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A Gaussian approximation to the effective potential Morgan, David C. 1987

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A G A U S S I A N A P P R O X I M A T I O N T O T H E E F F E C T I V E P O T E N T I A L by David C . Morgan B.Sc, The University of Waterloo, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA September 16, 1987 ©David C. Morgan 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library 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 Phys ics .  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 5th October. 1987. DE-6(3/81) A B S T R A C T This thesis investigates some of the properties of a variational approximation to scalar field theories: a trial wavefunctional which has a gaussian form is used as a ground state ansatz for an interacting scalar field theory - the expectation value of the Hamiltonian in this state is then minimized. This we call the Gaussian Approx-imation; the resulting effective potential we follow others by calling the Gaussian Effective Potential (GEP). An equivalent but more general finite temperature for-malism is then reviewed and used for the calculations of the GEP in this thesis. Two scalar field theories are described: <f>4 theory in four dimensions ((f)^) and <f>6 theory in three dimensions (</>!). After showing what the Gaussian Approximation does in terms of Feynman diagrams, renormalized GEP's are calculated for both theories. Dimensional Regularization is used in the renormalization and this this is especially convenient for the GEP in <j>% theory because it becomes trivially renor-malizable. It is noted that (f>% loses its infrared asymptotic freedom in the Gaussian Approximation. Finally, it is shown how a finite temperature GEP can be calcu-lated by finding low and high temperature expansions of the temperature terms in <j>% theory. Table of Contents Abstract ii Acknowledgements v 1 Introduction 1 2 Definition of the Effective Potential 4 3 Approximating the Effective Potential 3.0 Introduction 11 3.1 The Loop Expansion 12 3.2 Variational Approach - The Gaussian Approximation 15 3.3 Density Matrix/Finite Temperature Formalism 22 3.4 Renormalization of the GEP 25 3.5 Comparison and Discussion 30 4 G E P and <j>% Theory 4.0 Introduction 32 4.1 Power Counting in <f>% Theory 33 4.2 The One-Loop Effective Potential in <j>% Theory 35 4.3 The GEP for <j>% Theory 38 i i i 4.4 Finite Temperature Corrections to the 4>z GEP 5 Conclusion References iv Acknowledgements I would like to thank of course my supervisor, Gordon Semenoff, who suggested the project and made many helpful comments and suggestions along the way. My special thanks to Ian Lawrie who was always there to answer my questions and who often gave me new and insightful ways of looking at things. This work was supported in part by the Natural Sciences and Engineering Research Council. V C H A P T E R 1 Introduct ion Although the loop expansion has long been the standard method of studying the effective potential, there has recently been a resurgence of interest in certain non-perturbative approximation schemes. In particular, P.M. Stevenson1-3 has re-cently published a series of papers on what he calls the Gaussian Effective Potential (GEP). The GEP is the result of a variational calculation: we hypothesize a param-eterized gaussian trial wavefunctional for the ground state of the theory and then minimize the expectation value of the Hamiltonian in this ground state. Stevenson is the first to admit that this idea is not new; there has been work done on the problem several times during the development of modern field theory techniques. For example, Jackiw4 and more recently Barnes and Ghandour5 have studied or reviewed variational methods based on an initial, postulated, gaussian form. Stevenson's work is an analysis of the usefulness of the GEP first in quantum mechanics (a realm in which it appears to perform rather well) and then in scalar field theory - specifically in (f>\ theory (<f>4 theory in four spacetime dimensions) and theory ((j>6 theory in three spacetime dimensions). In studying the field theories he performs detailed numerical calculations of the GEP. This report will not have any such numerical calculations. It is more of an informal look at the GEP: where it comes from, how it does what it does, and how to go about calculating what it does rather than detailed calculations of what it predicts for a particular theory. Like Stevenson, we study the GEP in the context of 4>\ a n ^ <£l theories, but unlike him we adopt a different formalism - one developed by Bill Unruh.6 Unruh's approach is maybe conceptually more complicated than the conventional one, but it does incorporate finite temperature effects in a very natural manner. These effects are of course important when studying the effective potential. Since the effective potential 1 gives us information on symmetry breaking, these finite temperature effects can tell us about the possibility of restoration of a symmetry which is spontaneously broken at zero temperature. We shall begin this report by defining the conventional effective potential, first in the usual manner, and then in a more physically meaningful one. We shall also note its most salient characteristics; namely, reality and convexity. We then consider the problem of approximating the effective potential. Ideally we desire an approximation scheme which is physically motivated, which reflects the fundamental charateristics of the object it is trying to approximate, which can be systematically improved, and which is easy to compute. The loop expansion is not all that well physically motivated nor is it guaranteed to be real or convex. However, being a perturbative approximation, it is conceptually straightforward to systematically improve. It is also reasonably straightforward (but often tedious) to compute. The GEP is always real but not always convex, and since it uses a trial ground state to approximate an interacting theory which is a slight generalization of the ground state of the free theory, it is fairly well motivated. Now, we will see that the GEP (at least for <f>\ and 4>t theories) sums up a certain subset of Feynman diagrams in the loop expansion to all orders of perturbation theory. Even though it naturally omits certain other classes of diagrams, the fact that we are obtaining non-perturbative information about the effective potential is rather a nice feature of the approximation - the GEP is fairly easy to compute in the bargain. However, it is difficult to see how to systematically improve upon it. After this initial look at what the GEP does, we then turn to Unruh's finite temperature formalism and show how it is equivalent in the limit as the temper-ature goes to zero to the variational trial wavefunctional method. We make this comparison by examining (f>\ theory and are subsequently led to the problem of renormalization. Here, we emphasize a conceptual difference between Stevenson and Unruh in the renormalization procedure. Both approaches seem valid although 2 it is not obvious that Unruh's is a priori possible. We choose to follow Unruh but deviate from him by using dimensional regularization as opposed to a momentum cut-off in the renormalization. We continue the analysis of the GEP by calculating it for <f>% theory. This theory has been found to be asymptotically free in the infrared limit for the <j>6 coupling.7 We stress the relationship of the GEP to the loop expansion by using standard methods to calculate the one-loop approximation and then recover this same result by cutting off the GEP's infinite sum of diagrams at one loop. Renor-malization of the <t>% GEP using dimensional regularization is shown to have the nice feature of being trivial; dimensional regularization does not produce poles at one loop in odd dimensions and this is good enough to render the GEP totally pole free. We also remark on a limitation of the GEP for 4>% theory: it does not reflect the infrared asymptotic freedom of the theory. We finish the report by showing how finite temperature effects are incorporated using Unruh's formalism. We give a high and low temperature expansion of these effects and note how they would affect the computational procedure. 3 C H A P T E R 2 Def in i t ion of the Effective Potent ia l To begin, we define the effective potential via the effective action in the usual manner; we then show how it is equivalent to a more physical definition. It is from this latter definition that we get most of our intuitive understanding of the effective potential, and it is from this definition that objects such as the Gaussian Effective Potential are a natural outgrowth. First consider the functional-integral definition of the generating functional for a scalar field theory, where we are using units such that h = c = 1 and where j(x) is the source and S(<f>, j) the action: Since Z(j) =< O;out | 0; in >y, (| 0; in > being the vacuum at / —> —oo and 0; out > being the vacuum at t —> +oo ) (2.1) d4 x^-d^{x)d^(x) - V(<j>{x))+j{x)<j>{x) Z(0) = < 0 | 0 > = 1 (2.2) (| 0 > being the well-defined physical vacuum) and so we get (2.3) 4 Z(j) is the generating functional of the full Green's functions; the generating functional of the connected Green's functions is defined by W{j) = -ilnZ{j) (2.4) Notice that 4>{x) = f D<j>eis(<t><i) ' .SlnZjj) 6j(x) (<f>(x) is clearly the expectation value of the field in the presence of the source j(x)) or 6W(i) = 1M (2'5) Note that this equation can be solved to give j(x) as a functional of 4>(x). Now define T(<^ ), called the effective action, as the Legendre transform of W(j) T($) = W(j) - j d4xj(x)${x). (2.6) Functionally differentiating (2.6) and using (2.5), we get T(^) can be expanded as r(^) = E ^ / d 4 x 1 . . . d 4 x n T ( n H x 1 , . . . , x n ) i ( x 1 ) , . . . , t ( x n ) . (2.8) n T^(xi,... ,xn) are the one-particle-irreducible (1PI) Green's functions (ie. their Feynman diagrams cannot become disconnected by cutting only one internal line; 5 these diagrams are also distinguished in that they have no propagators for the external lines). Since we are only interested in translationally invariant field theories, <£(x) will be independent of x, and so we write it as <^>. We define the effective potential Ttf) = - I d4xV($), or V{$) = - (/ d4*) r(^ ). (2.9) We now write T^(xi,... ,xn) in terms of its Fourier transform explj2pixi)T^(Pl,...,pn) (2.10) (the (2TT)4S^ (]C"=iPt) is included for the sake of overall momentum conservation in keeping with translational invariance). Substituting (2.10) into (2.8) we get r(#) = E^/-'.,...^gr...gi(^'(E«)-exp(^p i i t ) r ( , l ) (p 1 , . . . ,pn), 1=1 = E ^ T / d4x1d4Pl...d4pn6W(P2)...6W(pn)6W(p1 + ---+pn)-n e , p , I i r W ( p , , . . , P n ) , = E ^ T / d4x1d4PlS^(p1)ei^T^(Pl,0,...,0), n = ( / ^ ) E ^ r ( n ) ( ° ' - - - ' ° ^ n - ( 2- n) 6 Comparing (2.9) with (2.11), we find v ^ ) = - £ ^ r ( n ) ( ° ' - - - ' 0 ^ n - (2.12) n (2.12) will be useful for us later on when we try to calculate the effective potential. As it stands, the effective potential as defined seems somewhat of a technical, non-intuitive object. It turns out that we can bring it back down to earth by defining it in terms of a variational principle. We here follow the arguments of Coleman and others.8-9 Consider first this quantum mechanical problem: we wish to find a state | a > such that it minimizes < a \ H \ a > given the constraints < a | a >= 1 and < a | A | a >= Ac for some Hermitian operator A and c-number Ac. We can rewrite the variational problem in terms of the Lagrange multipliers E and j: vary < a | H | a > —E <a\a>—j<a\A\a> < a | (H - E - jA) | a > unconstrained with < a \ a >= 1 and < a j A \ a >= Ac. The variational equation is (H - E-jA)\ a >= 0. (2.13) This has a solution | a(E,j) >. Imposing the normalization condition < a{E,j) | a{EJ) >= 1 gives E = E(j) and using < a(EJ) | A | a{EJ) >= Ac 7 we get j = j(Ac). Writing (2.13) as (H-jA)\a(j)>=E(j)\a(j)>, (2.14) we see that E(j) is the eigenvalue of the Hamiltonian H — j A belonging to the normalized eigenstate | a(j) >. Now in field theory, the only normalizable eigenstate of the Hamiltonian is the ground state or vacuum state. Thus, with an eye ahead to field theory, we identify | a(j) > as the vacuum state of the Hamiltonian H — jA. From (2.14) we write < a{j) | {H - jA - E) | a{j) >= 0. (2.15) Differentiating (2.15) with respect to j we get \){H-jA-E)\eU)> + <a(j) \{H-jA-E) ( £ | « ( J - ) + <o(j) \(-A-^-)\a(j) >=0. The first two terms vanish by (2.14) and we are left with dE(j) dj So (2.15) implies - A c . (2.16) < ofj) | H | a(j) >= E(j) - jd^-. (2.17) Therefore E(j) — jdE(j)/dj is the ground state energy of the Hamiltonian H in the space of states satisfying < a \ A \ a >= Ac. 8 The jump to field theory and the effective potential now comes from comparing (2.17) with (2.6). (2.6) can be represented schematically as r ( * ) ^ M - , « w i t h ^ = * . OJ OJ Now Z(j) = eiw^ = < 0;out | 0;m >^ (2.18) If the source is adiabatically turned on at time 0 and adiabatically turned off at time T and localized in a volume V (not to be confused with the effective potential V(4>)), then (2.18) becomes eiW(j) = e-iVTe(j) where e(j) is an energy density. — W(j) is therefore the ground state action of the theory with source term, and in analogy with (2.17) we see that — T(4>) is the ground state action of the sourceless theory in the space of states satisfied by < a | <f> | a >= <{>. Thus we have (dividing by — VT) V($) = min < a \ )i | a >, (2.19) a (M is the Hamiltonian density) with < a \ a >= 1 and < a | (f> \ a >= <j> (see Brandenburger9 for details of this derivation). Clearly the state | a > which minimizes < a \ M \ a > is the exact ground state of the theory; equivalently, minimizing V(<^ ) with respect to 4> gives the exact ground state energy. Clearly V(4>) is the free energy density of the theory. Before concluding this section, we make two observations about the effective potential. First, it is real; this is clear from (2.19) since the expectation value of 9 the Hamiltonian must be real. Second, it is convex, i.e. d2V , , ( 2 - 2 0 ) A proof of this fact may be found in a paper by Calloway and Maloof.10 10 C H A P T E R 3: Approx ima t ing the Effective Potent ia l 3.0: Introduction The most widely used approximation to the effective potential is the so-called loop expansion. This is a perturbative procedure which is done order by order in the number of loops present in the Feynman diagram representation. It is well known that the loop expansion is equivalent to an expansion in powers of h (ie. for scalar field theory in four spacetime dimensions, we have for 1PI diagrams that P = L — 1 where P is the power of h and L is the number of loops). Note that the N-loop approximation contains the set of all Feynman diagrams of N'th order or less in the coupling constants as a subset; thus, the loop expansion is no worse than an expansion in powers of coupling constants. As Coleman and Weinberg11 point out, since the loop expansion parameter multiplies the whole Lagrangian density (£,) , it is unaffected by shifts in the fields or divisions of £ into free and interacting parts. This preserves our ability with the loop expansion to examine all candidate ground states (including modifications due to higher order effects) before choosing the actual vacuum of the theory. As already mentioned, the loop expansion is not guaranteed to be either real or convex. This is somewhat of a black mark against it since, as we saw in Chapter 2, the true effective potential does have these properties. Of course, we are used to seeing a non-convex effective potential: whenever we have a classical double-well potential giving spontaneous symmetry breaking, the resulting one-loop approxi-mation will be non-convex with at least two separate minima. The fact that the true effective potential does not allow this scenario seems strange. Actually, it has been shown (see, for example Ingermannson12 who discusses the whole situation in 11 detail) that the true effective potential in this case will be a flat-bottomed well, and this is not inconsistent with symmetry breaking. Jackiw 4 has remarked that although perturbative approximation methods such as loop or coupling constant expansions have been very successful in cer-tain applications, there are many interesting phenomena which cannot easily be described with this technique. Spurred on by their success in quantum mechanics, some physicists have sought to generalize variational techniques to field theory as an alternative approximation method to the standard loop expansion. The feeling is that a variational approximation probes the underlying physics more profoundly than the systematic perturbative approximations. One of the drawbacks of a varia-tional or non-perturbative approximation is that it is often difficult to systematically improve; the loop expansion, of course, does not have this problem. In addition the variational approximation to the effective potential in, say, the Rayleigh-Ritz method does not have as its minimum the vacuum state of the theory; as we shall see, it gives an upper limit. In this chapter, we shall discuss the Rayleigh-Ritz method, i.e. the specific realization of the variational method in terms of a gaussian ground state ansatz -the Gaussian Approximation - mostly in the context of cj>\ theory ((j)4 scalar field theory in four dimensions). First, however, we shall make a detour and describe in more detail the loop expansion and some of the ways there are to calculate it. 3.1: The Loop Expansion One of the most conceptually straightforward ways of expressing the effective potential in terms of a loop expansion is to use (2.12): v ^ ) = - E ^ r ( n ) ( ° ' - " ' 0 ^ n -n 12 All we have to do to calculate the one loop effective potential is to sum up the 1P1 functions at one loop order evaluated at zero external momenta. Consider <f>\ theory; that is a scalar field theory with Lagrangian density r(2)(o,o)= + O Q r(4)(o,-,o)= -|_ + >0< r(6)(o,...,o) = etc. The result is an infinite series which we can easily sum to give a logarithm. This is easy, but when we try to compute the two and three-loop contributions etc. , we are faced with a rapidly increasing plethora of diagrams all of whose combinatorial factors must be calculated. An alternative method (suggested by Ian Lawrie) is to consider the theory with a source term j{x), and make a change of variables in the action of the form <j>(x) — 4> + ip. For the n-loop effective potential we then compute up to n-loop order the function j($) such that < V >= 0 when j = j{4>)- Since from (2.7) we have 13 and we can antidifferentiate j(4>) to find V(<j>). In <f>\ theory, the shifted action gives rise to a modified propagator and linear, cubic and quartic interactions. To set < tp >= 0 it is sufficient to set the 1PI part of the one-point function to zero: —*+ -O " ( X represents the linear term in the action). So this gives us our condi-tion on j which can be antidifferentiated to give the effective potential. Note that this method has reduced the task to the calculation of just two diagrams as com-pared to the infinite number of them that we had to sum using the first approach. We shall see an example of this technique in Chapter 4. Jackiw 1 3 has formulated a method which also boils down to calculating just a few diagrams and which can be easily applied to gauge theories. His formula for scalar field theories involves an explicit zero and one-loop term and an expectation value of a specially defined action for the higher loop terms (this action is obtained from shifting the original action in a well-defined manner). Jackiw makes the in-teresting remark that the one-loop term is atypical because it explicitly involves a logarithm. Randall Ingermannson1 2 has recently dealt with the effective potential in terms of the Dyson-Schwinger formalism. He also reduces the problem to the cal-culation of just a few diagrams (the diagrammatics here are based on the Dyson-Schwinger equation). He is also able to obtain the Gaussian Approximation plus other interesting approximations (ones that are guaranteed to be convex) easily from his formalism. We refer the interested reader to his work. 14 3.2: Variational Approach - The Gaussian Approximation Consider the Hamiltonian operator for a scalar field theory in u spatial di-mensions H = jd'x QTT2(X) + l- |V^(x)|2 + U($) where we have explicitly put hats over the variables to emphasize that they are operators. We want to find the state | xp > such that 6 < xp | H | xp >= 0 (3.2.1) (i.e. < xp | H | xp > stationary) given that < xp | ip >= 1 (3.2.2a) < xp | <£(<,x) | xp > = 0(x) (3.2.26) and < xp | 4>(t,y)4>{t,y) \ xp >= ^(x)^(y) + /(x,y). (3.2.2c) We work in the fixed time Schrodinger picture with wavefunctionals, ie. each state | xp > being associated with a wavefunctional xp{<f)} (xp is a functional of (j>) such that | xp{(p} | 2 is the probability density for the field to have the value 4>(x). In this fixed time picture, we say £ ( X , O I 0 > = # X ) W > , ( 3 - 2 - 3 ° ) and TT(X,0I 0 > = \ 4rr^^W (3-2.36) (these relations come from seeing that the commutator <£(x), TT(X')1 \xp>=i 6{x - x>{^} 15 can be realized by the relations (3.2.3) with <^ (x) a c-number function). Inner products of states are now performed using functional integration: (3.2.4) For example the matrix element of the Hamiltonian operator between states | rb > becomes the functional integral < 1> | H | V >= j D<W*{4>} (^J dux 1 6' 2 6<j> 2+-{V4>Y + U{4>) ]) (3.2.5) or in momentum space 6(j>(p) 6<j>(-p) 2 + t-^p^-p) + U(<f>) (3.2.6) The essence of the Rayleigh-Ritz approach to solving the variational problem (3.2.l)/(3.2.2) is to postulate a trial wavefunctional of a specific form satisfying the constraints (3.2.2). If we call < ip \ H \ tb >, E(<f>,f), then (3.2.1) becomes 6E e - 6ECT lj6* + j j 6 f = 0 (where 4> is now independent of x). / and <j> are independent parameters so this becomes = 0, aq> 6_E Sf (*,f) =0. (3.2.7a) (3.2.76) In the ideal case, if we were unable to carry out the process of varying | xb > unconstrained, we would find that | ip > would be equal to | 0 >, the true ground 16 state of the theory. Since for purposes of calculation, we must obviously constrain the trial wavefunctional in some manner, we would like it to be a reasonable ap-proximation to the ground state. It is easy to verify that, in v spatial dimensions, | tp0 >= Nexp{-^ j dux<f>{x){m2 - V2x)V2<j>{x)) is the ground state wavefunctional for a free field theory with the familiar zero-point energy £ duk 1 / — — E0 = V -—-—uk ; wfc = \/k2 + J (27r)"2 ' k m2 (V is the spatial volume). Motivated by this, we follow Jackiw4 and particularly Barnes and Ghandour5 by using the trial ground state wavefunctional, | 0; >= xp{cp} = JVexp(~ j d3xd3y(<j>(x) - $) f{x - y){<p[y) - $)), (3.2.8) for <p\ theory (that is U(<p) = l/2m2<p2 + X(f>4). Substituting (3.2.8) into (3.2.6), it can be shown that E{f,$) _ < 0 f \ H \ 0 f > V .-m2p + \4>4 + 3X4>2 f d3P^A  _o J M 2 " W ( 2 T T ) 3 / ( P ) HPY where w2 = P2 + m2, and f ( x - y ) = j ^ e *<*-v>/(p) , 17 (it is implicit here that p is the three-dimensional vector p). The conditions (3.2.7) become * ( m * + 4 A ^ + 6 A / ^ J - ) = 0 , (3.2.10) / («) 2/(,)27 (2T) 3/(P) Now (3.2.10) gives either 0 = 0, (3.2.12) m 2 3 /" d3p 1 4 A _ 2 7 ( 2 T T ) 3 / ( P ) 2 m 2 3 f d3po r * =-^--"J ? ^ 7 w - ( 3 ' 2 , 1 3 ) (3.2.11) gives We consider the two cases: case(i): ^ = 0 and /[tf^^ + exj^j^. (3.2.15) This is the special case of non-broken symmetry where the minimum of the effective potential is fixed at the origin. case(ii): 4> ^ 0 and given by (3.2.13) at the minimum of E(f,4>); /(g) is given by (3.2.14). By (2.19) VG{$) = ^E{$). (3.2.16) Here we have suppressed the f dependence of E and V because of (3.2.14) which relates / and We have called the functional minimum of E/V =< Of \ H | 0/ > /V with respect to / (ie. (3.2.9) subject to the relation (3.2.14)) VG which we 18 will henceforth refer to as the Gaussian Effective Potential or G E P because of the gaussian form of the original ground state ansatz (3.2.8). We note at this point that the variational calculation will no longer give the true ground state as its minimum. Rather, we have ^ ) = m i n < ^ l ^ l ^ > > ^ ) . Also although VQ is obviously real, it is not necessarily convex and therefore does not do any better in this regard than the loop expansion. Again following Barnes and Ghandour 5 we look at (3.2.14) a little more closely; we write it as f{q)2 = q2 + M2, (3.2.17) where M2 = m2 + 12A<£2 + 6A / .f* * , (3.2.18) J (27T) d f(p) so M is given implicitly by the equation M1 Therefore what our variational calculation has told us is that the best vacuum Gaussian trial wavefunctional is of the form of the vacuum wavefunctional of the free theory but with a mass shifted from the original one and given by (3.2.19). (3.2.19) is therefore the key to understanding what the Gaussian approximation to the effective potential actually is. But before looking at the effective potential, what about the Gaussian ap-proximation for the ordinary 4> = 0 theory? We look back at (3.2.15): f(q)2 = q2 + M2 19 where this time Mo = m 2 + 6A J (27T )3(p2 + M2)1/2 (3.2.20) To make predictions about, say, the propagator in (j>\ theory in the conventional manner, we rewrite the integral in (3.2.20) as a four dimensional one: Ml = m 2 + \2i A / (2TT)V - M 2 + JV m 2 + / (A,M 0 ) . Now the propagator is basically the expectation value < 0/ | <b{x)<j>(y) | 0f > In a free field theory of mass Mo, this is exactly p2 — M2 + it However, we want to see the diagrammatic expansion in terms of m, the original mass in our <$>\ action: p2-M2+ie p2 - m2 - / (A,M 0 ) + ie ( p2 — m2 + ie r / (A,M 0 ) V p2 — rn2 + it J p2 — m2 + i £ + p 2 — m 2 + i e 12 d4Pl { - L X ) J <> ) > 2 - M 2 + t e + 20 The second term in the expansion may be expanded again: / = I £1 J (27T d4pi i (27r)4 p| — m2 + t£ d4pi i )4 p\ - m2 + it What we have so far is 1 + /(A, Mp)  2 - m 2 + i e ) i ml + i e 12 V ' J 2 7 T 4 (2TT)4 p\ - Ml + i e + + p2 — m2 -tie \p2 - m2 + ie 2 — m2 + ie + p2 — m2 + i e m1 + ic i P2 — m + t 6 Q (3.2.21) (3.2.22) We note that this expansion is exactly the same as the one we would get by ex-panding the bare <f>4 theory using the usual Feynman rules in Minkowsi space. It should be clear from (3.2.22) and the steps leading up to it that the expansion will not include diagrams like e i.e. the ones with overlapping divergences. The same omission will apply were we to calculate the expansion of the GEP (where now (3.2.17) - (3.2.19) apply). In expanding /(A, Mo) in (3.2.9) we will only be able to generate products of loops and not overlapping ones, (this should be clear from the way in which we always expand an integral containing f(p) in terms of products of integrals; it is impossible 21 to intermix the loop momenta from two different loop integrals). The failure of the GEP to pick up these types of diagrams can have dramatic qualitative consequences; we will see an example of this when we study (f)® theory which is free in the infrared limit - see Chapter 4. 3.3: Density Matrix/Finite Temperature Formalism We now study an alternative formalism devised by Bill Unruh6 which gen-eralizes the previous method by incorporating finite temperature effects. The free energy density (or finite-temperature effective potential) F of a system is given by the well-known formula F= (E -TS)/V (3.3.1) ( where E is the energy, T is the temperature, S is the entropy and V is the volume of the system). If we describe the state of the system by a density matrix p, (3.3.1) becomes Fp = {tr{pH) + Ttr{plnp))/V, (3.3.2) since E is the expectation value of the Hamiltonian < H >= tr(pH), and —tr(plnp) is the entropy associated with the state p (the trace can be thought of as a functional integral over the field or fields). The equilibrium true free energy density is given as the minimum value of Fp having varied over all density matrices. If we restrict the class of density matrices over which we vary by requiring that the scalar field variable satisfy < 4> >= tr{p<f>) = <£, (3.3.3) we have that F{T,4>)< min Fp. (3.3.4) {p \ tr(p^) = $} 22 Thus our result will be an upper limit for the true equilibrium free energy density. Unruh's approach in studying (j>\ theory is to use a gaussian ansatz for the restricted class of density matrices: >(T,M\j)= «*i-m*M ( 3 . 3 . 5 ) (/? = 1/T since we are using units where Boltzmann's constant has been set to one). R(M2,<f>) is a free Hamiltonian of mass M R(M2,4>) = \J d3x (n2(x) + (V0(x)) 2 + M2(<b(x) - 4>)2) . (3.3.6) Note that < <b >= tr(p4>) = I DfaWx) exp (-0/2 f d*x(n2(x) + (V0(x)) 2 + M2(<j>(x) - ^)2)) tr(exp(-PR)) 1 ' ' ' Changing variables in the functional integral <j> —»• <f> — <j>, we see that <(b>=4>. (3.3.8) The free energy density or what we now identify as the generalized two-parameter G E P (note the distinction from the previous section where we used the label G E P to refer to the one-parameter quantity Vc(4>) which was what was obtained after the minimization process was performed) is therefore given by VG(T,M2J) = [tr(p(T,M2,4>)H) + Ttr(p(T,M2J)ln(p(T,M2,$)))] /V (3.3.9) For details of the set-up of this method, the interested reader should consult Unruh's paper. 23 Now if we define f(T,M2) by f{T,M2) = {-Tlntre-PR)/V, (3.3.10) (the free energy associated with the free Hamiltonian R where p is given by (3.3.5)) and consider H = j d 3 x (V(z) + i ( V 0 ( x ) ) 2 + \m2<t>2{x) + \<b4(xfj , we get VG(T,M2,4>)= f+±m24>2 + \4>4 .2 , , o w 2 j i / f 2 \ df + {rnz + 12\<fiz - M )• dM2 ( df x 2 + 1 2 A U H > (3-3-n) {tr(P(b4) = 3{tr(p<b2))2 has been used here). f(T,M2) can be calculated straight-forwardly to give (the second term is the finite temperature part). Atr = 0 (0 -» oo), f{T,M2) ->./(M 2) where / ( M 2 ) = 5 / T ^ ^ 7 ^ ^ ' ( 3 - 3 - 1 3 ) 4 7 (2TT)3 Vfc2 + M2 1 ' d M 2  V / P T 2 ' 24 If we compare (3.3.11) at T = 0 with (3.2.9), and use (3.3.13), (3.3.14) and (3.2.17), we can see that the two expressions are identical. This suggest that (3.3.11) at T 7^  0 is a finite temperature generalization of (3.2.9). Note that we started with a general form f(p) in (3.2.9) and deduced variationally that it must have the form f{p) = y/k2 + M 2 , with M 2 given by (3.2.19). In (3.3.11) we've in fact already assumed this and reduced the problem to that of variationally deducing the form of M 2 . Now the form predicted for M2 in this density matrix scheme should of course agree with (3.2.19). We check: dV/dM2 = 0 gives us (m2 + m * 2 - M 2 + 2 4 A S a 0 r = ° -This implies either ^ = ^ + 1 2 A ^ + 6 A / ^ ( f c a + ^ 2 ) 1 / a , d2f = = n = 1 f d*k 1 °r d{M2)2 8J (2TT)3 (Jfc2 + M 2 ) 3 / 2 ' Now M2 > 0 so the integrand in (3.3.16) is always positive. We therefore reject (3.3.16) and adopt (3.3.15). (3.3.15) is identical to (3.2.19) and so M 2 has the same form at T = 0 in both methods. We have thus established that (3.3.11) is indeed the finite temperature GEP in that it agrees with our zero temperature trial wavefunctional approach as T —• 0. 3.4: Renormalization of the G E P There are two basic approaches to renormalizing the GEP. The first method is used by Stevenson2'3 and entails using the standard perturbation theory renor-(3.3.15) (3.3.16) 25 malization definitions m R and d2V{$) d<p2 1 d4V{4>) V.~df4 4>=o (3.4.1a) (3.4.16) <t>=o The important thing to notice about this approach is that we use the condition dVG{M2,4>) dM2 = 0 (3.4.2) in the renormalization process. Thus M2 depends implicitly on <j> via (3.4.2). This is the reason we have written V(<j>) and not V(M2,4>) in (3.4.1). Thus taking a derivative with respect to 4> in (3.4.1) could be more clearly written as dj> 4>=o dVG{M2J) d4> dVG{M2J) dM2{<j>) aM 2 !>=0 d4> (3.4.3) 4>=o The second method of renormalization is used by Unruh in his paper and involves calculating a more general object than the standard renormalized effective potential VR($). With this approach we demand that our renormalized parameters m2R and XR be independent of both ^ and M2 (in contrast, Stevenson's renormalized parameters are M2 dependent). Thus we calculate a renormalized two-parameter GEP VGR. If we then wish to locate the vacuum of the theory, we are, free to minimize this function with respect to M2 and ^. A priori, it is not obvious that such a two-parameter effective potential should be renormalizable. Nevertheless, Unruh has shown that for <j>\ theory it is indeed possible. The ability to study a renormalized GEP in which both <f> and M2 are independent variables is a nice feature. It allows us greater freedom to probe the structure of our approximation to the theory. 26 Unruh uses a momentum cut-off to regularize the theory. In this section, we shall renormalize the theory using his approach but with dimensional regularization instead of a momentum cut-off. This form of regularization seems to make the algebra a little simpler. We consider for the moment the zero-temperature case and write VG(T, M2, <p) as VG(M2,$) in (3.3.11). (3.3.12) becomes !{M2) = \j ^V^TW2. . (3.4.4) We have two divergent integrals in (3.3.11): / and df /dM2. We will be using the standard formula: ddk 1 /jdJ 0 )d (k2 + M2y oo l.d-1 dk-(k2 + M 2 ) a ^ , M - r ™ ° r " l 2 \ (3.4.5) 2 T(a) where Sd = l/(27r)d-(surface area of unit sphere in d dimensions) and T(x) is the usual Gamma function. We handle f(M2) first. Following the prescription of dimensional regulariza-tion, we analytically continue the integral (3.4.4) to d = 3 — e (e > 0) dimensions, use (3.4.5) and then take the limit e —> 0 to recover the result for d — 3 dimensions. = 11 ( g ^ + M * ) - - / ' = ^ ( M - 1 / 2 ) = > - r ( 3 / 2 - ; f _ ) i r / ( 2 , 2 + e / 2 ) (3.4.e) 27 Using the basic Gamma function formulae T{z+l) = zT{z), (3.4.7) and T{2z) = {2n)-^222z-^2T{z)T{z + l/2), (3.4.8) we can write (3.4.6) as We can get df /dM2 from this by differentiating: df sdM< dM2 8 To deal with (2/M) £ in these expressions in the limit as e —> 0, we are forced to introduce an arbitrary scale Mo with dimensions of mass: 2 V ( 2 M ° M = l M ^ M ° £ — £ = e x p ( Z n ( ^ ) ) M o f » - ( l + rf-(^)) < 3 - 4 1 1 ) (keeping only those terms linear in e). (3.4.9) becomes f(M2) = ^ M 4 M 0 - £ - — M 4 ^ — - ^ M 4 M 0 - £ / u f ^ ) . (3.4.12) ^ ; 32 0 16 e 16 0 V M / (3.4.10) becomes = ^ M 2 M 0 - - SJLM2M^ _ — M 2 M 0 _ £ / n , f . (3.4.13) d M 2 32 ° 8 £ 8 0 \ M J 28 We now substitute (3.4.12) and (3.4.13) into (3.3.11) and arrange the resulting expression as sums of terms involving different functions of M and 4> (since we want our renormalization definitions independent of these) VG(M2,<£) = 16 0 16 e 16 e2 32 e 256 0 / (&d 2 U - A » ^ 2 , ^ 2 M 0 M 3«d, W _ A 7 2 , , 2 , / 2 M o + ^m 2<£ 2 + A<£4. (3.4.14) Upon inspection of (3.4.14), we see that the renormalization definitions A = €2A* - — | — ( 3 . 4 . 1 5 a ) 3sdM0 m2 = em2R, (3.4.156) or A* = ^ + * , (3.4.16a) m | = — , (3.4.166) 29 make it completely finite. The resulting expression for VGR{M2,<f>) in terms of XR and mjj as e —• 0 is 3.5: Comparison and Discussion The most obvious feature of the implicit definitions (3.4.15) is that both the bare mass and the bare coupling constant go to zero as e —* 0 (equivalently - as the cutoff or regularization device is removed). That the bare coupling constant is zero is consistent with both Stevenson and Unruh. In fact, 3s d M 0 - £ ' so A —> 0 - i.e. A goes to zero from the negative side. This is precisely the scenario in which Stevenson defines his "precarious" (f>4 theory. The fact that the bare mass goes to zero (or is infinitesimal) seems to be a peculiar result particular to dimensional regularization. In an appendix to his <b\ paper,2 Stevenson shows that the bare mass goes to zero in his scheme also. The results have apparently been confirmed by Bollini and Giambiagi.14 Thus, we have shown that the renormalizations necessary for the more general object VGR(M2,q\) are qualitatively similar to those used by Stevenson in his calculation of the conventional renormalized GEP VGR{4>). If we look at (3.4.17) at its minimum with respect to M2 (i.e. impose the condition BVGRIdM2 = 0) we recover the conventional effective potential VGR{4>). We look (3.4.17) (3.5.1) 30 for its minima with respect to <f> to find the approximate vacuum state: dVGR{M\4>) _ dVGR dVGRdM2 _ q dcp d<j> dM2 d<p ' [ But we require dVGR/dM2 = 0 so (3.5.2) becomes d<f> d(j> This implies that there is only one extremum and that it is at the origin. Now since VGR is bounded below, this must be a minimum. This result agrees with both Unruh and Stevenson. In fact, by suitable finite (and M2 and 4> independent) redefinitions of A# and m#, (3.4.17) can be put into exactly the same form as Unruh's. 31 C H A P T E R 4 G E P and <j>\ Theory 4.0: Introduction In this chapter we shall discuss a scalar field theory in 2 + 1 dimensions with interaction U{(j>) = \m2<p2 + \<f>4 + (<j>6. (4.0.1) Hopefully, we shall be able to learn more about how the Gaussian approximation works by studying it in this new context. We shall begin by using the usual power counting type arguments to look at the divergence structure of the theory; we will discover what sort of diagrams are divergent. We shall then calculate the usual one-loop effective potential for comparison purposes. We then embark on the G E P calculation and show that it contains the one-loop result. As we did with (p4 theory we shall discover what it is that the Gaussian approximation actually does. In tackling the renormalization problem we shall run into a rather nice feature of using the Gaussian approximation in 3 dimensions: poles do not occur. This renders the whole renormalization process trivial. To conclude we show how to calculate the finite temperature effects by expanding the finite temperature integral in both the high and low temperarature limits. 32 4.2: Power Counting in <b% Theory Let D = superficial degree of divergence, E — number of external lines, J = number of internal lines, V4 = number of four-point vertices, VQ = number of six-point vertices, n = 3 = number of spacetime dimensions, L = number of loops. We use the usual arguments: 4V4 + 6V6 = 21 + E. (4.1.1) Each four-point vertex generates four lines and each six-point vertex generates six lines; each internal line ends on two different vertices and therefore counts twice. L = J-{V4 + V6-l) = I-V4-V6 + l. (4.1.2) The number of loops is given by the number of internal lines minus the total number of four and six-point vertices because for each vertex we have a momentum conserv-ing delta function which reduces the number of integrations by one. In addition, we must add on one because one of these delta functions conserves overall momentum and does not reduce the number of independent internal momenta. D = nL- 21. (4.1.3) The superficial degree of divergence is given by the sum of all powers of momenta in the measure of each loop integration (in three dimensions - d3p) minus the sum 33 of powers of momenta in the propagator associated with each internal line (notice that external propagators are not included, i.e. we are dealing with 1PI diagrams here). Putting (4.1.1), (4.1.2) and (4.1.3) together, we find that £> = 3 - | - V 4 . (4.1.4) Clearly, we will only find divergent diagrams when E < 6 (otherwise D < 0); that is, divergences are only to be found in the two, four and six-point 1PI Greeen's functions. The one and two loop contributions to the two-point function are Q Q O "'" (D=l) (D=2) (D=0) These are all divergent. In the four-point function, the important divergent contri-butions to two loops are (D=l) (D=0) (D=0) 34 For the six-point function, the divergent contribution to two loops is (4.1.7) (D = 0) The particular importance of this (D = 0) divergent contribution to the six-point function will become apparent when we study the Gaussian approximation. 4.2: The One-Loop Effective Potential in <f>% Theory We shall use dimensional regularization to get the one-loop result; to one-loop no poles are produced and the renormalization process will be trivial. The Lagrangian density with source in Euclidean space is We have explicitly included the source term here because we shall calculate the effective potential using Lawrie's method as described in section 3.1. We make the change of variables (4.2.1) (f>(x) = $ + xb{x) (4.2.2) in (4.2.1) and, discarding the terms independent of rb(x), we find L = \ M ) 2 + \{m2 + 12A<p + 3 0 ^ 4 ) V 2 + {m24> + 4\$3-r6Z4>5-j)ib + (4A<£ + 20£<£3)V>3 + (A + 15£<£2)V4 35 (4.2.3) This gives the following set of Euclidean space Feynman rules: _ 1 ~* k2 + m 2 + 12A<£2 + 3 0 ^ 4 ' = - ( m 2 ^ + 4A^3 + 6 ^ 5 -j), = -4(A^ + 5 ^ 3 ) , < = - ( A + i 5 ^ 2 ) , = (4.2.4) We require < xp > = < <f> > -<f> = 0. (4.2.5) For the one-point function (or tadpole) to vanish, it is sufficient that its 1PI part vanish, i.e. 1PI tadpole = 0 < xp >= 0 So to one loop, we get the expression -o * + ( ) = 0 (4.2.6) or - {m24> + 4A<£3 + 6£<£5 - j) d3k /*k 1 36 where M 2 = m 2 + 12A<£2 + 30£tf>4. The 3 in front of the second term corresponds to the three ways of combining the three legs of the vertex to form the tadpole. Solving for j in (4.2.7), we have /d3k 1 { 2 n ) 3 k 2 + M 2 - I 4" 2' 8) We now calculate Id{M,a) = sd dk-Jo (k2 + M 2 ) a ' where d = 3 — t and a = 1. (3.4.5) gives us T /1 * \ 1 £ r ( 3 / 2 - e / 2 ) r ( - l / 2 + e/2) J d ( M , l ) = -SdM1-*-^ r(I) • (4-2-9) Using (3.4.7) and (3.4.8), we find 7 d ( M , l ) = - ^ (4.2.10) 47T (notice that we don't have any poles). (4.2.8) becomes j = m2 + 4A<£3 + 6£<p5 - — (A<£ + 5 ^ 3 ) , (4.2.11) 7T or upon writing in the expression for M j = m2(p + 4\<p3 + 6£(£5 - - ( m 2 + 12A<£2 + 30^ 4 ) 1 / 2 (A<^ + 5 ^ 3 ) . (4.2.12) 7T 37 By (3.1.1), V(<p), the one-loop effective potential, is given by the antiderivative of j with respect to 4>: V{4>) = \m2^2 + \<t>4 + f<£6 - — (m2 + 12A<£2 + 30^ 4 ) 3 / 2 . (4.2.13) 2 127T 4.3: The G E P for $j Theory We shall calculate the GEP for </>| theory in three spacetime dimensions using Unruh's gaussian density matrix approach. The starting point is equations (3.3.9) and (3.3.5): VG(T,M2J) = (tr(pH) + Ttr(plnp)) /V, (4.3.1) (the V on the right hand side of the equation is the volume - it is not to be confused with VG(T,M2,4>) which is the effective potential). We can rewrite the second term in (4.3.1): -PR / .-PR Ttr(plnp) = -tr [ ^ p ^ l n ) ) = ~lntr[e-pR)-tr{pR) = fV - tr{pR) (4.3.3) (where we have used (3.3.10) in the last line). Substituting into (4.3.1), we have VG = f + ytr{p{H-R)). (4.3.4) 38 Using the 4>% Hamiltonian for H and (3.3.6) in three dimensions for R, we have 1 VG = f + tr Vtr(e-PR) -PR m2^2+ \<f>4 + - -M2(<b - 4>f . (4.3.5) We now make the change of variables, xp(x) = <b(x) — 4> (4.3.6) and (4.3.5) becomes VG = f+-m2<{tP + $)2> 1 + A < (0 + <£)>+£< (0 + (f>)b > --M < tb2 >, (4.3.7) where we have now replaced the trace operation by angular brackets < . . . > de-noting "average" or "expectation value of". We now expand (4.3.7) and use < tb > = < xb3 >=< ib5 >= 0, (4.3.8) and < tb4 > = 3< xb2 > 2 , < ib6 >= i5< tb2 >3, (4.3.9a) (4.3.96) ((4.3.9) follows from Wick's Theorem). If we also notice that df 3M2 vpt 1 a D tr (-?- [ d2xib2e-?R r{e-PR) \ 2 J W = - < ^ 2 > (4.3.10) 39 (we have used translational invariance here by assuming < xb2 > \s independent of x - we were thus able to cancel the volume factor), we can write (4.3.5) in the compact form VG(T,M2,$) = f + \rn2$2 + A<£4 + f<£6 + (m2 + 12A<£2 + 3O£04 - M 2 ) ^ + ( m + ! » £ * • ) i j ^ j + . 2 0 ? ( ^ ) 3 . (4.3.1!) We consider now 8VG/dM2: dVG = (rn2 + 12A<£2 + 30£<£4 - M 2 dM2 + (24A + 360£<£2) 2> df dM2 df_\ \ d2f dM2 + 3 6 0 « I ^ U a ( A # - ( 4 - 3 1 2 » We repeat our analysis of what the Gaussian approximation does by considering the T = 0 case and setting dVG/dM2 = 0. Now for T = 0 / ( M 2 ) = ^ / ( 2 ^ ( f c 2 + M 2 ) 1 / 2 ' ( 4 - 3 ' 1 3 ) df(M2) 1 f d2k and 2  1 j (2TT)2 (k2 + M 2 ) 1 / 2 " ( 4 " 3 ' 1 4 ) dM2 AJ ( 2 T T ) 2 (k2 + M 2 ) 1 / 2 ' dVG/dM2 = 0 implies M 2 = m 2 + 12A^2 + 30^ 4 + (24A + 3 6 0 ^ 2 ) ^ + 360C ( ^ ) ' (4.3.15) 40 (we reject d2f/d(M2)2 = 0 as before). We can see from this that the same sort of analysis applies as it did for <j>\ theory: the propagator of the free theory of mass M will be in Euclidean space 1 p 2 + M 2 ' if we expand loopwise with the expression for M2 given by (4.3.15) and (4.3.14), we see that we will only be able to generate products of single loops (see the steps leading up to (3.2.21) and (3.2.22)). Again all diagrams with overlapping diver-gences (two or more loops with intermixed integration variables) will not appear in the expansion. Clearly, the same argument holds true when we expand VG or, say, when we expand the six-point 1PI Green's function. In the latter case, most noticeably missing will be the logarithmically divergent diagram that we have seen already in (4.1.7). This omission means that when we come to renormalize the six-point function, we will not have to contend with this diagram. In the normal course of events in perturbation theory, we would be forced to in-troduce a non-trivial six-point coupling renormalization constant Z^. We would then use the usual renormalization group arguments and calculate a /? function, /?(£), which is dependent explicitly solely on Z^ and the coupling itself f. We have already mentioned that it has been shown that ^3 theory is asymptotically free in the infrared limit. This means that /?(£) is positive. However, we clearly don't need a renormalization constant Z^ in the Gaussian approximation - there are no divergences to remove - and we get a trivial /? function. Therefore, in the Gaussian approximation to theory, we lose the property of infrared freedom. This is a drawback to using this approximation for <f>% theory. 41 Now let us try to renormalize the generalized two-parameter GEP again using dimensional regularization (that is we renormalize independently of the condition dVG/dM2 = 0). The divergences in (4.3.11) are due to the integrals / and df /dM2. We use dimensional regularization to calculate / and then differentiate to get df /dM2. f is given by ^ = i / S<*2+M2>,/2 = \ / j | | r ( * 2 + M 2 ) ' / 2 + / P \ M 2 ) (4.3.16) (we will conssider the non-divergent finite temperature contribution / in detail in section 4.4). We consider the integral f°° 1^,-1/2)= uJo <ft(t,+Jtf,)-,/, with d = 2 - e. We get 7 < ( M . - l / » ) = l ^ - ' r < 1 - ' ^ - ; f - ^ . (4.3.1S) We evaluate (4.3.18) using the standard formulae (3.4.7) and (3.4.8) and expand it in powers of e to get M 3 J d (M,-1/2) = - — (4.3.19) as e —• 0, and so in this limit M3 / = - £ - + / , (4-3-20) 127T df -M df . a n d d M 2 ^ ^ + d M 2 - ( 4 - 3 - 2 l ) 42 Since dimensional regularization produces no poles, substitution of (4.3.20) and (4.3.21) into (4.3.11) gives us the finite temperature, renormalized, two-parameter GEP VGR{T,M2,<}>). We calculate the expression for the usual one-parameter ef-fective potential VG{4>) by setting T = 0 and substituting (4.3.15) into (4.3.11) and using (4.3.20) and (4.3.21). VG{4>)= im 2^ 2 + A^ + ^ 6 + @ - i i ) M 3 - ^ A + 1 5 ^ 2 - < 4- 3- 2 2' The resulting expression still contains M 2 ; to eliminate it explicitly we must solve the equation 4 5 i) M> 8TT 2 /3A 45 -2\ + — + — i<t> M \ 7T IX / -(m2 + 12A<£2 + 30^4) = 0. (4.3.23) For detailed numerical calculations of the GEP, VG{4>), the interested reader should consult Stevenson.3 We should mention here that, in the numerical calcu-lation, we must check the endpoints of M 2 (0 and oo) before using the mimimum condition (4.3.15). If this became necessary, then, of course, neither (4.3.22) nor (4.3.23) would apply. Rather, we would use M 2 = 0 (rejecting M 2 —• oo) and get VG to be the tree potential. We now check that the one-loop effective potential calculated in section 4.2 agrees with the one-loop reuslt obtained from the GEP; it should, after all, because we have seen that the Gaussian approximation sums up all the "easy" diagrams (those without overlapping divergences) to all orders of perturbation theory - the one-loop effective potential is one such easy diagram and should be recoverable from the GEP by terminating it at one loop. 43 To calculate the one-loop approximation, we must therefore cut-off (4.3.11) and (4.3.15) at one loop. (4.3.11) becomes VG = \m2^>2 + \<f>4 + & 6 + f + (m2 + 12A^2 + 30^ 4 - ^ 2 ) ^ , (4-3-2 4) and (4.3.15) becomes M 2 = m 2 + 12A<£2 + 30£<£4. (4.3.25) We have three remarks here: - we have set T = 0 because we want the zero temperature effective potential. - we have used the condition dVG/dM2 — 0 because we want to calculate the conventional one-loop effective potential and not its generalized two-parameter version VG{M2 ,<f>). -(4.3.25) is the zero loop contribution because its higher loop contributions generate greater than one loop terms when substituted in (4.3.24). Upon substituting (4.3.25) into (4.3.24), we obtain V <t , )(#)=im J* 2 + A^ + f#6 ^ / ( 0 ( i : 2 + M 2 ) , / 2 > M - 2 6 ) where M2 is given by (4.3.25) and VQ^ denotes the one-loop approximation to the GEP. The result of calculating the integral in (4.3.26) by dimensional regularization is given by (4.3.19). Substituting this result into (4.3.26) gives = \ m H 2 + \<t>4 + &G — (m2 + 12A<£2 + 30^ 4 ) 3 / 2 . (4.3.27) 127T 44 Comparing (4.3.27) with (4.2.13), we see that they are exactly the same. 4.4: Finite Temperature Corrections to the <f>% G E P In this section, we consider in detail the finite temperature term f(T,M2) in (4.3.16), If we make the change of variables x = (3k and let a = /?M, (4.4.1) becomes 1(0,M2) - ^ dxxln(e-^^-) . (4.4.2) We consider the integral 1(a) = j dxxln (l - e-\A 3+ a 3) . (4.4.3) We would first like to calculate 1(0): f oo 1(0) = I dxxln (l - e~x) . (4.4.4) Jo We notice first that dx for I x |< 1 which implies oo X n=0 ° ° n+1 ln(l -x) = -Y* . v ' ^ n + 1 45 Now since | e 1 |< 1 is in the range 0 < x < oo, we write ~ c - ( » + l ) x ln(l - e = ^j—j— for 0 < x < co. n=0 (4.4.4) therefore becomes 7(0) ' J o n=0 °° 1 fC 71+1 rfxxe-(n+1)l. Making the change of variables t — (n + l)x in (4.4.5), we get 7(0) °° 1 7C - t n=0 oo = "f (3) where c(x) is the Riemann Zeta function. Now dl_ i 2 /0 2 V dx x 1 - e V*2+a2 \/x 2 + a 2 i _ e - \ A 2 + a V* 2 + °2 - i / n ( l - e - a ) 2 v ' 46 Now since o = /?M, we can get a high temperature expansion for (4.4.7) by expand-ing for small a: di da2 --In (1 a? a3 a H 2! 3! 1, 1, / a a -In a In 1 + - - + 2 2 V V 2 6 (4.4.8) Expanding in small a again (this time for the logarithm) we get dl 1, 1 ( a a? 1 / a a 2 \ 2 da2 2 2 \ 2 6 2 V 2 6 i / r i a + i - i 8 ~ + 0 ( a 3 ) - ( 4 - 4 - 9 ) Antidifferentiating (4.4.9) with respect to a 2 , we get a 2 , 2 a 2 a 3 a 4 5 /(a) = Zn a1 + h h 0(a5) + const. (4.4.10) v ' 4 4 6 96 v ; v 7 For small a, the constant can be fixed using the 1(0) condition, (4.4.6). We get 1(a) = -c(3) ~-lna2 + - + - - - + 0(a5). (4.4.11) v ' s v 7 4 4 6 96 v ' v ; Putting (4.4.11) back into (4.4.2) and writing a explicitly as /3M, we finally get (4.4.12) as the high temperature expansion of (4.4.2) and it is from (4.4.12) that all temperature effects arise. 47 We can also calculate a low temperature (large a = /?M) expansion by ex-panding (4.4.7) for large a. Letting x = e~a in (4.4.7), we see that we want to expand (4.4.7) for small x: _ = - - M l - , ) a B - ^ + T + T + ...j (4.4.13) or Antidifferentiating (4.4.14) with respect to a 2 we get 1(a) = - e _ a ( l + a) - - e _ 2 a ( l + 2a) - ^-e~ 3 a(l + 3a) + • • • + const. (4.4.15) 8 27 For a large, we can neglect the terms independent of a in the brackets in (4.4.15), leaving 1(a) = -a (e~a + \~2a + ~e~3a + • • + const. (4.4.16) To obtain the constant, we consider the limit lim 1(a) = lim I dxxln (l - e-^x2+a2) a—»oo a—»oo JQ \ / and it is clear that it goes to zero. Now we have that lim 1(a) =0=> const. = 0. (4.4.17) a—too Using (4.4.17) and substituting a = 0M in (4.4.16), we get from (4.4.2) the full low temperature expansion: !) = -Ji- (e-W + -e'2^M + -e~^M + •••). (4.4.18) ' 2TT/?2 V 4 9 / K ' 48 To incorporate these effects into the calculation of the GEP, we must sub-stitute / back into (4.3.20) and (4.3.21). Consider the high temperature case for example. We terminate (4.4.12) after the first two terms. (4.3.20) and (4.3.21) become M 3 f(3) M2 ' = - 1 2 7 - 2 ^ + 8 ^ <4-4-19) 9 1 M + "V (4-4.20) dM2 8TT 8TT/3 Substitution of (4.4.19) and (4.4.20) into (4.3.22) and (4.3.23) gives us VG($) VG(4>) = ±m24>2 + A<£4 + £06 M 3 M2 + 127T 87T/3 - (12A + 180£<£2) M 1 x 2 8TT 8n/3 M 1 x 3 240£ + — - , (4.4.21) ^ 1 8TT 8TT/3/ ' ^ ' with M 2 given by 4 5 f \ w 2 i ; M 8TT /3A 45f J 2 45£ . - (m 2 + 12A02 + 3O£04 + - 3-(A + 15f02)+ 4 5 ^ 7f/? v ' ' ' 8 T T 2 / ? 2 , = 0. (4.4.22) We see that in this case the numerical calculation would still be quite easy since we still have a quadratic for M (again, it may be necessary to consider the M = 0 endpoint in which case (4.4.21) and (4.4.22) would not apply). Including more 49 terms in the temperature expansion or trying to compute the effects in the low temperature regime would entail more complicated numerical calculations (witness the exponentials in (4.4.18)). 50 C H A P T E R 5 Conclus ion In this report, we have examined one of the alternatives to studying scalar field theories in terms of a conventional perturbative expansion. We saw that the effective potential could be written as the expectation value of the Hamiltonian in the ground state of the theory. This prompted us to translate the calculation of the effective potential into a variational problem in which the expectation value of the Hamiltonian is minimized with respect to the parameters in a gaussian ground state ansatz (gaussian because of the gaussian form of the ground state wavefunctional in the free theory). The postulation of a gaussian ground state wavefunctional for an interacting theory we called the Gaussian Approximation; the resulting effective potential we called the GEP. We saw that since the true effective potential would be the minimum over all trial ground states (unconstrained), the GEP formed an upper limit for it. We first performed the gaussian variational calculation for <$>\ theory and found the gaussian form of the trial ground state to be that of another free field theory but with a new mass M. M could be related to the m and A in the original <p\ Lagrangian by a simple expression. It was from the nature of this expression that we were able to deduce what the Gaussian Approximation does in terms of Feynamn diagrams. We found that the Gaussian Approximation sums all diagrams without overlapping divergences to all orders of perturbation theory. So we realized that this approximation was indeed giving us non-perturbative information about the field theory. 51 We then introduced a new formalism phrased in terms of gaussian density matrices. We showed that this was the same Gaussian Approximation that we had just seen but, naturally incorporated within it, were finite temperature effects. We considered the problem of renormalizing the GEP and found that with <j>\ and <£f theories it was possible to renormalize the two-parameter quantity VG(M, 4>)\ that is, we were able to renormalize independently of the minimum condition on M. The ability to survey a renormalized VQ(M, $) as a function of two variables seems desirable. We used dimensional regularization in the renormalization procedure as opposed to a momentum cut-off. This made the algebra a little simpler especially in the case of <p% theory where the renormalization was trivial. This was because the Gaussian Approximation which omits all diagrams with overlapping divergences is the sum of diagrams which are basically products of single loops. Dimensional regularization does not produce poles at one loop order in odd dimensions and so the whole GEP was found to be trivially renormalizable. One of the less desirable features of the Gaussian Approximation in 0| theory was that it destroyed infrared asymptotic freedom. This we linked again to the omission, in this approximation, of diagrams with overlapping divergences. Since we did not need a six-point coupling renormalization constant to renormalize the theory (no renormalization constants at all were needed as already explained) we didn't find a positive /? function for the six point coupling. Hence, no infrared freedom. Lastly, we were easily able to show how finite temperature effects could be included by calculating low and high temperature expansions whose terms could easily be included in a numerical calculation if desired. It would be possible to improve upon the Gaussian Approximation in field theory in the same way variational calculations are improved upon in quantum mechanics; that is, choose a more complicated trial ground state wavefunctional with maybe a greater number of parameters. We did consider this but found very 52 quickly that we became bogged down in algebraic complexity. This does not seem a very systematic way of improving upon the Gaussian Approximation (we do not know what we will get until we finish the calculation), and, given the complexity encountered, we are doubtful as to whether this is a very fruitful avenue to pursue. 53 References 1 P.M.Stevenson, Phys. Rev. D30 (1984), 1712. 2 P.M.Stevenson, Phys. Rev. D32 (1985), 1389. 3 P.M.Stevenson, Phys. Rev. D33 (1986), 2305. 4 R.Jackiw in "Theories and Experiments in High-Energy Physics" (Orbis Scientiae II), edited by B.Korsunoglu, A.Perlmutter and S.M. Widmayer, (Plenum, New York), 1975, p.371. 5 T.Barnes and G.I.Ghandour, Phys. Rev. D(22) (1980), 924. 6 W.G.Unruh, private communication. 7 R.D.Pisarski, Phys. Rev. Lett. 48 (1982), 574. 8 S.Coleman in "Laws of Hadronic Matter" (Subnuclear Series volume 11, Part A), edited by A.Zichichi, (Academic Press, New York and London), 1975, p.138. 9 R.H.Brandenburger, Rev. Mod. Phys. 57, (1985), 1. 10 D.J.E.Calloway and D.J.Maloof, Phys. Rev. D27 (1983), 406. 11 S.Coleman and E.Weinberg, Phys. Rev. D7 (1973), 1888. 12 R. Ingermannson, Ph.D Thesis LBL-21916. 13 R. Jackiw, Phys. Rev. D9 (1974), 1686. 14 C.G.Bollini and J.J.Giambiagi, Report No. CBPF-NF-046/84, 1984 (unpub-lished). 54 

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