OPINION
MOELLER, Vice Chief Justice.STATEMENT OF THE CASE
This is our third review in this case. Earlier appeals have upheld all of defendant’s convictions and all of his sentences except the death penalty imposed on the murder count. Following the last remand, defendant was again sentenced to death for first degree murder. This is an automatic appeal pursuant to Ariz.Rev.Stat.Ann. (A.R.S.) § 13-4033, and Rules 26.15 and 31.2(b), Ariz.R.Crim.P., 17 A.R.S. We have jurisdiction pursuant to Ariz.Const. art. 6, § 5(3) and A.R.S. §§ 13-4031, -4033.
FACTS AND PROCEDURAL BACKGROUND
Because the facts are set out in detail in our earlier opinions upholding defendant’s convictions for first degree murder, attempted first degree murder and first degree burglary, it is necessary to recount them only briefly here. See State v. Rossi (Rossi II), 154 Ariz. 245, 741 P.2d 1223 (1987); State v. Rossi (Rossi I), 146 Ariz. 359, 706 P.2d 371 (1985).
On August 29, 1983, defendant went to the home of Harold August to sell him a typewriter. Following negotiations for the sale of the typewriter, defendant robbed and murdered August and then shot and injured a neighbor who entered August’s home to offer assistance. A jury convicted defendant of first degree murder of August, attempted first degree murder of the neighbor, and first degree burglary. The trial court sentenced defendant to death for murder and to prison terms for the other crimes.
In Rossi I we affirmed all of defendant’s convictions, as well as the prison sentences imposed for attempted first degree murder and first degree burglary. We vacated defendant’s death sentence for murder and remanded for resentencing because the original trial judge erroneously determined that evidence offered in mitigation had to be sufficient to constitute a defense to the prosecution before it could be considered a mitigating factor under A.R.S. § 13-703(G). Id. 146 Ariz. at 367, 706 P.2d at 379.
On remand, defendant was again sentenced to death. At this second sentencing, the trial court concluded that defendant had not proved that he could be rehabilitated or that his cocaine abuse had significantly impaired his capacity to appreciate the wrongfulness of the murder or to conform his conduct to the requirements of law. On appeal in Rossi II, this court disagreed with the trial court and found that defendant had shown that he was capable of rehabilitation. With respect to defendant’s cocaine use, this court agreed with the trial court’s finding that defendant’s addiction did not significantly impair his capacity to appreciate the wrongfulness of his conduct. However, this court, contrary to the trial court, found that defendant’s cocaine addiction had significantly impaired his capacity to conform his conduct to the requirements of law. Because two of this court’s findings varied from those of the trial court, this court remanded, directing the trial court to consider at the third sentencing the two additional miti*278gating circumstances that had been found by this court. In remanding, this court expressly noted that “[w]e express no opinion ... on whether the trial judge should impose a life sentence or a death sentence.” Id. 154 Ariz. at 251, 741 P.2d at 1229.
Because the original trial judge retired, a different judge presided over the third sentencing hearing. This judge heard numerous witnesses and also considered earlier evidence to which the parties stipulated. Finding that the mitigating circumstances were not sufficiently substantial to call for leniency, the trial court resentenced defendant to death.
ISSUES
Defendant raises five issues in his opening brief and a sixth in a supplement to citation of legal authority. They are:
1. Whether the trial court erred in its findings of mitigation and in evaluating previously found mitigating factors.
2. Whether a finding of a mitigating circumstance under A.R.S. § 13-703(G)(1) (significant impairment of capacity to conform conduct to law) negates a finding of an aggravating circumstance under A.R.S. § 13-703(F)(6) (especially heinous, cruel or depraved manner).
3. Whether application of Arizona’s heinous, cruel or depraved aggravating circumstance to this case violates the eighth amendment of the United States Constitution.
4. Whether Arizona law violates the sixth amendment of the United States Constitution by denying defendant the right to trial by jury on the factual basis of his eligibility for the death sentence.
5. Whether A.R.S. § 13-703(C) violates the eighth and fourteenth amendments of the United States Constitution by establishing a mandatory presumption of death and by shifting the burden of proof of mitigation to the defendant.
6. Whether the mitigating circumstances previously found by this court preclude reimposition of the death penalty under principles of collateral estoppel or double jeopardy.
DISCUSSION
1. The Trial Court Did Not Err in Its Findings of Mitigation and in Evaluating Previously Found Mitigating Factors.
This court independently reviews the aggravating and mitigating circumstances to determine whether a death penalty was properly imposed. State v. Mauro, 159 Ariz. 186, 207, 766 P.2d 59, 80 (1988). The defendant bears the burden of proving the existence of mitigating circumstances by a preponderance of the evidence. State v. McMurtrey, 143 Ariz. 71, 73, 691 P.2d 1099, 1101 (1984).
In Rossi II, this court held that defendant’s cocaine addiction significantly impaired his capacity to conform his conduct to the requirements of law. At resentencing, the trial court expressly noted that “defendant’s capacity to conform his conduct to the requirements of law was impaired by cocaine addiction.” Defendant first argues that the trial court erred by not stating that defendant’s impairment was “significant.” He then argues that if the trial court would have so stated, case law would require the trial court to grant leniency and impose a life sentence rather than death. We reject this argument because it is clear that the trial court understood and followed Rossi II and because the very language of Rossi II acknowledged that a significant impairment does not preclude a death penalty.
In addition to the impairment factor, the trial court, following its updated sentencing hearing, also found as mitigating factors: (1) defendant’s lack of a prior felony record and his history as a productive member of society; and (2) recommendations of leniency and attestations to defendant’s good character. Although we concluded in Rossi II that defendant was capable of rehabilitation, the trial court, after a full updated hearing, concluded that defendant’s prospects for a successful rehabilitation were not good.
Before the murder, defendant had no prior felony convictions and generally *279had been a productive member of society, working most recently as a bookkeeper. While defendant had no felony convictions, he had been arrested for theft, forgery, drug offenses and possession of a firearm, facts that a court may legitimately consider when lack of a conviction record is advanced as a mitigating factor. State v. Lavers, 168 Ariz. 376, 395, 814 P.2d 333, 352, cert. denied, — U.S. -, 112 S.Ct. 343, 116 L.Ed.2d 282 (1991). The record also contains recommendations of leniency and attestations to defendant’s good character from co-workers, friends, prison guards and medical personnel sufficient to support the trial court’s findings, with which we agree.
Defendant’s principal argument concerning mitigation involves his impairment resulting from cocaine use. He supports his argument by comparing his case to other cases in which the existence of an impairment or inability to control or conform conduct to the requirements of law was considered sufficiently substantial to call for leniency. State v. Mauro, 159 Ariz. 186, 766 P.2d 59 (1988) (defendant suffered from a mental disease caused by a chemical disorder in the brain); State v. Stevens, 158 Ariz. 595, 764 P.2d 724 (1988) (defendant ingested large quantities of drugs and alcohol up to forty-five minutes before the murder); State v. Graham, 135 Ariz. 209, 660 P.2d 460 (1983) (defendant had been taking prescription drugs since age seven); State v. Brookover, 124 Ariz. 38, 601 P.2d 1322 (1979) (defendant suffered from a brain lesion); State v. Doss, 116 Ariz. 156, 568 P.2d 1054 (1977) (court found defendant’s mental illness was a substantial factor in causing the murder).
Our reading of the cases relied upon by defendant indicates that in those cases the mitigating factor of impairment was only one of many factors considered in sentencing. Further, defendant’s case is factually dissimilar to the cases on which he relies. In each of the other cases, evidence was presented that, in addition to suffering from drug or alcohol addiction, the defendant was either seriously mentally ill or was intoxicated when the offense was committed. In the instant case, defendant produced evidence focusing on his drug addiction — not on any alleged mental illness or intoxication. Dr. Tatro, defendant’s psychologist, testified that defendant was not a sociopath, but that his personality was disrupted by his chronic cocaine addiction. Defendant claimed that he used cocaine the day before the murder, but presented no evidence of drug intoxication at the time of the crimes. Defendant is of superior intelligence, does not suffer from mental illness, and was not intoxicated when he committed the crimes.
The trial court considered all of the mitigating factors and also considered the defendant’s behavior while in prison which “was neutral, neither aggravating or mitigating.” The trial court found that the “aggravating factors outweigh any and all mitigating factors. The mitigating factors are not sufficiently substantial to call for leniency.” Our review leads us to the same conclusion.
2. A Finding of a Mitigating Circumstance Under A.R.S. § 13-703(G)(1) Does Not Negate a Finding of an Aggravating Circumstance Under A.R.S. § 13-703(F)(6).
A.R.S. § 13-703(F)(6) provides: “The defendant committed the offense in an especially heinous, cruel or depraved manner.” The trial court’s findings under this statute have previously been approved by this court. Rossi I, 146 Ariz. at 365-66, 706 P.2d at 377-78; Rossi II, 154 Ariz. at 246, 741 P.2d at 1224. At the third sentencing, the trial court made the following findings under A.R.S. § 13-703(F)(6):
The offense was cruel, heinous and depraved. The offense was cruel because Mr. August was in pain before his ultimate death, he was conscious for a short time after the second shot, and he had time to reflect as to the uncertainty of his fate which is shown in his statement “you have my money, you shot me, what more do you want.” This evinces the victim’s mental anguish. Heinousness and depravity include:
*2801. The defendant was totally without regard for human life;
2. The defendant used bullets designed to inflict greater tissue damage on a human body;
3. The defendant bragged to friends about his act;
4. The defendant gave three of the bullets to Bill Nelson as a souvenir;
5. The defendant relished the murderous act;
6. The defendant complained that the bullets did not make as big a hole as they were supposed to;
7. The act was senseless and August was helpless, being 66 years old and in failing health. August was not in a position to thwart the robbery, and the defendant could have escaped without firing the fatal shot.
Defendant argues that the finding that his cocaine use significantly impaired his ability to conform his conduct to the law must negate the finding that the murder was cruel, heinous and depraved. We reject this argument because it runs counter to Arizona’s statutory death penalty scheme and because it also runs counter to the implicit holding of Rossi II.
Arizona’s death penalty scheme, the constitutionality of which has been upheld repeatedly, belies defendant’s argument. The statutes provide that if one or more of the aggravating circumstances set forth in A.R.S. § 13-703(F) is present, the perpetrator of a first degree murder who would otherwise be sentenced to life imprisonment becomes eligible for the death penalty. If one or more statutory aggravating circumstances exist and there are no mitigating circumstances, the death penalty is to be imposed. If, however, mitigating circumstances are found to exist, they must be weighed by the court to determine whether they are “sufficiently substantial to call for leniency.” A.R.S. § 13-703(E). Contrary to defendant’s assertion, there are no “mandatory” mitigating circumstances. Having due regard to the Arizona statutory scheme, we necessarily reject defendant’s argument that a finding of a mitigating circumstance under A.R.S. § 13-703(G)(1) negates the finding of an aggravating circumstance under A.R.S. § 13-703(F)(6).
We also note that, while the argument presently made on this point was not made in Rossi II, it was implicitly rejected in Rossi II. In Rossi II, after finding the mitigating circumstance of impaired capacity, we remanded for the purpose of having the trial court reweigh this circumstance against the aggravating circumstance which had been found. Such reweighing would be unnecessary if the mitigating circumstance negated use of the aggravating circumstance.
3, 4 and 5. Arizona’s Death Penalty Statute is Constitutional.
Defendant argues that Arizona’s death penalty law is unconstitutional for three reasons: (a) defendant’s case cannot be distinguished from those of defendants convicted of first degree murder who did not receive the death penalty; (b) the judge, and not the jury, determines whether aggravating and mitigating factors exist; and (c) should one or more aggravating factors be found, a mandatory presumption of death exists precluding individualized sentencing and shifting the burden of proof to the defendant to show why death should not be imposed.
Defendant’s first argument is fully answered by the United States Supreme Court’s decision in Pulley v. Harris, 465 U.S. 37, 104 S.Ct. 871, 79 L.Ed.2d 29 (1984). While a majority of this court believes proportionality reviews are desirable, the Supreme Court has made clear that they are not constitutionally required. See id. at 44-51, 104 S.Ct. at 876-79, 79 L.Ed.2d at 33-34. Defendant’s argument fails because it is based on the faulty premise that proportionality reviews are constitutionally required.
Second, it is not unconstitutional to allow the judge rather than the jury to determine the existence of aggravating and mitigating circumstances. Defendant raised the same issue in a more general context in Rossi I by arguing that the *281Arizona death penalty statute is unconstitutional because the jury takes no part in sentencing. This court noted that the argument had been previously considered and rejected in State v. Bracy, 145 Ariz. 520, 536, 703 P.2d 464, 480 (1985), cert. denied, 474 U.S. 1110, 106 S.Ct. 898, 88 L.Ed.2d 932 (1986).
The third constitutional argument was also addressed by this court in Rossi I. Defendant argued that the Arizona death penalty statute is unconstitutional because it requires imposition of the death penalty when one aggravating factor exists and there are no mitigating factors. This court rejected the argument, relying on Bracy and State v. Jordan, 137 Ariz. 504, 672 P.2d 169 (1983). See Rossi I, 146 Ariz. at 366, 706 P.2d at 378. In Jordan, this court relied on the discussion in State v. Gretzler, 135 Ariz. 42, 659 P.2d 1, cert. denied, 461 U.S. 971, 103 S.Ct. 2444, 77 L.Ed.2d 1327 (1983), for its holding. Jordan, 137 Ariz. at 508, 672 P.2d at 173. Gretzler held that Arizona’s death penalty statute provides constitutionally adequate standards to guide sentencing discretion. See 135 Ariz. at 53-56, 659 P.2d at 12-15.
Further, the United States Supreme Court recently addressed defendant’s second and third constitutional arguments in Walton v. Arizona, 497 U.S. 639, 110 S.Ct. 3047, 111 L.Ed.2d 511 (1990), aff'g State v. Walton, 159 Ariz. 571, 769 P.2d 1017 (1989) , and held that Arizona’s capital sentencing scheme is constitutional. We have also had occasion to reconsider and again reject defendant’s arguments in State v. Vickers, 159 Ariz. 532, 543-44, 768 P.2d 1177, 1188-89 (1989), cert. denied, — U.S. -, 110 S.Ct. 3298, 111 L.Ed.2d 806, (1990).
6. The Mitigating Circumstances Previously Found by This Court Do Not Bar Reimposition of the Death Penalty Under the Doctrines of Collateral Estoppel or Double Jeopardy.
After briefing and before oral argument, defendant filed “Supplemental Citations of Legal Authority.” He contended, for the first time, that the mitigating circumstances found by this court in Rossi II constitute the “law of the case” and that he may not again be sentenced to death because of the doctrines of collateral estoppel or double jeopardy. Failure to argue a claim in a brief constitutes abandonment and waiver of the claim. State v. McCall, 139 Ariz. 147, 163-64, 677 P.2d 920, 936-37 (1983), cert. denied, 467 U.S. 1220, 104 S.Ct. 2670, 81 L.Ed.2d 375 (1984). Because this is a death penalty case, we consider the late claim, but find it meritless.
This court did not undertake to sentence or resentence defendant in Rossi II. Instead, we vacated the previously-imposed death sentence and remanded for resentencing with directions to the trial court to consider the two additional mitigating factors that we found to exist. As we have already pointed out, there are no “mandatory” mitigating circumstances under Arizona law. In remanding, we expressly recognized the possibility of another death sentence being imposed. A remand would have been futile if the death penalty was foreclosed. Defendant has cited no authority that supports his argument under either collateral estoppel or double jeopardy and we know of none.
DISPOSITION
The death penalty was properly imposed and is affirmed.
CORCORAN, J., concurs.