
560 F.3d 299 (2009)
Rigoberto AVILA, Jr., Petitioner-Appellant-Cross-Appellee,
v.
Nathaniel QUARTERMAN, Director, Texas Department of Criminal Justice, Correctional Institutions Division, Respondent-Appellee-Cross-Appellant.
No. 07-70028.
United States Court of Appeals, Fifth Circuit.
February 17, 2009.
*302 Chris K. Gober, Law Office of Chris K. Gober, San Antonio, TX, Robin Norris, Hart & Norris, El Paso, TX, for Avila.
Thomas M. Jones, Postconviction Litigation Div., Austin, TX, for Quarterman.
Before WIENER, GARZA, and BENAVIDES, Circuit Judges.
BENAVIDES, Circuit Judge:
Petitioner Rigoberto Avila, Jr., ("Avila"), convicted of capital murder in Texas and sentenced to death, appeals the district court's denial of federal habeas relief with respect to his conviction. Respondent cross-appeals the district court's grant of habeas relief with respect to Avila's sentence.
Avila contends that the State suppressed a pathologist's expert opinion in violation of his due process rights. He also contends that his counsel's failure to discover the suppressed evidence constituted ineffective assistance in violation of the Sixth Amendment. Finally, Avila contends that because the jury was not required to find the mitigation issue at punishment beyond a reasonable doubt, his right to a jury trial was violated. Respondent cross-appeals, asserting that the district court erred in holding that the State's suppression of evidence during the punishment phase of trial violated Avila's due process rights. Concluding that the state court's adjudication of Avila's claims was not an unreasonable application of clearly established Federal law and that Avila has not made a substantial showing of the denial of a constitutional right, we AFFIRM in part, REVERSE in part, and DENY a Certificate of Appealability.

I. BACKGROUND
An El Paso grand jury charged Avila with the capital murder of Nicholas Macias, an individual younger than six years of age. Tex. Penal Code Ann. § 19.03(a).[1] Avila pleaded not guilty and was tried by a *303 jury. At trial, the evidence established that sometime between 6:00 and 6:15 p.m., on February 29, 2000, Marcelina Macias left her home to attend a class, leaving her 19-month-old son, Nicholas Macias, and his four-year-old brother, Dylan Salinas, in Avila's care. At 7:02 p.m., Avila called "911" and told the operator that the infant boy he was babysitting had stopped breathing. When the paramedics arrived, they administered emergency treatment to the child before transporting him to the hospital. While treating the boy, paramedics found a bruise on Nicholas's abdomen in the shape of a foot print. When they asked Avila about the bruise, he denied any knowledge of the marking. At the hospital, surgical attempts to save Nicholas's life by repairing the injury to Nicholas's intestines and other abdominal injuries were unsuccessful, and Nicholas died.
An autopsy revealed that major organs in Nicholas's body had been split in two by considerable blunt-force trauma consistent with being stomped by an adult. Specifically, the medical examiner reported that Nicholas "died of internal bleeding due to massive abdominal trauma resulting from blunt for[ce] injury." The surgeon's testimony likened Nicholas's injuries to those caused by such events as exiting an automobile traveling at sixty miles per hour or being dropped twenty feet.
Officer Jose Lopez testified that on February 29, 2000, he was dispatched to the home of a child who had stopped breathing. Avila told Lopez that he had been watching the television when Dylan came into the room and told him that Nicholas was not breathing. Dylan told Avila that "he had held [Nicholas'] mouth" and then Nicholas stopped breathing. Lopez then allowed Avila to drive to the hospital.
Detective Tony Tabullo arrived at the hospital to assess the situation. Because Avila was the last adult known to be with Nicholas, Tabullo asked him if he would be willing to discuss the incident with him at the Crimes Against Persons (CAP) offices. Avila initially gave a statement in which he denied injuring Nicholas. Subsequently, Tabullo received from other detectives Polaroid photographs which appeared to show an adult-sized footprint on Nicholas's stomach. Tabullo confronted Avila with the photographs, after which Avila orally admitted to stomping Nicholas. Tabullo typed the confession, which Avila signed. The confession was admitted at trial. During the guilt-innocence phase of trial, Avila testified that he did not injure Nicolas.
The jury found Avila guilty of capital murder. After the punishment phase of trial, the jury affirmatively answered the first special issue regarding whether Avila would be a continuing threat to society. The jury answered negatively the second special issue regarding whether mitigating circumstances warranted a sentence of life imprisonment. Avila appealed, and the Texas Court of Criminal Appeals affirmed his conviction and sentence. Avila v. State, 2003 WL 21513440 (Tex.Crim.App. July 2, 2003) (unpublished). Avila filed an application for writ of habeas corpus in Texas state court, which was denied by the Texas Court of Criminal Appeals. Avila then filed a petition for federal habeas corpus in district court. The district court denied his petition as to his conviction, but granted relief as to the punishment phase of trial. Avila v. Quarterman, 499 F.Supp.2d 713 (W.D.Tex.2007). Avila appeals the denial of habeas relief, and Respondent cross-appeals the grant of habeas relief.

II. STANDARDS OF REVIEW
Avila filed his 28 U.S.C. § 2254 petition for a writ of habeas corpus after *304 the effective date of the Antiterrorism and Effective Death Penalty Act (AEDPA). The petition, therefore, is subject to AEDPA. See Lindh v. Murphy, 521 U.S. 320, 336, 117 S.Ct. 2059, 138 L.Ed.2d 481 (1997). Pursuant to the federal habeas statute, as amended by AEDPA, we defer to a state court's adjudication of a petitioner's claims on the merits unless the state court's decision was: (1) "contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States"; or (2) "resulted in a decision that was based on an unreasonable determination of the facts in light of the evidence presented in the State court proceeding." 28 U.S.C. § 2254(d). A state court's decision is deemed contrary to clearly established federal law if it reaches a legal conclusion in direct conflict with a prior decision of the Supreme Court or if it reaches a different conclusion than the Supreme Court based on materially indistinguishable facts. Williams v. Taylor, 529 U.S. 362, 404-08, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). A state court's decision constitutes an unreasonable application of clearly established federal law if it is "objectively unreasonable." Id. at 409, 120 S.Ct. 1495. Further, pursuant to section 2254(e)(1), state court findings of fact are presumed to be correct, and the petitioner has the burden of rebutting the presumption of correctness by clear and convincing evidence. See Valdez v. Cockrell, 274 F.3d 941, 947 (5th Cir.2001).
Additionally, under AEDPA, a petitioner must obtain a Certificate of Appealability (COA) before he can appeal the district court's denial of habeas relief. See 28 U.S.C. § 2253(c); see also Miller-El v. Cockrell, 537 U.S. 322, 335-36, 123 S.Ct. 1029, 154 L.Ed.2d 931 (2003) ("[U]ntil a COA has been issued federal courts of appeals lack jurisdiction to rule on the merits of appeals from habeas petitioners."). As the Supreme Court has explained:
The COA determination under § 2253(c) requires an overview of the claims in the habeas petition and a general assessment of their merits. We look to the District Court's application of AEDPA to petitioner's constitutional claims and ask whether that resolution was debatable among jurists of reason. This threshold inquiry does not require full consideration of the factual or legal bases adduced in support of the claims. In fact, the statute forbids it.
Miller-El, 537 U.S. at 336, 123 S.Ct. 1029.
A COA will be granted only if the petitioner makes "a substantial showing of the denial of a constitutional right." 28 U.S.C. § 2253(c)(2). "A petitioner satisfies this standard by demonstrating that jurists of reason could disagree with the district court's resolution of his constitutional claims or that jurists could conclude the issues presented are adequate to deserve encouragement to proceed further." Miller-El, 537 U.S. at 327, 123 S.Ct. 1029 (citation omitted). "The question is the debatability of the underlying constitutional claim, not the resolution of that debate." Id. at 342, 123 S.Ct. 1029. "Indeed, a claim can be debatable even though every jurist of reason might agree, after the COA has been granted and the case has received full consideration, that petitioner will not prevail." Id. at 338, 123 S.Ct. 1029. Moreover, "[b]ecause the present case involves the death penalty, any doubts as to whether a COA should issue must be resolved in [petitioner's] favor." Hernandez v. Johnson, 213 F.3d 243, 248 (5th Cir.2000) (citation omitted).

III. SUPPRESSION OF EVIDENCE
Avila contends that the State failed to disclose certain evidence in violation *305 of his due process rights. The district court granted a COA as to this issue. The State has a duty to disclose evidence favorable to the accused that is material to guilt or punishment. See Brady v. Maryland, 373 U.S. 83, 86-87, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963). To establish this due process violation, an accused must show that the State withheld evidence, that the evidence was favorable, and that the evidence was material to the defense. Little v. Johnson, 162 F.3d 855, 861 (5th Cir. 1998). Additionally, the petitioner must show that his belated "discovery of the allegedly favorable evidence was not the result of a lack of due diligence." Rector v. Johnson, 120 F.3d 551, 558 (5th Cir. 1997).[2]
The Supreme Court has explained that a new trial is not "automatically require[d] ... whenever a combing of the prosecutors' files after the trial has disclosed evidence possibly useful to the defense but not likely to have changed the verdict." Giglio v. United States, 405 U.S. 150, 154, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972) (internal quotation marks and citation omitted). The standard for determining "materiality is a `reasonable probability' of a different result." Kyles v. Whitley, 514 U.S. 419, 434, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995). In assessing Brady materiality, "[t]he question is not whether the defendant would more likely than not have received a different verdict with the evidence, but whether in its absence he received a fair trial, understood as a trial resulting in a verdict worthy of confidence." Id. In other words, "[a] `reasonable probability' of a different result is accordingly shown when the [State's] evidentiary suppression `undermines confidence in the outcome of the trial.'" Id. (quoting United States v. Bagley, 473 U.S. 667, 678, 105 S.Ct. 3375, 87 L.Ed.2d 481 (1985)).
Avila contends that the prosecution team violated his due process rights by failing to disclose the expert opinion of Dr. Harry Wilson, a pathologist. In his capacity as a physician at Memorial Hospital, Dr. Wilson prepared a pathology report regarding the victim's injuries around the time of the victim's death.[3] The State designated Dr. Wilson as an expert witness based on this involvement in the victim's medical care. During the investigation of the instant offense, the State contacted Dr. Wilson and met with him to discuss the case. Dr. Wilson was present when Dr. Fausto Rodriguez, the defense's pathologist, conducted a second autopsy on the victim.
The State did not, however, call Dr. Wilson to testify at either phase of Avila's trial. Instead, the State called Dr. Juan Contin, the medical examiner in El Paso County. Dr. Contin testified that he performed an autopsy on the victim and that the victim had several bruises, with the largest one on the right side of the abdomen. Dr. Contin explained that the injury was caused by either a "blunt force instrument *306 or object." The shape of the injury was roughly oval and consistent with the shape of a shoe. The blunt force applied to the child's abdomen entailed "considerable force" and compressed his internal organs against the spine, resulting in the organs' detachment from the spine. The duodenum and the large bowel were detached, and the pancreas was "cut in two." There was a second bruise on the left side of the abdomen that "was not very big." The third bruise was in the upper lumbar region in the center of the back.[4] Dr. Contin further testified that the three injuries were likely caused by separate blows. There are "no specific techniques to age bruises"; however, the abdominal bruises were "very fresh because you cannot survive that injury for long." In Dr. Contin's opinion, the abdominal bruises were inflicted "probably within an hour" of the victim arriving at the hospital. After sustaining these injuries, a child would not be walking or playing because such injuries would send a child into shock. These types of injuries are usually seen in children who were not buckled during a traffic accident. Dr. Contin did not think the injuries happened accidentally because "they are too big and they are too many." On cross-examination, Dr. Contin testified that the victim's mother was capable of inflicting the injuries.
The State also called Dr. George Raschbaum, a board certified pediatric surgeon. On the night of the offense, Dr. Raschbaum saw Nicholas in the emergency room and described him as non-responsive and having a distended abdomen. X-rays showed that there was "free air" inside the abdomen, and the likelihood of survival was minimal. Nonetheless, because the child was only 19 months old, Dr. Raschbaum performed surgery. The surgery was not an attempt to "fix things permanently but fix things so that we could achieve stability if there was any way humanly possible." Dr. Raschbaum testified with respect to observing the duodenum and pancreas being "split into two." Dr. Raschbaum, like Dr. Contin, testified that this type of injury is not uncommon in "high speed accidents." He had observed "almost identical injuries" when a person "had jumped out of a vehicle going 60 miles an hour." Despite resuscitative efforts, the victim expired. In his opinion, a four-year old was not capable of inflicting these injuries in a normal household. He further opined that "[t]he magnitude of impact is so dramatic that I can't imagine that this could be any kind of accident." "It would be a strong force, like a stomping force."
The defense called Dr. Fausto Rodriguez, the pathologist who had conducted a second autopsy on the victim. Dr. Rodriguez testified that he reviewed Dr. Contin's autopsy report, Dr. Raschbaum's surgical report, Dr. Wilson's report, and the microscopic slides from the hospital. Dr. Rodriguez testified that "all the [fatal] injuries  internal injuries in this case could be explained by a single traumatic event." Dr. Rodriguez testified that the injuries could have been inflicted by an adult accidentally falling on the child but he could not make such a determination. He further opined that the older bruises on the victim's skull were "[p]robably just the daily activities of the child." It was very unlikely that the older bruises were caused by blunt force trauma. On cross-examination, Dr. Rodriguez reiterated that the injuries could have been the result of a single blow.

A. Guilt-Innocence Phase of Trial
Avila contends that the prosecution suppressed Dr. Wilson's opinion that *307 the fatal injuries were inflicted by one stomp, as opposed to the State's contention at trial that the victim was stomped or kicked three times. Dr. Wilson's opinion, Avila argues, would have enabled counsel "to argue persuasively that [his] conduct resulted from a sudden loss of control, not a conscious desire to kill." In an affidavit submitted during post-conviction proceedings, Dr. Wilson stated that:
The nature of the child's injuries, the manner and circumstances under which I understand that they were inflicted, and the reported response of Rigoberto Avila to those injuries are all consistent with cases in which caretakers of children, under conditions of stress, lose control and act without reflection in a manner which endangers the safety of the infant or child under their care and control.
As set forth above, Avila must first show that the State withheld the evidence. The state habeas court made a finding that "[b]efore and during trial, the State was not aware of the opinion now expressed by Dr. Harry Wilson that a single blow caused the fatal injuries sustained by the victim." Avila has the burden of rebutting this presumed correct factual finding with clear and convincing evidence. Both trial prosecutors submitted affidavits stating that Dr. Wilson never stated to them or any member of the prosecution team during the trial of the case that a single blow caused all the injuries to Nicholas. Also, Dr. Rodriguez, the defense's pathologist, submitted an affidavit stating that "Dr. Wilson told me that it was his opinion that the fatal injuries to Nicholas could have been caused by a single blow or they could have been caused by multiple blows." Further, there is no written documentation of Dr. Wilson's single-blow opinion from the time of the trial.
On the other hand, Dr. Wilson has submitted an affidavit indicating that he communicated his single-blow theory to the prosecutors. This affidavit, which was not executed until long after the trial, does not constitute clear and convincing evidence sufficient to rebut the presumption of correctness afforded the state court's finding under § 2254(e)(1). See Pippin v. Dretke, 434 F.3d 782, 792 (5th Cir.2005) (explaining that a state trial court's credibility determination based on conflicting evidence is "virtually unreviewable" by a federal court).[5]
Nonetheless, Avila contends that Dr. Wilson's opinion should be imputed to the prosecutors because Dr. Wilson was part of the prosecution team. It is well settled that if a member of the prosecution team has knowledge of Brady material, such knowledge is imputed to the prosecutors. United States v. Antone, 603 F.2d 566, 569 (5th Cir.1979); Schneider v. Estelle, 552 F.2d 593, 595 (5th Cir.1977). In Antone, the issue was whether knowledge held by state law enforcement agents could be imputed to the federal prosecutors. 603 F.2d at 569-70. In that case, FBI agents and Florida state law enforcement agents had formed a joint task force to investigate the murder of a former police officer. Id. at 568. The government *308 argued that the knowledge should not be imputed because the two agencies represent entirely separate sovereigns. Id. at 570. This Court declined to impose a bright line rule, explaining that such a rule "would artificially contort the determination of what is mandated by due process." Id. Instead, we concluded that the determination should be made on a "case-by-case analysis of the extent of interaction and cooperation between the two governments." Id. After recognizing the extensive cooperation between the agencies, we concluded that the state investigators acted as agents of the federal government pursuant to the principles of agency law. Id.
Thus, in the instant case, we must determine whether Dr. Wilson was part of the "`prosecution team[,]' which includes both investigative and prosecutorial personnel." Id. at 569. "The question is not whether any expert witness can be treated as `an arm of the prosecution,' but whether [the expert] played such a role." United States v. Stewart, 323 F.Supp.2d 606, 618 (S.D.N.Y.2004).[6] In Hill v. Johnson, this Court addressed a claim that an expert witness acted as an arm of the prosecution. 210 F.3d 481, 488-89 (5th Cir.2000). During the sentencing phase of Hill's trial, a psychiatric expert testified for the State with respect to the probability that Hill would be a danger to society in the future. Id. at 488. Relying on language in Estelle v. Smith, 451 U.S. 454, 467, 101 S.Ct. 1866, 68 L.Ed.2d 359 (1981), Hill contended that because the psychiatric expert testified for the prosecution that Hill would be a future danger, the expert transformed into an agent of the State. Hill, 210 F.3d at 489. Estelle v. Smith involved a court-ordered psychiatric exam given without advising the defendant of his rights. 451 U.S. at 466, 101 S.Ct. 1866. The psychiatrist subsequently testified with respect to the results of that examination. Id. at 459-60, 101 S.Ct. 1866. This Court quickly rejected Hill's argument, recognizing that the psychiatrist did not examine Hill but rather testified in response to hypotheticals with respect to future dangerousness. Hill, 210 F.3d at 489 n. 5. Unlike the petitioner in Hill, Avila is not relying on Estelle v. Smith. Nonetheless, Hill suggests that merely testifying as an expert witness for the State does not necessarily transfer an expert witness into an "arm of the state." Hill, 210 F.3d at 488-89. Further, deeming every expert witness to be an arm of the state would be inconsistent with our previously cited precedent declining to impose a bright line rule and, instead, directing that a case-by-case analysis is better suited for determining whether an individual is deemed part of the prosecution team pursuant to the principles of agency law. See Antone, 603 F.2d at 570.
Similarly, in United States v. Stewart, in the context of a use-of-perjured-testimony claim, the Second Circuit rejected a defendant's argument that the expert witness's knowledge should be imputed to the prosecution. 433 F.3d 273, 297-99 (2d Cir.2006). There, the witness was an expert with respect to ink, and the district court had concluded that the expert witness had "acted as an ordinary expert witness and not as part of the `prosecution team.'" Id. at 298. The Second Circuit stated that the district court's findings "demonstrate that [the expert's] role was limited to matters *309 concerning his area of expertise." Id. The expert witness's duties included analyzing a document, explaining the forensic ink tests that had been conducted, discussing possible testimony that the defense expert would give, assisting the prosecutors in developing cross-examination questions with respect to technical aspects of testing, taking part in a mock examination prior to trial, and testifying at trial regarding the tests and his resulting conclusions. Id. The Second Circuit held that none of those duties "suggest[ed] that [the expert] was in any way involved with the investigation or presentation of the case." Id. at 298-99. Concluding that the witness had "acted only in the capacity of an expert witness, ... and not as a `fully functioning member of the prosecution team,'" the Second Circuit held that the expert's knowledge should not be imputed to the prosecutors. Id. at 299.
Here, we are not persuaded that Dr. Wilson became part of the prosecution team. Indeed, Dr. Wilson's own affidavit does not portray his role as anything but a pathologist. Because Dr. Wilson did not become part of the prosecution team, his opinion is not imputed to the State. Thus, we hold that the district court erred in finding the state court's conclusion was "an objectively unreasonable application of clearly established federal constitutional law." Avila, 499 F.Supp.2d at 748.[7]
In the alternative, even assuming arguendo that Dr. Wilson did become part of the prosecution team and thus his opinion would be imputed to the trial prosecutors, we conclude that the state court's conclusion that his opinion was not material is not unreasonable. The Supreme Court has explained that to determine materiality with respect to a Brady claim, "the reviewing court may consider directly any adverse effect that the prosecutor's failure to respond might have had on the preparation or presentation of the defendant's case." United States v. Bagley, 473 U.S. 667, 683, 105 S.Ct. 3375, 87 L.Ed.2d 481 (1985). More specifically, this Court "should assess the possibility that such effect might have occurred in light of the totality of the circumstances and with an awareness of the difficulty of reconstructing in a post-trial proceeding the course that the defense and the trial would have taken had the defense" been aware of the suppressed evidence. Id.
To the extent the undisclosed evidence is cumulative, "no Brady violation occurs." Spence v. Johnson, 80 F.3d 989, 995 (5th Cir.1996). At trial, the defense called a pathologist, Dr. Rodriguez, who testified that "all the [fatal] injuries  internal injuries in this case could be explained by a single traumatic event." Dr. Rodriguez never testified that the injuries were caused by multiple blows. Indeed, in his affidavit, even Dr. Wilson admits that he "cannot say with certainty that multiple blows did not occur." Dr. Rodriguez's opinion was consistent with Dr. Wilson's opinion and, thus, to some extent cumulative.
Avila acknowledges Dr. Rodriguez's opinion but nonetheless contends that Dr. Rodriguez did not have as much experience or prominence in the field as Dr. Wilson. Avila also points to Dr. Wilson's theory that caregivers of small children sometimes find themselves in stressful situations and act violently but without the intent to kill. Thus, he argues that the *310 opinion is material to the guilt phase because it might have persuaded a juror that he was not guilty of capital murder but instead a lesser offense. At trial, however, Avila testified before the jury that he did not stomp the baby. Dr. Wilson's theory therefore directly conflicts with the defense at the guilt-innocence phase of the trial. Avila admits as much but contends that, had his defense attorneys known of Dr. Wilson's stressed-caregiver theory, they perhaps would have been able to persuade him not to testify.[8] Avila fails to point to any evidence supporting this speculative contention. Avila further contends that, had counsel been aware of Dr. Wilson's opinion, "it is likely they would have altered their defensive strategy to focus more on the question of Avila's mental state than on his conduct." Again, Avila points to no evidence in support of this contention. Further, this contention ignores defense counsel's affidavit that was submitted during the post-conviction proceedings. In his affidavit, Avila's trial attorney stated that his strategy was not to put on psychological or psychiatric evidence to prevent the State from examining Avila. Although counsel's affidavit did not specifically address Dr. Wilson's opinion, it certainly does not offer support for the contention that counsel, with hindsight, would have changed his strategy to focus on Avila's psychological or mental state.
Finally, we are mindful that Dr. Wilson's opinion regarding the stressed care-giver pertains to Avila's intent or state of mind. Dr. Wilson's expertise is pathology, not psychology or psychiatry. Under these circumstances, the state court's conclusion that the evidence was not material at the guilt-innocent phase of trial is not objectively unreasonable. The district court's denial of habeas relief on this claim is affirmed.[9]

B. Punishment Phase of Trial
The State cross-appeals the grant of habeas relief based upon the district court's ruling that the State suppressed Dr. Wilson's opinion with respect to the punishment phase of trial. As set forth above, we hold that Dr. Wilson's actions were not sufficient for him to be deemed part of the prosecution team.
In the alternative, again assuming arguendo that Dr. Wilson was part of the prosecution team and thus knowledge of his opinion must be imputed to the prosecutors, we find that it was not material. At the conclusion of the punishment phase of trial, the jurors had to answer two *311 questions: (1) do you find from the evidence beyond a reasonable doubt that there is a probability that the defendant Rigoberto Avila would commit criminal acts of violence that would constitute a continuing threat to society?; and (2) do you find from the evidence, taking into consideration all the evidence including the circumstances of the offense, the defendant's character and background and the personal moral culpability of the defendant, that there is a sufficient mitigating circumstance or circumstances to warrant that a sentence of life imprisonment rather than a death sentence should be imposed? The jury unanimously answered the first special issue regarding whether Avila would be a future danger "yes." The jury unanimously answered the second special issue regarding whether mitigating circumstances warranted a life sentence "no."
Although the court below ruled that Avila had not shown materiality with respect to the guilt phase of trial, the court ruled that Avila successfully demonstrated materiality with respect to the punishment phase of trial. Based on this conclusion, the district court conditionally granted habeas relief, ruling that Avila was entitled to a new trial on the issue of punishment.[10] We conclude that the district court erred in finding unreasonable the state court's conclusion that Avila had failed to satisfy the materiality prong. As previously discussed, to determine materiality, this Court should assess what course the defense would have taken during the punishment phase if Dr. Wilson's opinion and theory had been known. See Bagley, 473 U.S. at 683, 105 S.Ct. 3375.
The State argues that the district court erred in "rewriting the defense's trial strategy." We agree. The district court found that "Dr. Wilson's experienced professional opinion would have lent support to the single-blow theory, and undermined the prosecution's multiple-blow theory" and that his opinion was relevant to the special issues. Avila, 499 F.Supp.2d at 758. In finding that the evidence satisfied the materiality prong, the district court explained as follows:
Dr. Wilson's opinion regarding the potentially unpremeditated nature of petitioner's violent actions (based on Dr. Wilson's extensive experience with injuries to children resulting from violent, albeit unpremeditated, misconduct by care givers) dove-tailed remarkably well with petitioner's description of his offense contained in petitioner's second written statement to police. In that regard, Dr. Wilson could have furnished petitioner's defense counsel with a rational explanation for petitioner's violent conduct which was vastly more beneficial to petitioner's efforts to obtain a favorable verdict on the future dangerousness and mitigation special issues than petitioner's trial counsel's dubious suggestion to the jury that petitioner had murdered Nicolas in a jealous rage.
Id.
Although the district court concludes that Dr. Wilson's opinion would have afforded a better defense strategy at the punishment phase, Avila has offered no evidence demonstrating that counsel would have changed strategies. As referenced above, Avila's defense counsel executed an affidavit during post-conviction proceedings. In pertinent part, the affidavit provides:
In the investigation of the case, and in preparation for the presentation of mitigating evidence at the punishment phase, I made the strategic decision to not put on any psychiatric or psychological evidence to prevent the State from *312 being able to examine the defendant under Soria v. State, 933 S.W.2d 46 (Tex. Crim.App.1996), and gain possible damaging evidence. That strategy worked as neither side presented such evidence at trial.
* * *
During my cross-examination of Marcelina Macias (Avila's girlfriend and mother of the deceased, Nicholas Macias), I made the strategic decision to bring out the pressure that she had put on Avila in her relationship. She knew that he was jealous of her and of her alleged relationships with other men, and I had her admit that she knew he knew about those relationships, she knew he was jealous, and he was a ticking time bomb at the time of the offense. I did this in order to give the jury a motive for the offense, namely, to show that Avila was not a cold-blooded killer who committed the offense for no reason. Bringing out the pressure he was under in his relationship with Marcelina was done in hopes of showing that the alleged offense was spur-of-the-moment, and something that was a total aberration in Avila's life. This evidence, I hoped, would show a one-time error in judgment with a likelihood that he would not be a danger in the future. In my opinion, if I had not brought that evidence out, the jury would have just thought that Avila was a cold-blooded killer who committed the offense with deliberation and would likely be a danger in the future. I believe it was a valid strategical decision at the time that it was made.
Avila had the burden of demonstrating how the evidence would have changed the course of the punishment hearing, and defense counsel's affidavit does not indicate that he would have changed his strategy at the punishment phase. Moreover, as previously noted, Dr. Wilson's expertise is pathology, not psychology or psychiatry. Simply put, under these circumstances, our confidence in the verdict is not undermined by the State's alleged suppression of a pathologist's opinion as to Avila's intent at the time of the murder. Avila has not shown that the state court's conclusion was an unreasonable application of clearly established Federal law. We therefore reverse the district court's grant of habeas relief with respect to Avila's Brady claim during the punishment phase of trial.

IV. INEFFECTIVE ASSISTANCE OF COUNSEL
Avila contends that defense counsel's failure to discover Dr. Wilson's opinion constituted ineffective assistance of counsel during both phases of the murder trial. To establish ineffective assistance of counsel, Avila must show (1) defense counsel's performance was deficient and (2) this deficient performance prejudiced the defense. Strickland v. Washington, 466 U.S. 668, 687, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). We must find that trial counsel "made errors so serious that counsel was not functioning as the `counsel' guaranteed... by the Sixth Amendment." Id. The Supreme Court instructs courts to look at the "norms of practice as reflected in the American Bar Association and the like" and to consider "all the circumstances" of a case. Id. at 688, 104 S.Ct. 2052. While "[j]udicial scrutiny of counsel's performance must be highly deferential," Avila can demonstrate deficient performance if he shows "that counsel's representation fell below an objective standard of reasonableness." Id. at 688, 104 S.Ct. 2052. However, "[t]here is a `strong presumption that counsel's conduct falls within the wide range of reasonable professional assistance.'" United States v. Webster, 392 F.3d 787, 793 (5th Cir.2004) (quoting *313 Strickland, 466 U.S. at 689, 104 S.Ct. 2052). Strickland's "prejudice" prong requires a reasonable probability that, but for the deficient performance of his trial counsel, the outcome of his capital murder trial would have been different. Strickland, 466 U.S. at 694, 104 S.Ct. 2052. "A reasonable probability is a probability sufficient to undermine confidence in the outcome." Id.

A. Guilt-Innocence Phase of Trial
Avila contends that counsel rendered ineffective assistance by failing to discover the opinion of Dr. Wilson during the guilt-innocence phase of the trial. The district court denied a COA with respect to this issue. We must therefore determine whether Avila has made a substantial showing of the denial of effective assistance during the guilt-innocence phase.

1. Performance Prong
Avila faults counsel for not discovering Dr. Wilson's opinion that the victim's fatal injuries most likely were inflicted by one stomp as opposed to three stomps or kicks. Avila contends that Dr. Wilson's one-stomp theory would have impeached the testimony the State elicited from Dr. Contin that there were multiple blows. Of course, defense counsel was aware of the theory that Avila could have inflicted the injuries with only one stomp based on defense expert Dr. Rodriguez's testimony that the victim's internal injuries could be explained by one single event of trauma.[11] Counsel, though undisputably aware that the injuries could have been inflicted with one stomp, never argued that theory to the jury. We now consider whether Avila has made a substantial showing that counsel's performance was deficient.
Trial counsel's defense strategy is made clear in his closing argument. During the closing argument at the guilt-innocence phase, defense counsel urged the jury to find Avila not guilty, which was consistent with Avila's trial testimony. Defense counsel argued that there was no evidence that Avila had any motive to kill the 19-month old child. Counsel spoke of the four character witnesses who testified that Avila always treated children well. Counsel highlighted Avila's lack of any felony convictions or crimes involving moral turpitude. Counsel reminded the jury that a Department of Public Safety crime lab employee testified that his comparison of the soles of Avila's shoes did not match the impression of the victim's body. Counsel sought to create reasonable doubt by asking why the State had not called the victim's mother to testify regarding either the night of the offense or the older bruises on her son's head. Counsel also questioned why the State had not had compared the victim's mother's shoes to the victim's body. Counsel discounted the testimony of Dylan, the victim's brother, asserting that Dylan's testimony indicated he had no independent memory of what happened. Counsel argued that Avila's call to the emergency number 911 and his efforts to revive Nicholas would be inconsistent with guilt. Counsel emphasized that Avila did not flee the scene but instead, drove the victim's brother, Dylan, to the hospital.
Counsel pointed out that Avila's first statement after interrogation provided that he did not hit Nicholas and had no knowledge of the bruise. That statement, counsel noted, bore Avila's initials next to the paragraphs. The second statement, in which Avila admitted that he "stamped on [Nicholas] with his right foot," contains no such initials on the individual paragraphs. *314 Counsel stated to the jury that the second statement was "generate[d]" by Detective Tony Tabullo and signed by a sleepless Avila at 6:00 the next morning. Counsel argued that Detective Tabullo told Avila that he "had only made some grammar changes, ... just sign it, you don't need to read it." Counsel further argued that the incriminating statement, unlike the first statement, contained no initials and no notation of the time when it was completed. Counsel reminded the jury that the State, not the defense, had the burden of proving the offense beyond a reasonable doubt. Counsel concluded by asking the jury for a verdict of not guilty.
Counsel's chosen strategy comported with Avila's trial testimony in which he denied striking the victim.[12] We are mindful that "Strickland does not allow second guessing of trial strategy and must be applied with keen awareness that this is an after-the-fact inquiry." Granados v. Quarterman, 455 F.3d 529, 534 (5th Cir. 2006). Dr. Wilson's opinion regarding Avila's intent to kill would have been in direct conflict with Avila's testimony that he did not kill the victim. Simply because counsel's strategy did not successfully convince the jury does not mean counsel's performance was deficient. Counsel's strategy falls within the range of reasonable professional assistance. We conclude that counsel's "failure" to discover Dr. Wilson's opinion did not constitute deficient performance in light of the defense used at trial.

2. Prejudice Prong
Because we find no deficient performance, we need not reach the prejudice prong. Nonetheless, "[t]he test for prejudice under Brady and Strickland is the same." Gonzales v. Quarterman, 458 F.3d 384, 390 (5th Cir.2006). Having already found that Dr. Wilson's opinion did not show prejudice under Brady, we would also find no prejudice under Strickland. Avila has not made a substantial showing with respect to ineffective assistance of counsel during the guilt phase. We therefore DENY a COA with respect to this claim.

B. Punishment Phase of Trial
Avila next contends that counsel's failure to discover Dr. Wilson's opinion constituted ineffective assistance during the punishment phase of trial. The court below granted a COA with respect to this issue. As previously set forth, we are constrained under AEDPA to determine whether the state court's conclusion is "contrary to, or involved an unreasonable application of, clearly established Federal law." § 2254(d).

1. Performance Prong
We must first consider whether defense counsel's failure to discover Dr. Wilson's opinion constituted deficient performance.[13] Again, although defense counsel was aware of the theory that all three bruises could be a result of one blow, counsel did not argue that to the jury *315 during the punishment phase. Instead, counsel's argument focused on the prosecutor's failure to demonstrate that Avila would commit further acts of violence. Counsel argued that Avila's actions immediately after the killing showed that he had a conscience. Counsel emphasized that the prosecution was not able to find evidence of any other bad act that Avila had committed. Counsel argued that the killing was an isolated incident and that even the victim's mother testified that Avila had been good to her children. Counsel referred to the evidence that Avila was a good father to his son and paid child support. Counsel pointed to the respected members of the community who testified on behalf of Avila. Counsel closed by asking the jury to vote "no" with respect to the future dangerousness special issue.
Counsel's decision not to argue the one-stomp theory fell within the range of reasonable professional assistance. We conclude that counsel's "failure" to discover Dr. Wilson's opinion did not constitute deficient performance in light of the strategy at the punishment phase.[14]

2. Prejudice Prong
As previously set forth, "[t]he test for prejudice under Brady and Strickland is the same." Gonzales, 458 F.3d at 390. Having already found that Dr. Wilson's opinion did not show prejudice under Brady, we would also find no prejudice under Strickland. Accordingly, Avila has not shown that the state court's ruling was an unreasonable application of Federal law.

V. RIGHT TO JURY TRIAL
Finally, Avila contends that allowing a sentence of death without a jury finding beyond a reasonable doubt that there were no mitigating circumstances sufficient to warrant a sentence of life imprisonment violated his Sixth and Fourteenth Amendment right to due process and a fair trial. The district court denied a COA with respect to this issue.
This Court previously has rejected this claim. Rowell v. Dretke, 398 F.3d 370, 378 (5th Cir.2005) (opining that "[n]o Supreme Court or Circuit precedent constitutionally requires that Texas's mitigation special issue be assigned a burden of proof"). We are bound by our precedent and must conclude that Avila has not made a substantial showing with respect to the denial of his right to a jury finding of beyond a reasonable doubt. Scheanette v. Quarterman, 482 F.3d 815, 828-29 (5th Cir.2007) (ruling that reasonable jurists would not debate the district court's rejection of petitioner's claim that the jury was required to find the mitigating issue beyond a reasonable doubt).

VI. CONCLUSION
For the above reasons, the district court's judgment is AFFIRMED in part and REVERSED in part. Avila's application for a COA is DENIED. Avila is denied any relief on his section 2254 claims.
NOTES
[1]  The facts are taken largely verbatim from the Texas Court of Criminal Appeals opinion on direct appeal. Avila v. State, No. 74142, 2003 WL 21513440, at *1-3 (Tex.Crim.App. July 2, 2003) (unpublished).
[2]  We note that "[t]he State's good or bad faith in withholding favorable evidence is irrelevant." Id.
[3]  In the clinical history section of his report, Dr. Wilson stated that the victim had "severe abdominal trauma allegedly occurring in a care-taking setting as possible non-accidental inflicted trauma." The report described the surgery performed and the described the injuries in detail. The report provided that "[m]ultiple sites of acute serosal and surface mural hemorrhage are identified in the specimen." The report also provided that "[i]n general the surface sites of hemorrhage on these various gastrointestinal portions are considered to be a result of surface impact trauma with the diffuse mucosal hemorrhage representing a widespread effect of diffuse ischemic injury secondary to a traumatically interrupted blood supply."
[4]  The victim also had five bruises on his skull. Those appeared to be two or three days old.
[5]  Avila complains that the factual findings were made without an evidentiary hearing. This Court has stated that a "full and fair hearing does not necessarily require live testimony." Clark v. Johnson, 202 F.3d 760, 766 (5th Cir.2000). "Indeed, we have repeatedly found that a paper hearing is sufficient to afford a petitioner a full and fair hearing on the factual issues underlying the petitioner's claims, especially where, as here, the trial court and the state habeas court were one in the same." Id. We note that Judge Woody Densen presided over both Avila's capital murder trial and the state habeas proceedings.
[6]  The State relies on Summers v. Dretke, in which this Court rejected a Brady claim involving an expert witness. 431 F.3d 861 (5th Cir.2005). In Summers, the petitioner did not argue that the relevant knowledge should be imputed to the prosecutors and therefore, its analysis is inapposite. Id. at 874 (explaining that petitioner "has made no such claim" after citing a case that involved imputation of knowledge).
[7]  Although the district court found that Dr. Wilson acted as an arm of the prosecution, the district court ultimately denied relief on this claim because it found that the state court's conclusion that the evidence was not material was not an unreasonable application of clearly established Federal law. Avila, 499 F.Supp.2d at 757.
[8]  Pursuant to AEDPA, a district court is precluded from granted relief with respect to an unexhausted claim. 28 U.S.C. § 2254(b)(1)(A). "To satisfy the exhaustion requirement, the petitioner must fairly present the substance of his federal claim to the highest state court." Ries v. Quarterman, 522 F.3d 517, 523 (5th Cir.2008). The State contends that Avila, in his state court habeas pleadings, did not assert that the prosecutor withheld evidence of Dr. Wilson's stressed care-giver theory. In his reply brief, Wilson does not claim that he did assert the stressed care-giver theory in his state writ application; instead, he relies on "Dr. Wilson's state court affidavit and related exhibits" as urging the stressed care-giver theory. We are not necessarily satisfied that Avila exhausted the stressed care-giver theory in state court. Nonetheless, because we ultimately deny relief, we may address the claim even if it is unexhausted. See Granados v. Quarterman, 455 F.3d 529, 536 (5th Cir.2006).
[9]  In light of our determination with respect to materiality, we need not reach the question of whether the failure to discover the evidence was a result of a lack of due diligence. See Rector v. Johnson, 120 F.3d 551, 558 (5th Cir.1997). Nonetheless, we note that Avila's claim that counsel rendered ineffective assistance during the guilt-innocence phase by failing to discover Dr. Wilson's opinion, discussed infra, tends to support a conclusion of lack of due diligence.
[10]  This Court granted the Respondent's motion to stay the ruling pending appeal.
[11]  Additionally, Avila's confession arguably indicates that it was one blow: "I walked over to [Nicholas] and stamped on him with my right foot."
[12]  We note that "[t]he decision of whether to testify belongs to the defendant and his lawyer cannot waive it over his objection." United States v. Mullins, 315 F.3d 449, 454 (5th Cir.2002).
[13]  Avila contends that the prosecution's case for death was based on the theory that Avila stomped and kicked the victim repeatedly. We believe Avila overstates the importance of the number of blows described by the prosecutor in closing argument. The prosecutor did assert that Avila kicked the child twice and then stomped his abdomen. Although the prosecution's closing arguments fill 32 pages of transcript, the prosecutor's description of the killing is only about one-half of one page.
[14]  We note that counsel's decision to not put on any psychiatric evidence at the punishment phase successfully kept the State from discovering potentially aggravating evidence that surfaced during post-conviction proceedings regarding Avila having a suicidal ideation and his diagnosis of Major Depressive Disorder.
