this would be a definite signal of the creation  29  Chapter 4.  Signatures  of an Induced 9-Vacuum  State  30  of a 0-state. In the following we w i l l provide further evidence that this is the case and could also account for the unexplained large number of low momentum dileptons seen at C E R N [3, 4]. T h i s idea was originally proposed in [24]. In order to support this speculative conjecture more in depth calculations must be performed.  4.1  P r o p e r t i e s of the pseudo-Goldstone Bosons i n the non-zero ^ - B a c k g r o u n d  As was shown i n [23, 24], the properties of the pseudo-Goldstone bosons are altered in 1  the presence of a non-zero ^-background. T h i s has been suggested as a possible signature of the formation of a non-trivial E s t a t e .  In the #-world, the masses of the octet of  Goldstone bosons and the rj singlet are decreased. mass m a t r i x for an arbitrary value of 6 w i t h m  = m^ = m  u  „r  4  2  Ji m\  =  I  For instance, a calculation of the q  gives [23, 24]:  ° \ 2  A(2M,|cos^4-2M ),  (4.1)  a  2  A  EN,  4  1 . „ . r  9.  ,  A s well, these particles become a mixture of pseudoscalar/scalar rather than pure pseudoscalars as for 9 = 0. The mixing angle between the singlet and octet combinations depends on the ^-parameter [23]. This fact means that the C P - o d d decays 77 —>• i]' —>•  7T7T  7T7T  and  are no longer suppressed in the ^-background and become of order 1 [24]. These  matrix elements were recently calculated [24] to give the following results: r> ("-™»  r  2  = ~  1  - 4rnl(9)\ ~  9  [  Ji  ) •  Q 0.5Mel/(sin-) , 2  F r o m now on the pseudo-Goldstone bosons will be referred to as simply Goldstone bosons.  (4.2)  Chapter 4. Signatures  of an Induced 9-Vacuum  6 r(r/  31  State  / m s i n f (0|gg|0) g  —>• TTTT)  (4.3) The full w i d t h of the rj', rj decays in our world are much larger than the above decays in the 0-world. T h e particle data booklet gives T {rj) total  ~ 118 keV  and  r^(?/) ~ 0.2  MeV  as the full experimental widths. Therefore, the widths could be increased by as much as an order of magnitude in the 9-world. Thus observation of an increased rate of decay of rj', rj at masses slightly shifted from the accepted values would be a signature of the ^-vacuum. However, this signal may be a difficult one to distinguish in the large amounts of data generated in a heavy ion collisions. Therefore we would like to discuss a signature that would be easier to observe experimentally.  4.2  Signature of the Creation of  9-Vacua  In this section we consider an additional new signature which could possibly verify if a restate can be created in heavy ion collisions. A s was discussed briefly in the introduction, the creation of a 0-state could greatly enhance the production of low momentum (~ 25 MeV)  Goldstone bosons.  Before going into the details we would like to discuss the concepts behind this idea. We have already presented evidence that a 0-state can be formed in heavy ion collisions if protected from the exterior 9 = 0 world by the shell of debris. A t some point, however, this shell ceases to isolate the interior and the influence of the exterior world will be felt. T h e way in which this happens is not well understood, but we would like to use an instantaneous approximation in order to obtain some approximate results. There are basically two time scales present in the system: the lifetime r h u of the shell separating s  e  Chapter 4. Signatures  of an Induced 9-Vacuum  32  State  the two worlds and the scale associated w i t h strongly interacting particles,  (AQCD) -1  If these differ by orders of magnitude, the disappearance of the shell can be considered either as an instantaneous perturbation or an adiabatic perturbation. Realistically, we expect T h ii to be about the same order of magnitude as ( A Q C D ) s  - 1  e  and therefore the disappearance of the shell is somewhere between an  instantaneous  process and an adiabatic one. However, we do not yet know how to treat the process properly so we w i l l use the instantaneous approximation i n order to obtain an order of magnitude estimate of the spectrum of emitted particles. The instantaneous perturbation is an approximation that is well known i n quantum mechanics. We assume that the shell separating the 0-vacuum state from the vacuum state w i t h 9 = 0 instantaneously blows apart.  Therefore, all. states which had been  formed in the induced 0-state will suddenly find themselves in a new vacuum state w i t h 0 = 0. The 0 - s t a t e s are now forced to transform to asymptotic states of our world md  with 0 = 0. We do not know how to treat this complex transformation exactly but if the process is instantaneous things simplify considerably and we can expand the initial state in terms of the asymptotic states of the 0 = 0 world. The above approximation is applied to our calculation i n the following way. The field values, (pi, obtained for the 0-state in Chapter 3 is embedded into a larger grid where the field values take their 0 = 0 vacuum values. T h i s data now contains a plateau of field values corresponding to 0-state values surrounded by zero field values. T h i s field configuration must now resolve itself into asymptotic free particles. W i t h this process in m i n d we determine the momentum spectrum of free particles corresponding to this distribution by considering the Goldstone boson fields as true quantum fields at this instant. In order to calculate the spectrum, we must obtain the quantities a(k) and  at(k).  These are obtained by performing the Fourier transform of the field configuration at  Chapter 4. Signatures of an Induced 9-Vacuum  State  33  time t = T heii when the shell breaks down, s  a(k) = J d xip(x)ex.p(-ik  • x),  3  (4.4)  where ip(x) is the distribution amplitude of the fa fields obtained from evolving the equations of motion (Eq.(3.4)) on a cubic lattice, as was done in Chapter 3. Eq.(4.4), the number operator is given by N(k)  = N a(k)a^(k) 0  where N  0  From  is an overall  constant that w i l l be determined using the following argument. A s we mentioned earlier, the 9-vacuum state differs from the true vacuum state by a small amount of energy ~  m. q  T h e amount of energy that is available when the 9-vacua decays is obtained by analyzing the ^-dependence of the vacuum energy density E: E  9  = m \^)\N cos^). q  (4.5)  !  Therefore, the amount of energy AS available due to the formation of the #-state is given by AS  = =  {E -E )-V 0=0  e  \(W)\N (l-cos(^-))-V  mq  (4.6)  f  where V is the the volume of the created ^-region. If we take for example a 0-state w i t h volume V = (8 fm)  3  the amount of energy available is AS = 10 GeV.  This represents  only a small amount of energy compared to cm energy, ^/s = 40 TeV, expected in heavy ion collisions: AS v ^ 7  20 ( ^ )  GeV  40 TeV  (  y  \(10  ^ fm) J 3  If the total amount of available energy is known, the constant N can be fixed by enforcing 0  the following conservation of energy constraint: r  ri k 3  Chapter 4. Signatures  of an Induced 9-Vacuum  34  State  where the sum is performed over a l l types of particles considered (ie. a l l neutral Goldstone bosons).  Once this constant i V is fixed, the total number of pions and etas can be 0  calculated by  (4.8) for the simplified case of only one collision of two gold nuclei. If we consider that R H I C w i l l collide gold nuclei at a frequency of 1 kHz,  we can estimate the total number of  these low momentum particles produced by simple multiplication. A l l the results that follow are presented for just one event.  4.3  Results  A s we have already mentioned, we proceed as follows. A t some time t before the fields fa settle down to the constant field configuration (see F i g . 4.1), we take the position space data and embed this in a larger square grid where the field values are zero (our world). We assume an instantaneous perturbation and then take the Fourier transform of this field configuration i n order to obtain the operators a(k) and af(k).  It may be argued  that the instantaneous approximation is not valid in this case as we expect  ~  TQCD  and therefore this calculation gives us a rough estimate at best. In F i g . 4.1, we show at what point in the evolution of the fa fields we chose for T heiis  In order to calculate the spectrum of the diagonal (neutral) components of the matrix U  (-7T , 0  rj, rj'), we must make the following correspondence: U  =  diag(expifa) a \a  2  rj'  (4.9)  to obtain the TT , rj, and rj' in terms of the fa's: 0  {4>u -  fa)  (4.10)  Chapter 4. Signatures  of an Induced 0-Vacuum  State  35  4fc 0  1  2  Time  3  (10"  4  sec)  2 3  Figure 4.1: \fa(k = 0)| is plotted as a function of time for the up, down, and strange quark. Notice that the zero momentum modes of the fa fields settle to a non-zero value in a time on the order of 1 0 ~ s. T h e times t\ and t represent the value we chose for r heii, the the time when the shell separating the two regions disappears. 23  2  S  V=^=(fau  +  fa-2fa),  V' = ^(<f>u + fa +  (4.11)  fa)-  For a l l numerical calculations, we can take the constant  (4.12)  to be ~ f . n  One of the  crucial decisions which must be made is when to choose the time at which the shell breaks down. We chose several values for T u. she  A s r h u gets larger, the majority of s  e  the particles which are produced due to the formation of the \9) state have momentum k < 25 MeV. In F i g . 4.2, we show a plot of N o(k) as a function of \k\ for the neutral  Ao(fc) /  pion. T h e function  7r  7T  has dimensions M e V  - 1  , so that the total number of particles,  N^o, is a dimensionless number given by: r°° No = / dkN o(k) J-oo n  n  r°° 47T = / dk N a(k)a{k) . J-oo 2{2TT)' W) ]  n/n  0  i  :  x  (4.13)  Chapter  4.  Signatures  of an Induced 9-Vacuum  State  36  > CD  175  k| ( M e V ) Figure 4.2: We plot the number of pions produced, N o(k) as a function of the magnitude of the wave vector, |A;|. Above we show that the momentum distribution of the 7r°-mesons produced is p r i m a r i l y < 25 MeV for two different values of Tgheii- T h e solid line represents the earlier time t\ and the dotted line represents the later time t (see F i g . 4.1 for the positions of t\ and t in the evolution of the fa fields). T h i s is expected as the formation of the 0-state can be attributed to the enhancement of the low momentum modes. n  2  2  The two different lines represent the times at which we assumed the shell to break down. The solid line represents the earlier time T  shM  = ti = 1500/6000 time steps^  the dotted line represents T h ii = t = 3000/6000 time steps^ s  e  2  io-  io-  s while  s (this representation  for t i and t w i l l be used in all graphs that follow). For this calculation, we assumed the 2  created #-state had a volume of (8 fm)  3  which exists in a larger volume of (64  fm) . 3  T h e explanation of this phenomenon is simple: at time t\ the amplification of the zero mode is not as large as at time t . 2  Furthermore, at time t\ a considerable portion of the  energy goes to high momentum particles. We show the same graph for the ry-meson and the r/-meson. Notice that that the rj is produced in copious amounts compared to the 7r°-mesons and the 77-meson. In order to obtain the total number of each particle produced, we use Eq.(4.8) to calculate the  Chapter 4. Signatures of an Induced 9-Vacuum State  37  0 .25  > CD  175  k| ( M e V ) Figure 4.3: N (K) is plotted as a function \k\, at two different times t and t as i n F i g . 4.2. T h e higher peak represents the later time w i t h the main difference being that now a larger percentage of the produced particles lie i n the k < 25 MeV range. n  x  total number of particles produces per collision: N o = 0.1, N n  t = ti and N o = 0.5, iV„ = 4.3, n  v  = 2.8,  2  = 19.3 for  = 18.4 for t — t . We have also checked that i n the 2  limit where a l l quark masses are equal, only the 77' is produced, as expected. It should be noted that as demonstrated by these figures and the coarsening phenomenon (i.e. the phenomenon of amplification of the zero mode as time increases), we would expect that as  T h ii s  e  increases up to a m a x i m u m value, the majority of the particles would reside i n  the low momentum regime. We consider our analysis to be a lower bound on the number of produced particles due to the fact that we do not include the particles which would be produced during the formation of the #-state. T h e particles that are produced during this period w i l l have a higher momentum and therefore will be hard to detect i n such the background of high momentum particles that is expected i n heavy ion collisions. O u r main conclusion is that the production of an induced #-state will result i n the emission  Chapter 4.  Signatures  1  of an Induced 9-Vacuum  State  38  ' \  0  25  50  75  100  125  150  175  |k| ( M e V ) Figure 4.4: N '(k) vs. \k\ is shown above. T h i s graph shows that the rj clearly dominates the spectrum compared to the other neutral particles. v  of low-energy rj and //-mesons. In a recent paper by Baier et al. [34], the production of an excess of //-mesons is discussed in the scenerio suggested in [32]. In [32] they use the W W effective potential to show that upon cooling through the phase transition temperature T , metastable states c  may be formed where P and CP are violated. W h i l e the ideas presented i n [34] are similar to the work presented in this paper, there are important differences to be noted. Our approach is essentially a two step process. First, we let the chiral fields evolve and observe that a non-trivial vacuum state has been formed. Once the 0-state has been realized, we assume that it instantaneously decays and calculate the particle production due to this single event, not taking into account previous particle production. We also assume a quenched system where the temperature changes quickly from T ~ T to T = 0. Their c  approach is different from ours as their calculation of the number of produced //-mesons receives contributions from all time, not just one particular instant. In order to calculate  Chapter 4.  Signatures  of an Induced 9-Vacuum  State  the number of ry'-mesons produced when the CP  39  odd bubbles disappear i n [34], they  use the fact that at some temperature below the temperature T  c  the metastable state  disappears and becomes a saddle point. Once this saddle points forms the 7/-field begins to roll down towards the true m i n i m a where CP is restored. The number of produced rjmesons is calculated as a function of time by considering the quantum  fluctuations  about  the vacuum expectation value of the field and using a Hartree-type approximation. The work presented i n [34] relies on the assumption that the potential is changing slowly as a function of temperature compared to the motion of the field, so that the temperature can be regarded as frozen at a particular point T = T  .  saddle  The fact that their calculation  receives contributions from all times explains the difference in rates: for a domain with radius 5 fm,  they estimate that about 90 — 100 ?/-mesons would be produced while our  estimate is 4 — 5 times smaller. In the above calculations of N(k)  the geometry of the created E s t a t e is assumed to  be a cubical box. For the case of heavy ion collisions, the more realistic case is that of an ellipsoid or elongated rectangular box. We expect the collision to create a region of quark-gluon plasma which has an asymmetry in one direction. In order to model this situation, we evolve a rectangular grid and embed it in a larger square grid and compute the Fourier transform. Once again, we consider the angular averaged value of For a ^-region with dimensions 32 fm  x (8 fm) ,  N(k).  we show the spectrum of the rj at  2  the same time slices shown in F i g . 4.4. T h e side of length 32 fm is the direction parallel to the beam direction. The low momentum modes still dominate, but the peak is not as sharp and the higher modes show a stronger presence.  Also, since the ^-region is now  larger, from Eq.(4.7) we would expect more particles to be created since there is now more energy available. T h i s is evident when comparing F i g . 4.4 w i t h F i g . 4.5. For the two different times shown, we find that A ^ o = 0.3, N  v  N o = 1.7, N v  v  = 16.8,  = 73.8 for t = t . 2  = 12.7, N > = 76.3 for t = t\ and v  Chapter 4. Signatures  of an Induced 9-Vacuum  State  40  Ikl(MeV) Figure 4.5: T h e momentum distribution (N^(k)vs.\k\) is now shown for the case where the geometry of the 0-region is considered as a rectangle. We consider the case where the size of the 0-region is larger along the longitudinal directions compared to the transverse directions. T h e solid and dotted lines once again represent times T h u = t\ and t respectively. A l t h o u g h the majority of the spectrum still exists in the low momentum range, the higher momentum modes now become more evident. s  e  2  U p o n consideration of the above geometry, we must also examine the angular distribution of the system. We choose the coordinates such that the long edge of the 0-region is parallel to the z-axis. In F i g . 4.6, we show the angular distribution of N o(9) n  as a  function of the azimuthal angle 9. T h i s plot suggests that the majority of the particles would have larger momentum components in the direction perpendicular to the beam direction.  We also investigated the dependence on the polar angle 4> and found that  there is essentially no dependence, as expected. N(k)  has a constant value which is equal  to the peak value i n F i g . 4.6. As was shown in Figures 4.2, 4.3, 4.4, and 4.5, if a 0-vacuum state can be created in heavy ion collisions we should expect an excess of low momentum Goldstone bosons. In particular, the rj would be produced in large amounts.  The excess low momentum  Chapter 4.  Signatures  of an Induced 8-Vacuum  State  41  12  7i  TI  3n  4  2  4  e  Figure 4.6: For the scenario shown in F i g . 4.5, we consider the angular dependence on the azimuthal angle for the 7r° instead of the r(. T h e azimuthal angle is defined so that 9 = 0 coincides w i t h the beam axis. 7r°-mesons, 77-mesons, and ^'-mesons may eventually eventually decay to photons and e e~ pairs through such decays as 7r° —>• e e ~ 7 , 77 —>• e e ~ 7 , and 77' — > e e~j. +  +  +  +  T h e end  result would be an excess of long wavelength dileptons, which could possibly provide a solution to the observation of an excess of these particles seen at C E R N [3, 4].  Chapter 5  Conclusions and Future Considerations  In this thesis, we provided evidence through numerical simulations of a model based on the Halperin-Zhitnitsky effective potential for Q C D that it may be possible to create non-trivial induced 8-vacua  in heavy ion collisions.  To reiterate, as in [17], we saw  an initial growth of the low momentum (long wavelength) modes, followed by damped oscillations. T h e fact that the only surviving mode is the quasi-zero mode indicates that a nonperturbative condensate has been formed. In addition to this, our hypothesis that the formation of 0-vacua may be possible is supported by the test of independence of volume, as well as the observation of the coarsening phenomenon (amplification of the zero mode w i t h time). The next logical step would be to ask what kind of signatures should one expect if #-vacua can be formed in heavy ion collisions. We showed that the creation #-vacua in heavy ion collisions could result in the amplification of production of light Goldstone bosons in the (10 — 100) MeV  momentum range. In a l l calculations, we worked in the  instantaneous approximation where the shell separating the 0-state disappears in a time much less than any internal time scale. These low momentum particles would then decay to low momentum photons and dileptons, which could be easily detected. We would like to make the optimistic suggestion that this could possibly account for the unexplained abundance of low momentum dileptons observed at C E R N [3, 4]. Our calculations could be extended and improved i n several ways. In order to i n corporate the effects of expansion, we could introduce a time dependent lattice spacing  42  Chapter 5.  a(t).  Conclusions and Future  Considerations  43  If a(t) increases w i t h time, this w i l l model the dilution of energy i n the system.  Following an analogy w i t h an expanding universe, the coefficient of the damping term 7 goes like constant x a/a.  A l t h o u g h we have not performed the simulation taking this  into account, we can estimate the result as follows. T h e Laplacian term i n Eq.(3.4) is numerically proportional to 1/a, and if a(t) increases w i t h time, the system w i l l require more time to relax to the (9-state. If the coefficient, 7, is altered as mentioned above and a(t) increases at a constant rate, the damping term w i l l become smaller as time goes on. The overall effect of a time dependent lattice spacing is to increase the formation time of the 0-state. A l l calculations relied on the quenched approximation that the system could be evolved according to the zero temperature equations of motion. There exists the possibility that this approximation is not justified. In the case that this proves to be true, we could extend this analysis by introducing non-zero temperature effects.  We could also  ask how the presence of a non-zero chemical potential would affect our results. Looking back at Chapter 4, there exists the other possibility that the instantaneous approximation may be the incorrect physical picture. If this process is actually carried out adiabatically, then one must ask what the question: how does the the transformation of the asymptotic states from the 0 ^ 0-world to our world with 0 = 0 occur? presents itself as a very difficult problem which could have interesting results.  This  Bibliography  [1] I. Halperin and A . Zhitnitsky. Anomalous effective Lagrangian and 8 dependence i n Q C D at finite N . Phys. Rev. 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