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1 IMPORTANCE SAMPLING QFT ON A LATTICE

IflA--v-t(J CA tlN-~ Now that we have a firm grasp of QFT we can pursue CQRlFIlitatiofial eJqIerimeRshy

we first have to reformulate the path integral to make it tract~ Monte Car 0 ec es In doing so we perform a set of manipulations JIld

upon s on the lattice we then undo them The process can be imagined by first conceiving of the field 4gt in the continuum We are simulating a two-dimensional model so your imagination can easily grasp 4gt a quantum sea with out ffrcessMit metaplrors of waves and vibrations We then place this on a discretized lattice and aJiffiit to fluctuate for configuration with high amplitud~

We then measure certain quantitIes for each fluctuation However we are in a t sort-of fictionalized world where the field is limited bYfinite sized lattices and t-)

separated by a lattice spacing To return to you(theoreticaYreality we have to eJ IJff -1 W ~ extrapolate to infinite sPace-ti~land then to zero lattice spacing We then return a f J(r II-ltk ~1Af skt ) to the continuum limit and presjlt physical results The following sections present ~I-

this procedure V v(ll~ -~hI

11 Landau-Ginzburg Model Remember in our formulation of the path inteshy ~~M- gral a for quantum field we briefly referred to Wick rotation This is a rotation from Minkowski time to Euclidean time and amounts to t -+ it In the path integral doing so provided a semi-rigorous approach to ensuring the integral converged I repeat the result here

(11)

AfteF leokillog at this for iii while BfJlealli1 the thQught tbai it is similar to the parshytition function seen in Chapter 2 The partition function is a sum of Boltzmann factors (which have exponential form) Here we have an integral over an exposhynential of energy To make this connection more explicit we can rewrite things in terms of the action

(12)

Immediately striking about this is the appearance of 3 For the Ising model it obviously had to be related to the temperature for 4gt4 theory there is no obshyvious connection We will come back to this uncertainty soon The important ~r 3 (LfshyCOngefit hze is the maniplJ~ the path integral-te-a statistical mechanical system ~

1 (cn hlt ~Jd cent ~ sJbw ~

(rmiddot1I4 ~

2

Upon first sight it is surprising to see a reformulation of a quantum field in terms of a classical system But then we realize we extract information from quantum system in the form of probabilities the same language as a statistical system

12 Discretizing the Action We have the Landau-Ginzburg model but how n we jam that on~a lattice Wee6rio~ go from the continuum where derivashy

tives and integrals live to the discrete world inhabited by sums and differences We actually already did this before when evaluating the path integral for the free field ~ beelegrmud provides a comforting salllee 10 leaamp against and refl~c1

Discretizing the path integral in our classical statistical for~ion tlhe same l procedure we followed to evaluate the only soluble integralofquantTmfield~ Not only does casting QFT in terms of a statistical system allow for conceptual coherence but it is also suggested by the evaluation of the path integral Its always nice to see things converging in physics

To the discretization the integrals just become sums and differentials turn into the lattice spacing a We change the continuum of space-time indexed by x to a discrete index i The trickiest thing to change is the derivative We can simplify the span of the derivative to operating only on nearest neighbors so its discrete

- ~is rfi-~) ~~ c l-J voJ ~-- ---r4rAO~

(13) loW Y ocent -+ cent(i+~)cent(i~~(~~vVff -IIJ Lrlt ( ox aC__

By framing the derivative like so we are essentially restricting our consideration of the gradient to nearest neighbors The nearest neighbor interaction sounds like an Ising model and indeed we find something similar After p1ugging these In ~

implify and obtain S~X h it yt~t

(14) SE = Lcent(i)cent(j) +L [(2+ ~2) +~cent4] tNlt e~ (ij) N

13 Updating the Algorithm With just a few simple manipulations that can be reduced to some mumbling ana hand waving we have reached some familiar territory After doing all that work in the Ising model we find ourselves nodding our heads confidently at this new theory Now we have to go to practice The scalar field expands the Ising spin to a continuum of negative and positive values Turning to the algorithms we used for our Ising simulation we want to ensure they can probe the continuum of states at a field site The Wolff algorithm flips entire clusters and we cannot simply apply a change to the field value uniformly across an area This would violate detailed balance producing a state outside of

--- ~-~------------

3

a Markov Chain Instead we turn to the Metropolis Algorithm In addition to flipping the sign of our field at a site we can also change its value by a randomly selected increment It is exactly the same calculation as in the Ising model but there is a greater multiplicity of states to consider

The updated Metropolis algorithm goes as follows

(1) Randomize the lattice to form an initial state

(2) Randomly choose a site i

(3) Randomly choose an increment d in [~a aJ

(4) Assign a new value according to 4gtnew(i) -4gt + d

(5) CalculateLE resulting from flipping 4gt(i) and

If LE 0 accept the flip and increment

If LE gt 0 generate a random number r E (01) If r lt e-f3AE accept the flip

(6) Return to step 2 until iterations equal to the size of the lattice ha-ee~ completed ~ shy

We also apply the Wolff Algorithm to counter critical slowing down We cannot apply it blindly we must reconsider the conditions for detaile~nce now that we have a continuum of field values We simply replace the discrete values of the Ising value with a continuum of values The new probability for selecting a site to add to the cluster is

P4gt = 1 - e-2ltfgt(seed)4gt(i)(15)

Applying the Wolff Cluster algorithm has a great physical interpretationJL ~ theory that was lacking in the Ising model With only a Metropohs algorithm not only would the correlation time significantly diverge near criticality our field would also settle into one of the minima The importance sampling in our simulashytions would be biased to one of the minima depending on initial conditions The Wolff cluster algorithm produces sampling of both minima without having to run multiple simulations with different initial conditions

The combination of these two algorithms ensures ergodicity and has been impleshymented successfully[8] In our program we define a Monte Carlo step a 10 Metropshyolis scans and a Wolff cluster flip Taking measurements at this interval reduces the correlation between states so we are able to obtain better statistics

-___- - shy---------__~shy

Pjfi ~r

HI 95S

j fh~Jcshy

ytgtN~

~ CJS~h~ k~ cIt ~ ~S~ MoLl hJJy ~Ih

-Nonetheless futur~Isuit in attempt1Dg tG-infp1ement the Worm algorithm or an Invaded Cluster algorithm would prove rewarding on two fronts It would allow for analysis on how well these advanced algorithms reduce critical slowing down

lThe programming challenge would also require a lot of thought and possibly the implementation of new programming techniques

2 PHASE TRANSITION

The 4gt4 -toymode1 we are working with is frequently turned to for a simplified study of the Hig~ field The spontaneous symmetry breaking seen in 4gt4 theory is generalized to explain why certain particles have mass and others dont In the history of the physical universe this occurred at a specific temperature when the electroweak symmetry (SU(2) reg U(l)) was broken To probe that area of physics finite temperature quantum field theory is used where temperature becomes a parameter I only mention it in passing and will not delve into it as it is beyond the research area of this thesis

The Higfield is a complex scalar field with an electrodynamicgauge he Hia( Lagranglan is 3tcr f J) (21) t -~FF+1(8+ieA)q12-m2qmiddotq-AWq)2 J JJ

f f ftis rc J In four dimension 4gt4 theory has 4 massless odstone-N~osons from sponta-

I

J (flneous symmetry breaking However the adllit~~ of)lft gau mixes the massless bosons with the gauge vector boson This is the Hiig)( c anism It leaves only ~J a spin-1 W boson [J 5Y ~ ~~shy

(The connection between 4gt4 theory and the Higampfield is not addressable by our] IJ i(eJ researcg One extension we would have to include is to move from 2 dimensions to ~ 4 The physical Higgs field is throughout space-time The connection between a Higgs field in 1+1 dimensions to 4gt~ theory is not immediately apparent We could Sf k actually quite easily update our program to simulate the 4gt4 in 4 dimensions We f~ 1J WOU~de to grow our array we store our data in square our loops and update o~- the ollf-c uster algorithm build a 4-dimensional cluster It would probably only ~ take days to update my code and be confident the program is doing what ~ you want it to do This path was not pursued here due to lack of time but I stress J) it is only a simple modification of our code 3o-v-t-- b-Even if we were to run simulations in 4 dimensions Ming the phase transition 0 AI 4gt~ theory has no immediate connection to the Hig~eld What it can be u d

~ ~)fvt~f ~r II ~ VLr

fi is to inlom a nu_loa simulation 01 the effective p(ential )The effecti~ potential of ltgt4 theory has been used extrapolate a mass for tlie Higgs particle ~ ~ 5 I) I

The simulation is essentially the same but the data observed from it are different ----- ( The goal is to obtain the re~malized vacuum expectation value VR and relate that to the mass of the Rigs article Lattice simulations [1 2J have how~~ overestimated the mass compEL ed to the recently announced discovery of the HiV boson from the LHC cr-J The phase transition for a 1 + 4imensional ltgt4 theory was firsi~onte Carlo simulation by Loinaz and Willey[8] There had been many attempts to locate the phase transition by using analytical approximations but the variety of techniques led to a wide range of predictions A dimensionless critical coupling constant was reported[8J and then refined upon[13] being Ie = 105 and Ie = 108 respectively These are close to estimates from numerical calculation using the Gaussian effective potential reinforcing the applicability and reliability and Monte Carlo simulations Other studies have reported a consistent Ie [10 14J

In addition to knowing where the phase transition lies it is important to know what order phase-transition ltgt4 theory exhibits Chang[3] demonstrated that ltgt4 theory is in the same universality class as the Ising model through theoretical arguments This means that ltgt4 shares the Ising models second-order phase transition and also its critical exponents ltgt4 theory will have the same critical behavior as the Ising model near its critical point This connection between ltgt4 theory and the Ising model suggests the use of the same critical point analysis we performed in Chapter 2 in locating the critical point of the ltgt~ field

Despite the theoretical argument for the shared universality class between the two models it is of interest to verify this through experimental means Indeed multiple studies [10 14 5J have been conducted using Monte Carlo simulations that have found critical exponents to be near those for the Ising model These results ~ r-yrvf 1 ~ 0 that ltgt4 is indeed in the Ising universality class We can also test this~ using data from our simulations We address some of the critical points in the following I section and find they are in good agreement with the Ising values The consistent I -- lov (rlJ results for ltgt4 critical exponents support the claim that it does indeed exhibit a I t ~11~~ second-order phase transition Furthermore the finite size scaling analysis we r I- ~ discussed i~~apter 2 and apply ahead are only valid for second-order phase ~ - ti-- q- r- -II

transitions~hey are found to be applicable a second-order phase transition is f---tA supported ~ l f ~ Ja-k~ or ~ f n-$ h We could have also used a Binder Cl~nt on the energy averages to determine ~ if the phase transition was ideed second-order First-order phase transitions are ~ tI b t- rt (LSI w-r bull

indicated in the limit limL--+oo U4 = 1 for T =f Te and limL--+oo U4 = constant gt 1 C U k fhJ ~ It r for T To Second-order transitions are indicated by U4 always going to unity 0- tr

~ r I h ~ v-Ilo I) ~tV~ ~ kS~~c CJ

6

r Ibull

lttfJ) =0 bullbullS

06

0

lttfJ) 0 02

-Ll -LO -08 - -04

FIGURE 1 Phase diagram for 4gt~ in bare 2 - A space Results from three estimates of the critical points are plotted and are visually indistinguishable The phase transition line separates the symmetric phase on the right from the broken symmetry phase on the left Smaller values of A approach the origin representing moving to the continuum limit

for [7] We however did not accumulate data for E4 so were not able to conduct this analysis

Beyond locating the phase transition and determining its order it is also of interest to know the physical mechanisms behind spontaneous symmetry breaking In 44 theory De et al[6] find that as the field moves from the disordered to order state a single kink develops into a multitude of kink-antikink pairs They propose that ~y be the mechanism for spontaneous symmetry breaking To obtain this

z- result they implement anti-periodic boundary conditions where the field values at the end of the rectangular lattice interact with their toroidal neighbor with a I - b ~ sign opposite than for the non-edge field This promotes the formation of kinks i ~ as seen in Chapter 4 i ~

~SJb 3 FINDING THE CRITICAL POINT tt~

_ L~ ~ 7 We have cast the cent4 path integral as a classical statistical system making it ~IJN-

tractable to numerical simulation In this form it shares the same general behavior of the Ising Model We still have to address our lack of an obvious temperature Drawing from spontaneous symmetry breaking we have two parameters 112 and gt We choose 112 to act like a temperature and vary it for a given gt The value of the field at each lattice site fluctuates within a macroscopic state define by the parameters 112 and gt We are able to define the same quantities that indicated a phase transition in the Ising model the susceptibility specific heat and the Binder Cumulant

7

(31)

(32)

(33)

cent is the volume average of the field and (tJ) is the statistical average~ number of field configurations We normalize the susceptibility and specifiampy lattice area to compare these quantities for different lattice sizes l JI ~

Having calculated these quantities finding the critical value of J12 is ~a matter of locating the points prescribed in Chapter 2 There is not asingk method for doing so and finding the peaks of X and C or the intersection of U4 is a bit of an art With the freedom of choice for evaluating the critical value 12 it is vital to obtain go09 statistics and develltjp a robust method for error analysis I

( b s-I-h d ~M ~flJ First YIe baue the error in our calculated thermodynamic quantities This is mestlj 1 t depaaent on the number of Monte Carlo steps we dlQQS to iterate over ~~dth~Jev- VJ 5 the size of our data set But remember that there is correlation in time ~n ~ t L the states generated for each Monte Carlo step This necessitates a correction to (01_ W i the typical standard deviation dependent on the autocorrelation time We saw this in Chapter 2 in passing r 1

C f7v(~1 bull

To determine the error in thermodynamic quantities we apply the J~e~)~lJ method A thermodynamic quantity say X is calculated from the fluctuations -R-( p1r over all runs in the simulation This gives us only one value of X and th~0 I - work with for a statistical evaluation of the error To obtain a set X al e ftr-d calculate the susceptibility after each Monte Carlo step Essentially e cu ting (JIh up our data set into n blocks of increasing size We are then able calculate the mean and standard deviation The Jackknife standard deviatio IS given as

11it-cA~ t ~ N - 1 7 2 ~N jr

(34) UJ = ~ I)XJi - x) i=1

We then correct this to obtain the error due to correlations

(35)

8

Alternatively we could have only taken measurements of the field in a number of Monte Carlo steps greater than the auto correlation time This would ensure that each measurement is taken on a statistically independent configuration and the error is then given by the simple standard deviation

In addition to statistical errors there is another type of error A subtle but ~ pervasive error Systematic errors arise based on our methodology casting-- J c-lt quantum system as a classical one choosing certain algorithms over others the ~v-~ proceaure for finite size analysis These error are difficult to quantify Doing so It ~ wou require repeatmg ou e simulation or analysis using alternate methods and comparing results Bu the new methods chosen would also carry with them their own systematic errors f-r CauL 0 X f~ of ~dtl-chcu-nr One method for determine systematic errors proposed by Schaich would be to run the simulation with interactions beyond nearest neighbors This has been done for the Ising model and would actually be quite simple to implement We would simply add a term to our discretized Lagrangian Nearest neighbor interactions are also prevalent in our algorithms We then could also update our algorithms or use others that are not limited to nearest neighbor interactions

31 Simulation Det~ The lattice simulation codes were implemented in C++ and run on Amherst C lleges computing cluster 15000 runs were done on rectangular lattices of siz pound = 64 L = 128 L = 192 L = 256 L = 394 L = 512 r For each lattice 500 j bs were submitted per A value except for L = 512 where J ( ~ 400 jobs were submitted To determine the range of JL to simulate over I first ~ I-Jrl rr ran a sample simulation on a L = I visually determined where the susceptibility curve began to behav quadratcall from the s metri oken f (4 12f1111

tpoundgt symmetric phases and chose that the sim ion range To account for finite ~ ~ ~ ~~ ize effects on the critical point I re the symmetric JL upper bound by 01 1170 3V

for every two lattice sizes I observed from plots that the peak of the susceptishy-nOr Q f(r tgt J- 512 bility becomes sharper from the symmetric side more than the broken symmetric

oh w-f- ~~()- side Reducing the upper bound accounts for finite size effects and better samples

(SA ~~-e sharper peaks CgCr-rp- I wanted to also run simulations at L = 640 I ran one set but due to a miss- S 1 ing input JL set the rest of the input JL values were passed to the wrong value of A[I believe that such a large lattice is necessary to include in finite size analy- fti ~ yt-rCf yv- ~~j U nfort unately the time required to run simulating for such a large lattice aJ CI

5 l t was beyond what I would have gained from havlg the data After running the ( r ~ ~11f( ~1 simulation data was imported into M athematictmiddoto for analysis t11 J1 J) H~ ~ 32 Susceptibility The phase transition is indicated by a peak in the susyep- t ~ lvtrJ - tibility at some JL~t dependent on A and L From our simulation we obtain cal- ~

e-- )i culations of X for discrete series of JL2 This provides us with a description of the J ~fe~fl $4 y

~ ~ e J~ n0j a f1sJ

07-pIQf(jul

a ~ ~ f~C~i-] ~~f IJ

~ af~ I~ ~I- r~

wridv

9

x

Lx 10-

-0711 -0721 -0720 -0719 -0718 -0717 -0716 -0711

I FIGURE 2 Plot of susceptibility data points and fit for a 51i2 lattice at gt = 05 ~ Fit is done using an 5th order polynomial The maximum of the fit gives the (Moi~~ tial pnt ff JIbull f-Ifl ~r~

dependence of X on JL2bull Looking at i s platonic aussian shape we almost hear ~ it crying in despair and loneliness withou a accompanying fit Well provide itt with one to satisfy our data This allows us to extrapolate X to values of JL2 that - were not inputs of our simulation ensuring that we can still obtain an accurate ) ~ VM J(fI I 1-- critical value even if we didnt choose it as an input 0 ~ ~JS~

The shape of X is remarkably Gaussian but I chose to abandon that functional j t f S tJofl (shyform for a computationally and analytically simpler one I use the simple Taylor _L ov~~ expansion ~

-0- (La ftct f- Gr-sr I I

(36) Ix = a + bx + cx2 + dx3 + ex

4 + Ix5 t tj ~-t1 1-

to fit my data to This is a completely valid procedure as any function is well appr~ed by its Taylor expansion to a certain order It has also been done befo [] We can heuristically determine a cutoff order to our expansion by ~ g the error in our fit I did this by looking at the R2 value of the fit for

7 ~ ~tarting at x 2bull Even at this low order the value of R2 is high at ~ 93 R2 begrns to decrease by including x 6

so I stop there

7 To fit the correct area of data to the curve I first locate the position of the maximum in the data set I then use 50 data points above and below to use for

the fit To find the peak I then simply find the maximum of the fit To put an c error on the maximum I run jackknife analysis by applying the fit on data sets

v-~ with one point removed from both sides upon each iteration tIr f- () I - [-( ~ gt fc r ~ f

u v

0 1gt5

os

0

-074 0

(a) (b)

FIGURE 3 Binder cumulant plot for various lattice sizes 8lambda = 05 (a) and a zoomed in view near the intersection point (b)

33 Binder r4umulant Binder cumulant indicates a phase transition and proshyvides finite siie ling all in one It is a thermodynamic workhorse and in my work I found it pro ed a good est~rte of JL~t when analyzed for even relatively small lattice sizes IA~ ~M~ I C J

The intersection of the Bindeld1mulant for various lattice sizes is near impossible to estimate from the data plMnly To analytically obtain an intersection I first fit U4 with the same Taylor expansion as I used to obtain a suscepti~

~ieve a good fit you have to limit the size of the data set you fit toI r~~rai~ed ~ Jydata sets to include data for 6 ~ U4 ~ 2 This provides a good fit that ~ ~

captures the sigmoidal behavior of U4 and ensure the inclusion of the intersection III Qy~

~ ~-0- Imiddot We could consider alternative functional forms to use in our fit Any function that

fS JVft j C-~ is sigmoidal is a candidate however using such a function constrains the fitting M t1IIlpound pre procedure We can expand these functions and find that we can equally fit to a

~ f-lt- 11 ~ more free series

f I- We now have to locate the intersection point To do so we simply take a sum ofJvrl 0 r_r the difference between U4 for each lattice size

~ r bull v-r 1_ 51~-t ~l~ 7

(37)~o1I1

and minimize it The minimization process presented a small complication as there is frequently a local minima very close to the global minima To avoid finding the local minima you have to find a value to the left of the global minima to provide to the minimization function Doing so ensures that the minimization function spits out the global minima instead of stopping at the local minima

11

u

-ants -onll ont

(a) (b)

FIGURE 4 Intersection of the Binder cumulant fits at A= 05 for various lattice sizes (a) and the intersection fiction (b) The minima of the intersection function is not strongly differentiated A good starting point must be supplied to the minimization algorithm to obtain the correct critical point

Despite the lack of finite size scaling the intersection of the Binder cumulant estishymates the critical point in good agreement with the other methods Nonetheless perhaps a more rigorous procedure for estimating the critical point is to build up a series of intersections for increasing L To do so you could find the intersection of two only plots for closest lattice sizes You would repeat this for each pair U4(64) n U4 (128) U4 (128) n U4 (192) After doing this perform the same reshygression as done for the susceptibility and specific heat on L -1 The intersection would give the critical point with finite size correction H61yY-e-ef K miy turn o~ dUM tlie f8lY6ff frow this nuanced method may gat be worth the illfeF-t _______et

34 Specific Heat Similar to the susceptibility the specific heat indicates a ehase transition when it peaks However the specific diverges like C ex In IfL2 - fL~tI [] The divergence presents a very difficult behavior to fit to To complicate things even more the divergence is almost unnoticeable at small lattice sizes but quickly appears as the lattice size is increased The strong lattice size dependence of C eliminates a standard fitting procedure that can be applied to each data set for all L A logarithmic function can be used as a fit for only L 512 and only by carefully specifying constraints for the fitting parameters For smaller lattice sizes it is possible to fit the small range of data over which the divergence artifact appears to the expansion series

Despite these possibilities to work with I abandoned the use of a fitting procedure to locate the specific heat peak Applying good constraints to the data set and fitting procedure were heuristic and could not be captured by an algorithm to apply to all lattice sizes Instead I find the maximum value of the specific heat in my data set and claim the associated fL2 is the critical value The validity-Qf thjs proce4tte nnrot-8B-Sllspicious as it~uda The specific heat has a strong

( divergence for fL2 = fLt At that fL the susceptibility should dramatically jump

0-~~J-ro~

I fI f roJCL J (JVI~ h rJ- ~Jy~tishy

n f ftP-V-A shy

~ I 1 ~ JI- ~U ~

I

~ ~ L r-urJ 1(( cmiddot

tJ f f~t flN-l ~-

$1 III-) I

1J( Q tJ r J1VT ~ j( ~rwz 04 I _

Tr-lt-Smiddotj-vr I

12

C

14

L=64 12

L=SI210

08

06

04

02

-075 -070 -065 -060 t

FIGURE 5 Plot of specific heat for two lattice sizes at ) = 05 The divergence becomes stronger for larger lattices and the peak is easier to pick out

high above the values for 1L2 ~ lLt that surround it This is indeed the case for the data set from L = 512 and can be seen in Figure 5

This argument does not hold through the analysis After applying this method the critical points do not reflect an obvious linear relationship as they do from using the susceptibility We can correct for this by weighting a fit according to the large error in the critical point After doing so we find that the intercept of our fit agrees with the critical points found from the susceptibility and Binder cumulant ~- lt7 for all values ~7 c

Using the specific heat to determine lLt has been done by others but neither source describes their technique for locating the critical point Finding the peak in the data is the simplest procedure but when compared to our technique used on the susceptibility data it seems lacking The more robust technique of fitting to a logarithmic curve may be applicable for large lattice sizes But doing so would increase the time for our simulations to complete

35 Finite Size Scaling To adjust for finite size effects we use the analogue of the scaling correction function used in Chapter 2 It is here ]

2 2 ( -V) C- ~J7 (38) ILl = ILcrt 1 C3 lt-- J~ $ (

We fit the susceptibility and specific heat data separately Each plot can be seen in Figure 6 The thermodynamic critical point is simply the y-intersection of each plot The error is given by the standard error in that parameter from the fit

13

(a) (b)

FIGURE 6 Finite Size Scaling extrapolation for susceptibility (a) and speshycific heat (b) at A == 05 plotted against L-1 The y-intersection give the L- gt 00 limit The large error in specific heat is due to the inability to extrapshyolate for non-sampied p2 values They nonetheless provide agreeable results of -0720(963) amp -0720(823) respectively

There is also the possibility that finite size effects are not accurately corrected for by a single term If this is so we need to include higher order corrections such as in

(39)

Upon using the above equation I found that for data at mostgt values w = 0 or a value that does not alter the intersection point within the accuracy of the fitting algorithm However I did seey change in the critical point estimate at smSllI gt values The applicability of for small gt may indicate that finite size effects are ~ronounced for these parameters Despite this difference I did not include critical points from using in the following analysis I was unable to apply the fitting algorithm for all values of gt I received a machine overflow error in Mathematica I thus abandoned this fit model

36 The Ising Limit We can reformulate the Lagrangian to show that 4gt4 theory is the Ising model in the gt --+ 00 limit

~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i

1 I 1 64shyA constant term has been added to the potential part of the Lagrangian We are free to do this as we are free to label the ground state as whatever we wish Adding a constant does not change the physics

~ d 1 15middotJt-

~

if

14

I P I by x Al = 1000 -4457(42) A = 500 -22546(9) Al Al Al

= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01

FIGURE 7 Critical values for p2 on the lattice Results from using the susceptishybility specific heat and the Binder cumulant as phase transition indicators are reported Finite Size Scaling was used to obtain the L -+ 00 limit for estimates

from X and C -1J ~ middotrzjrvri (J-- (1Jrrlt2-~P-- - J

6-f lt-6 J- Z ( [iY1 Sl u- I I 1 I-t h --Ie0 r U lt- f - ~I ~

lim centgt(i) = plusmnJp2 2d j)i yh(311) gt--gt00 _ ~

This gives us the (3 analogous to the Is g Hamiltonian temperature

i f ~~hL- ~r- V P~ r (312) 1pound (3 = _p2 + 2d A

This provides us with a check on our simulation We can run simulations folshylowing the limit A -t 00 calculate the critical mass term and determine how well that matches to the known critical temperature for the two-dimensional Ising Model

For the Ising model Tc = 2269 and j3 is given by its inverse 0440723 If our simulations are to be trusted we should find that our found critical points approach the Ising critical temperature in the A -t 00 limit Since the critical points from our three technique are in good agreement I arbitrarily pick one to us ~y LaHl aiMed 4le the Bindel cUlffiilaBt data so I~ Figur 8 shows the limit is indeed being approached

15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894

FIGURE 8 Critical values for 1-2 on the lattice as ) -+ 00 f3c approaches the

critical temperature of the Ising model r4~ ftlt~ ~

37 Bare Ratio The coupling constant and mass te~e treat as parameters are dimensionless on the lattice We have ~QReile ~ wIth-the dimensional quantities in the continuum Since length is given in dimensions of mass both parameters can be considered to be measured in units of inverse lattice space squared

(313)

Although the lattice spacing disappeared after we d appear in the definition of the lattice values A fL We want to cover up the source of our data and extrapolate it to the continuum limit leaving the lattice as nothing but empty scaffolding We have already beguQ ooiQg so tsjCorrecting for finite size affects and extrapolating to the limit L - 00 But as seen above we still have discrete data in terms of the lattice spacing To take the limit a - 0 we first have to capture the two parapets in one term the dimensionless critical coupling defined as

(314) Ie = if) We can plot this for all of our critical points and then extrapolate to the origin giving Ie in the continuum limit Before taking this final limit we have to do one more thing make our theoretical values physics Enter renormalization

38 Renormalized Ratio In the previous sections we have worked only with the bare mass and bare coupling These are theoretically significant we have to make them physically meaningful The coupling constant A does not have a renormalized correction so AR A The renormalized mass is given by

(315)

16

FIGURE 9 The one-loop diagram the only divergent amplitude in two-dimensional 4gt4 theory J

as we saw in Chapter 3 We have to decod(ij2 by determining a renormalization scheme [81

In two dimensions there is only one ultraviolet divergent Feynman diagram (see Figure 9) The amplitude for this diagram in the continuum limit is

(316)

Consider the inverse propagator with renormalization corrections in the term L(p2) called the self-energy

(317) C-1(p2) = p2 + fL2 + L(P2)

(318) L(p2) 3gtAp 2 - OfL2 + two-loop

The self energy captures the divergent amplitude of the one-loop diagram along with convergent higher order terms in two-loop diagrams OfL2 is the renormalized correction on fL2 so we have

(319)

where Ap2 is given analytically by

(320)

and 10 is a Bessel function Numerically evaluating (319) gives us the renormalized mass term on the lattice fLL After finding the numerical solution we then follow the argument outlined above and determine the renormalized dimensionless critical coupling constant fRe To see that fRe does capture the phase transition we can look at the renormalized Lagrangian

17

106

104

102

98

96

94

FIGURE 10 Renormalized dimensionless critical coupling Large errors for small gt strongly suggest that larger lattices or higher order terms need to be used to accurately account for finite size effects

(321) c

For small fR we can again consider the classical potential V(4)) For small fR the A Ct- YP I

coefficient in the second term is positive and there is only a single minima How- ( $i f fflfever for large fR the coefficient becomes negative and the double-well potential

emerges [8] Jic p-e--hJ The renormalization procedure was run on data for each three methods and an hI Iflt error was obtained by taking their difference Figure 10 shows the averag~shy frv6 vk f mensionless coupling constant Extrapolating to the origin a value of 1044 ~ 16)~ was found for the dimensionless critical coupling This value is in the mid e-6f )

j previously calculat lues [8 13J The large error bars seen in Figure 10 are j

1 most likely due t poo nite size corrections I did run a simulation at L = 640 i1w dl C for) = 5 when inl 1 ly developing ym data analysis algorithms When using this yt- ob~~

r lattice I found that ecritical points obtained from finite size extrapolation were tt IZ in better agreement ) 1 ( vi - Kit- f Lt Jrn~-hpv-t(

s r~ Lfgtlotting fe using data from only U4 intersection points I obtain fe = 1077 plusmn 06 -= ~1--l This is in better agreement with the refined calculation [13] This suggests that

(tI the Binder cumUlant~a better estimate of the critical point for a limited set t(f I J of lattice sizes The nder cumulant may be favored for simulations on larger ~ ~ (( lattices in higher dime ions Although it is also possible that the promise of the j

-( t

~Ptvh c-It C ~J ~I ~ of- r~Vt~ -~ r fL

2-bull

18

Binder cumulant falls short at higher dimensions but there is obvious reason for such a failure --- I 1 bull fL f I shy

L-i ~ (r fgt( (JU J tn bull

4 BARYON NUMBER VIOLATION

Cornwall and Goldberg following the work of Ringwald and Espinosa found that there is a nonvansihing amplitude for the production of n particles for small gt in cent4 theory Monte Carlo simulations have provided evidence that this does not occur [11 4] but it is of interest to refine their calculations to better accuracy given the limits of computational power in the 1990s It may also be the case that their limited simulations did not accurately capture finite size effects due to the restraint to small lattices n _0

~v~ 41 Theory The explicit ~litude for the creation of n particles is given [9] as

gt ) (n-l)2 (41) a(N) 1)nfL2 shy( CfL2

If we simplify things and consider a(n) ex nlgtn-l we see that the amplitude becomes large for n ~ 1gt In this limit perturbation theory fails as the amplitude approaches unity The cross section for n particle creation may become large at high energies E ~ nr [9] (

~

To probe this possibility we have to relate somethino-4can measure in our simshyulations to the multi particle cross section Charngb~s that the mass and wave function renormalization are related to the inverse Euclidean two-point correlation function in momentum space

(42)

Charng then relates this to the decay rate r for a cent particle to decay into n cent particles

(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z

frrr r here is the physical decay rate leoking at (43)~it is simple to see that finding Z from Equation 42 will set an upper bound on the decay rate The idea is that

19

there may be some critical energy E where the integral in (43) goes exponentially If that is the case the bound of the integral found from Z should be high [4]

To ensure that our lattice simulations are sensitive to energies up to E we place a restriction on the parameters at which we choose to run the simulations The lattice requirement is )

lshy

e puc (44) (pr (J-+e)

where ~ is the correlation length Estimates for ~ have been done for 2 dimensions [l1J and are

m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~

When running our simulations we want to choose parameters that sat~we then move along a line of constant physics towards the continuum de~by a constant dimensionless coupling constant

42 Propagators The inverse propagator in 4yk defined in momentum space but we run our lattice simulations in coordina~e space To obtain(42)ve perform a Fourier transform on the spatial propagator given as

(47) G(x y) = (4)(x y)4gt(Oraquo)

The average is taken for all points as origins Since we are taking the Fourier transform of this function we first want to check that we are getting what we expect This also provides us an opportunity to check the one of the critical constants of the universality class associated with the divergent behavior of the correlation length at criticality Near the critical point the two-point correlation function 47 is proportional to an analytic solution ---

lt--- J- r 4 hlgt~

(48)

The form of(4~ given as a function of one variable so we must change our twoshyvariable G(x y) to a function of radius This is a simple procedure of calculating

20

G(r)

07

06

40 60 80

FIGURE 11 Two-point spatial correlation function plotted as a function of rashydius for a 1282 lattice near the critical line ( = 05 2 = -713) It is well fit by the analytic exponential function returning a value of T = 247 This is close to the Ising critical exponent T = 25

r = x2 + y2 and averaging over all equivalent distances This gives the function plotted in Figure

Having confirmed our algorithms are returning something sensible we can confishydently apply a discrete Fourier transform to G (x yl Chis goes like

(49) x x

where kx and ky are the lattice momenta These are restricted by the integer wavelengths that can span our lattice They are given by kx 1 and kx1 On our lattice Lx = Ly so they span the same values After applying the

transform we then plot the inverse of G(p) against the continuum momenta given by

(410) i 4 Lsin2(k~2) ~

The two-point correlation function in momentum space becomes highly non-linear for large momentum modes so we restrict our attention to low modes We are interested in the behavior near the origin so looking at a small subset of momenta is reasonable We then apply a linear fit to predict the intersection point and the slope giving m~ and Z respectively

21

REFERENCES

[1] P Cea M Consoli and L Cosmai New indications on the higgs boson mass from lattice simulations arXiv preprint hep-ph()1139 2002

[2] P Cea L Cosmai M Consoli and R Fiore Lattice effective potential of massless (lambda phi4)in four-dimensions triviality and spontaneous symmetry breaking Amv preprint hep-th9S()S()48 1995

[31 Shau-Jin Chang Existence of a second-order phase transition in a two-dimensional 1 4 field theory Physical Review D 13(10)2778 1976

[4] Y-Y Charng and RS Willey Nonperturbative bound on high multiplicity cross sections in 1 4 theory in three dimensions from lattice simulation Physical Review D 65(10)105018 2002

[5] Asit K De A Harindranath Jyotirmoy Maiti and Tilak Sinha Investigations in 1+ 1 dimensional lattice p4 theory Physical Review D 72(9)094503 2005

[6] Asit K De A Harindranath Jyotirmoy Maiti and Tilak Sinha Topological charge in 1+ 1 dimensional lattice p4 theory Physical Review D 72(9)094504 2005

[7] Zvonko Glumac and Katarina Uzelac First-order transition in the one-dimensional threeshystate potts model with long-range interactions arXiv preprint cond-mat98()7417 1998

[8] Will Loinaz and RS Willey Monte carlo simulation calculation of critical coupling constant for continuum phi 42 arXiv preprint hep-lat971()()8 1997

[9] Yu Makeenko Threshold multiparticle amplitudes in phi 4 theories at large n arXiv preprint hep-ph94()8337 1994

[10J Pablo J Marrero Erick A Roura and Dean Lee A non-perturbative analysis of symmetry breaking in two-dimensionalj i1 Pii1i sun 4jsuP1 theory using periodic field methods Physics Letters B 471(1)45-52 1999

[11J RD Mawhinney and RS Willey Nonperturbative lattice simulation bounds on high multishyplicity cross sections in 1- zy 4 Physical review letters 74(19)3728-3731 1995

[12] Istvan Montvay and Gernot Miinster Quantum fields on a lattice Cambridge University Press 1997

[13J Tadeusz Pudlik Lattice simulations of the 04 theory and related systems Tadeusz Pudlik 2009

[14] David Schaich and Will Loinaz Improved lattice measurement of the critical coupling inp_ 2y 4 theory Physical Review D 79(5)056008 2009

[15] Takanori Sugihara Density matrix renormalization group in a two-dimensional Aphi4 hamilshytonian lattice model JOtImal of High Energy Physics 2004(05)007 2004

[16J Raul Toral and Amitabha Chakrabarti Numerical determination of the phase diagram for the cphi4 model in two dimensions Physical Review B 42(4)2445 1990

2

Upon first sight it is surprising to see a reformulation of a quantum field in terms of a classical system But then we realize we extract information from quantum system in the form of probabilities the same language as a statistical system

12 Discretizing the Action We have the Landau-Ginzburg model but how n we jam that on~a lattice Wee6rio~ go from the continuum where derivashy

tives and integrals live to the discrete world inhabited by sums and differences We actually already did this before when evaluating the path integral for the free field ~ beelegrmud provides a comforting salllee 10 leaamp against and refl~c1

Discretizing the path integral in our classical statistical for~ion tlhe same l procedure we followed to evaluate the only soluble integralofquantTmfield~ Not only does casting QFT in terms of a statistical system allow for conceptual coherence but it is also suggested by the evaluation of the path integral Its always nice to see things converging in physics

To the discretization the integrals just become sums and differentials turn into the lattice spacing a We change the continuum of space-time indexed by x to a discrete index i The trickiest thing to change is the derivative We can simplify the span of the derivative to operating only on nearest neighbors so its discrete

- ~is rfi-~) ~~ c l-J voJ ~-- ---r4rAO~

(13) loW Y ocent -+ cent(i+~)cent(i~~(~~vVff -IIJ Lrlt ( ox aC__

By framing the derivative like so we are essentially restricting our consideration of the gradient to nearest neighbors The nearest neighbor interaction sounds like an Ising model and indeed we find something similar After p1ugging these In ~

implify and obtain S~X h it yt~t

(14) SE = Lcent(i)cent(j) +L [(2+ ~2) +~cent4] tNlt e~ (ij) N

13 Updating the Algorithm With just a few simple manipulations that can be reduced to some mumbling ana hand waving we have reached some familiar territory After doing all that work in the Ising model we find ourselves nodding our heads confidently at this new theory Now we have to go to practice The scalar field expands the Ising spin to a continuum of negative and positive values Turning to the algorithms we used for our Ising simulation we want to ensure they can probe the continuum of states at a field site The Wolff algorithm flips entire clusters and we cannot simply apply a change to the field value uniformly across an area This would violate detailed balance producing a state outside of

--- ~-~------------

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2 PHASE TRANSITION




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6

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x

Lx 10-

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13

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(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















--- ~-~------------

3















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Pjfi ~r

HI 95S

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2 PHASE TRANSITION




I











6

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(33)




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(35)

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(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

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~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















-___- - shy---------__~shy

Pjfi ~r

HI 95S

j fh~Jcshy

ytgtN~




2 PHASE TRANSITION




I











6

r Ibull

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(a) (b)



(39)



~ s wi-J-d

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gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



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lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

























6

r Ibull

lttfJ) =0 bullbullS

06

0

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(31)

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-0711 -0721 -0720 -0719 -0718 -0717 -0716 -0711





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L=64 12

L=SI210

08

06

04

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-075 -070 -065 -060 t








13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



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lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















7

(31)

(32)

(33)




C f7v(~1 bull


11it-cA~ t ~ N - 1 7 2 ~N jr

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(35)

8










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Lx 10-

-0711 -0721 -0720 -0719 -0718 -0717 -0716 -0711





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4 + Ix5 t tj ~-t1 1-



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u v

0 1gt5

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0

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11

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L=64 12

L=SI210

08

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-075 -070 -065 -060 t








13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















8










~ ~ e J~ n0j a f1sJ

07-pIQf(jul

a ~ ~ f~C~i-] ~~f IJ

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Lx 10-

-0711 -0721 -0720 -0719 -0718 -0717 -0716 -0711





(36) Ix = a + bx + cx2 + dx3 + ex

4 + Ix5 t tj ~-t1 1-



so I stop there




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0 1gt5

os

0

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(37)~o1I1


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L=SI210

08

06

04

02

-075 -070 -065 -060 t








13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















9

x

Lx 10-

-0711 -0721 -0720 -0719 -0718 -0717 -0716 -0711





(36) Ix = a + bx + cx2 + dx3 + ex

4 + Ix5 t tj ~-t1 1-



so I stop there




u v

0 1gt5

os

0

-074 0

(a) (b)










~ r bull v-r 1_ 51~-t ~l~ 7

(37)~o1I1


11

u

-ants -onll ont

(a) (b)






0-~~J-ro~


n f ftP-V-A shy

~ I 1 ~ JI- ~U ~

I


tJ f f~t flN-l ~-

$1 III-) I


Tr-lt-Smiddotj-vr I

12

C

14

L=64 12

L=SI210

08

06

04

02

-075 -070 -065 -060 t








13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















u v

0 1gt5

os

0

-074 0

(a) (b)










~ r bull v-r 1_ 51~-t ~l~ 7

(37)~o1I1


11

u

-ants -onll ont

(a) (b)






0-~~J-ro~


n f ftP-V-A shy

~ I 1 ~ JI- ~U ~

I


tJ f f~t flN-l ~-

$1 III-) I


Tr-lt-Smiddotj-vr I

12

C

14

L=64 12

L=SI210

08

06

04

02

-075 -070 -065 -060 t








13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















11

u

-ants -onll ont

(a) (b)






0-~~J-ro~


n f ftP-V-A shy

~ I 1 ~ JI- ~U ~

I


tJ f f~t flN-l ~-

$1 III-) I


Tr-lt-Smiddotj-vr I

12

C

14

L=64 12

L=SI210

08

06

04

02

-075 -070 -065 -060 t








13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















12

C

14

L=64 12

L=SI210

08

06

04

02

-075 -070 -065 -060 t








13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















13

(a) (b)



(39)



~ s wi-J-d

(310) C = - L 4gt(i)4gtU) + L ~ [4gt2(i) - -Jl2

gt+ 2df lt ~-)110 ~ (iJ) i


~ d 1 15middotJt-

~

if

14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















14


= = =

100 -4941(36) 10 -7768(31) 10 -12719 8

Al = 05 Al = 01









15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















15

~l f3a i

Al = 10 -776993 117699 Al = 100 -494247 0534247 Al = 500 -225509 0459018 Al = 1000 -445894 0449894




(313)




(315)

16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















16




(316)


(317) C-1(p2) = p2 + fL2 + L(P2)



(319)


(320)


17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















17

106

104

102

98

96

94


(321) c









-( t


2-bull

18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















18








~


(42)


(43) ~JdEr(E)7r E2

= Z( -1) lt -1 Z - Z


19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















19



lshy

e puc (44) (pr (J-+e)


m(45) (4)) 0 E ~ 186

m A

(46) (4))10 ~~54~



(47) G(x y) = (4)(x y)4gt(Oraquo)


lt--- J- r 4 hlgt~

(48)


20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















20

G(r)

07

06

40 60 80




(49) x x



(410) i 4 Lsin2(k~2) ~


21

REFERENCES

















21

REFERENCES

















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