POSNER, Chief Judge.
William Bracy and Roger Collins were convicted in an Illinois state court in 1981 of three murders committed the previous year. They were sentenced to death and after exhausting their state remedies (see People v. Collins, 106 Ill.2d 237, 87 Ill.Dec. 910, 478 N.E.2d 267 (1985); 153 Ill.2d 130, 180 Ill.Dec. 60, 606 N.E.2d 1137 (1992)) sought habeas corpus in federal district court. Judge Hart denied them relief, United States ex rel. Collins v. Welborn, 868 F.Supp. 950 (N.D.Ill.1994), and they have appealed, arguing that the state denied them due process of law both at their trial and in the sentencing hearing.
The victims had been taken, bound, from an apartment in a building on the south side of Chicago and had been driven to a viaduct and there shot to death with pistols and a shotgun. The main prosecution witness was Morris Nellum, an accomplice who testified for the government in exchange for being charged only with concealing a felony and promised that the state would recommend a sentence of only three years. (In fact he received only two and a half years — and of probation, not prison.) Nellum testified that Collins had summoned him to the apartment, where he had watched as the victims were led out of the apartment and into a waiting automobile by Bracy, Collins, and a third man, Hooper. (Hooper was tried separately, convicted, and sentenced to death. See People v. Hooper, 133 Ill.2d 469, 142 Ill.Dec. 93, 552 N.E.2d 684 (1989), affirming the conviction but vacating the death sentence. On remand, Hooper was again sentenced to death, and this time the Supreme Court of Illinois affirmed. 1996 WL 30547 (Ill. Jan. 25, 1996).)
Collins told Nellum to drive Collins’s car, which was parked near the apartment budding, to the viaduct. Collins and Hooper then got into the car that contained the three victims and drove away, followed by Bracy in another car. Nellum waited a few minutes and then drove to the viaduct as well. As he approached it, he heard shots. He stopped the ear. Collins jumped in and they sped off. Later the two drove to Lake Michigan and Collins threw two pistols into the lake. Nel-lum, after he was arrested, told the police where the guns had been dumped, and the police found them there. Bullets found in the bodies of the dead men were of the type fired by these guns, although the guns had so deteriorated as a result of their prolonged immersion in the lake that no positive ballistics identification was possible.
Nellum’s testimony was corroborated not only by the finding of the guns but also by testimony from a resident of the apartment building who saw the group leaving on the fatal night. She identified Collins, Nellum, and Hooper in court as resembling three of the men she had seen. She testified that one of the three had been wearing a wide-brimmed hat — and Nellum testified that Collins had indeed been wearing such a hat that night. Further corroboration of Nellum’s testimony came from another resident, who testified to having seen Bracy and Collins in the building that night, and from a witness who testified that Bracy had borrowed a pistol from her before the murders and that afterward, when they were in a bar and she asked for the pistol back, he had told her that he had murdered some people with it. One of the pistols found in the lake on the basis of Nellum’s tip turned out to be the pistol that she had lent Bracy. This witness also testified that in the same bar she had seen a woman give Bracy a sawed-off shotgun that Bracy had then handed to an employee of the bar, apparently for safekeeping. Bracy and Collins testified on their own behalf, denying any participation in the murders, and presented a parade of alibi witnesses of dubious credibility.
The evidence of guilt presented at the trial was compelling, and while there is a question, as we shall see, about the veracity of some of Nellum’s testimony, even if that question were resolved in the defendants’ favor we would have no basis for doubting the guilt of either Bracy or Collins. Hooper was tried separately because his confession [688]*688implicated them and the confession is further evidence, though of course not evidence presented to the jury in our case, that they really did, along with Hooper, commit the murders. Because this evidence was inadmissible it cannot be used to show that the errors of which Bracy and Collins complain are unlikely to have affected the verdict. Cf. Sullivan v. Louisiana, 508 U.S. 275, 278-79, 113 S.Ct. 2078, 2081, 124 L.Ed.2d 182 (1993); United States v. Ross, 77 F.3d 1525 (7th Cir. 1996). But the evidence that was admissible shows that they were guilty and this is important because, with a few exceptions, a person convicted in a state court may not obtain an order for a new trial from a federal court on the basis of constitutional errors committed at the trial unless the errors resulted in actual prejudice, or, equivalently, unless they substantially influenced the verdict, Brecht v. Abrahamson, 507 U.S. 619, 637-38, 113 S.Ct. 1710, 1722, 123 L.Ed.2d 353 (1993), or, in other words, were likely to have made the difference between conviction and, acquittal.
The only error that the petitioners argue requires a new trial regardless of whether it was prejudicial is that the judge who presided at their trial was later convicted of having accepted bribes from criminal defendants in several other cases (including murder cases) around the time when Bracy and Collins were tried. United States v. Maloney, 71 F.3d 645, 650-52 (7th Cir.1995). There is no suggestion that Bracy and Collins bribed or offered to bribe him. The argument rather is that Judge Maloney came down hard on criminal defendants in cases in which he was not bribed, to avoid suspicion that he was on the take, to cancel any bad impression that his acquittals might make on the voters— maybe even to make defendants desperate to bribe him, fearing he would punish them with adverse rulings if they did not. There is no evidence, but only conjecture, that Maloney actually did lean over backwards in favor of the prosecution in this or any other case in which he was not bribed; did, that is, rule against the defense only because he was taking bribes in other cases. Collins argues that evidence is unnecessary, and Bracy that if it is necessary their request for discovery should have been granted.:
A judge could be biased and yet the bias not affect the outcome of the case. But judicial bias is one of those “structural defects in the constitution of the trial mechanism,” as distinct from mere “trial errors,” that automatically entitle a petitioner for habeas corpus to a new trial. Brecht v. Abrahamson, supra, 507 U.S. at 629, 113 S.Ct. at 1717; see Sullivan v. Louisiana, supra, 508 U.S. at 278-79, 113 S.Ct. at 2081; Turney v. Ohio, 273 U.S. 510, 535, 47 S.Ct. 437, 445, 71 L.Ed. 749 (1927); Tyson v. Trigg, 50 F.3d 436, 442 (7th Cir.1995). What is bias? Defined broadly enough, it is a synonym for predisposition, and no one supposes that judges are blank slates. There are prosecution-minded judges, and defense-minded judges, and both sorts have predispositions— biases that place an added burden on one side or the other of the cases that come before them. Yet no one supposes that the existence of such biases justifies reversal in eases where no harmful errors are committed. The category of judicial bias is ordinarily limited to those predispositions, real or strongly presumed, that arise from some connection pecuniary or otherwise between thé judge and one or more of the participants in the litigation. Whether the present case even fits that mold may be doubted, but, in any event, for bias to be an automatic ground for the reversal of a criminal conviction the defendant must show either the actuality, rather than just the appearance, of judicial bias, “or a possible temptation so severe that we might presume an actual, substantial incentive to be biased.” Del Vecchio v. Illinois Dept. of Corrections, 31 F.3d 1363, 1380 (7th Cir.1994) (en banc); see Branion v. Gramly, 855 F.2d 1256, 1268 (7th Cir.1988); Margoles v. Johns, 660 F.2d 291, 296-97 (7th Cir.1981) (per curiam). In rejecting reversal on the basis of a mere appearance of partiality or bias Del Vecchio relied in part on a presumption, obviously inapplicable here, that judicial officers perform their duties faithfully. 31 F.3d at 1372-73. But that was not the core of the decision. The fundamental reason that an appearance of impropriety is not alone enough to require a new trial is that it provides only a weak basis for supposing the original trial an unreliable test of the issues [689]*689presented for decision in it. The fact that Maloney had an incentive to favor the prosecution in cases in which he was not bribed does not mean that he did favor the prosecution in such cases more than he would have done anyway.
Sometimes — this is the second half of the test that we quoted from Del Vecchio— the incentive to engage in biased behavior is so great that inquiry into the actuality of that behavior is pretermitted. Id. at 1372-73; see also In-re Murchison, 349 U.S. 133, 75 S.Ct. 623, 99 L.Ed. 942 (1955). This rule recognizes both the practical impediments to obtaining reliable evidence of a judge’s motives and the difficulty of overcoming public skepticism of judicial motives when the temptation to impropriety is great. But the automatic rule must be interpreted circumspectly, with due recognition of the cost to society of overturning the convictions of the guilty in order to vindicate an abstract interest in procedural fairness. The fact that the people for obvious practical reasons do not have judicially enforceable rights to the protection of the criminal laws (though they do have judicially enforceable rights against discriminatory withdrawal of that protection) does not warrant a court in disregarding their interests when the court is formulating rules of constitutional law. Accepting Collins’s contention would require a new trial in every case, jury and nonjury, capital and noncapital, in which a judge later found to be corrupt had presided and the defendant had been convicted, even though the judge had not been bribed by the prosecutor. (If the defendant had bribed the judge and been acquitted, the double jeopardy clause probably would not bar reprosecution, Benard v. State, 481 S.W.2d 427, 430 (Tex.Crim.App.1972)— the defendant would never have been in any actual “jeopardy.” The issue has not been definitively resolved, however, David S. Rudstein, “Double Jeopardy and the Fraudulently-Obtained Acquittal,” 60 Mo. L. Rev. 607 (1995), and obviously need not be in order to decide the present case.) Any judge who is on the take will have an incentive to adopt Judge Maloney’s alleged strategy and thus always do his best (or worst) to see to it that a defendant who does not bribe him is convicted. A principled acceptance of Collins’s argument would thus require the invalidating of tens of thousands of civil and criminal judgments, since Judge Maloney alone presided over some 6,000 cases during the course of his judicial career and he is only one of eighteen Illinois judges who have been convicted of accepting bribes. The fact that this is a death case magnifies the appearance of impropriety but is irrelevant to an issue that goes to the propriety of conviction rather than merely to that of the sentence.
The assumption underlying Collins’s argument is that a judge’s corruption is likely to permeate his judicial conduct rather than be encapsulated in the particular cases in which he takes bribes. The assumption is plausible but the consequences are unacceptable. If we were to inquire into.the motives that lead some judges to favor the prosecution, we might be led, and quickly too, to the radical but not absurd conclusion that any system of elected judges is inherently unfair because it contaminates judicial motives with base political calculations that frequently include a desire to be seen as “tough” on crime. See generally Steven P. Croley, “The Majoritari-an Difficulty: Elective Judiciaries and the Rule of Law,” 62 U.ChiL.Rev. 689, 726-29 (1995).
No precedent has been cited to us for invalidating a judge’s rulings in a case in which he is known not to have taken a bribe, simply because he took bribes in other cases. Teague v. Lane, 489 U.S. 288, 109 S.Ct. 1060, 103 L.Edüd 334 (1989), disapproved the use of novel grounds to grant relief on an application for habeas corpus. The state does not cite Teague, but we are free to apply it anyway. Caspari v. Bohlen, — U.S.-, -, 114 S.Ct. 948, 953, 127 L.Ed.2d 236 (1994); Eaglin v. Welborn, 57 F.3d 496, 499 (7th Cir.1995) (en banc). The argument for automatic reversal is not compelling even if its lack of a secure grounding in prior eases and its alarming potential irradiation of future cases are ignored. While a corrupt judge might decide to tilt sharply to the prosecution in cases in which he was not taking bribes — to right the balance as it were — it is equally possible that he would fear that by doing so he would create a [690]*690pattern of inconsistent rulings that would lead people to suspect he was on the take. When a severely prosecutorial judge sides unexpectedly with the defense in some arbitrary subset of eases, corruption is a possible explanation. If instead the judge maintains a generally pro-defendant stance, he may jeopardize his chances for reelection (Malo-ney was appointed to a vacancy, but he had to stand for election, and did so, when the term of his original appointment expired), and the number and size of the bribes he receives may be diminished because defendants will be less fearful of the consequences of not bribing him. But he may also still any suspicions that he is on the take, because his rulings in favor of defendants in cases in which he is bribed will not stand out.
This was a jury trial rather than a bench trial, moreover, and acquittals in jury trials are more likely to be blamed on the jury than on the judge. When as sometimes happens a judge campaigning for election is accused of never having convicted a rapist or sentenced a murderer to death, cf. Buckley v. Illinois Judicial Inquiry Board, 997 F.2d 224, 226 (7th Cir.1993), the reference is to bench trials, where the decision to convict or acquit is the judge’s, and to sentences handed down by judges rather than, as in capital eases in Illinois, by juries unless the defendant waives a jury. It is in cases tried to the bench that the judge as decision-maker must shoulder full responsibility for the decision. When he merely presides, his responsibility for the outcome is less. We do not understand Bra-ey and Collins to be arguing that Maloney was more likely to sentence them to death, as distinct from being more likely to rule against them during the trial, as a consequence of his taking bribes in other cases.
We are, it is true, speculating about the likely impact of Maloney’s corruption on the rulings that he made at the trial of these petitioners. We also acknowledge the possibility that the cumulative effect of those rulings was greater than we imagine. Tyson v. Trigg, supra, 50 F.3d at 439. But the defendants are speculating too. Some of Malo-ney’s rulings went against the defendants and obviously those are the ones they complain about, but they have not shown that there were so few rulings in their favor that the judge must have been biased in favor of the government. To show this would not have required an investigation, but merely a review of the transcript of the trial. It is unlikely that the specific rulings of which the defendants complain either were the product of a corrupt backward bending in the government’s favor or influenced the jury’s verdict. The Supreme Court of Illinois did not find any errors in the rulings.
The argument that a judge who accepts bribes in some eases is corrupt in all is not a sufficiently compelling empirical proposition to persuade us to treat this case as if Judge Maloney had taken a bribe from the government to convict. If the argument is rejected, ours is a ease in which there is merely an appearance of impropriety in the judge’s presiding, and an appearance of impropriety does not constitute a denial of due process. Appearance of impropriety there was. We know this because if a judge were under indictment for accepting bribes he would not be permitted to hear any eases. Ill. S.Ct. R. 56(a)(1). But without more a defendant’s conviction cannot be set aside.
The petitioners also seek discovery, so that they can try to find out whether there was actual bias by Judge Maloney at their trial. Discovery is available in a habeas corpus proceeding not as a matter of course as in an ordinary civil litigation but only if the district judge finds “good cause” to order discovery. Rule 6(a) of the Rules Governing Section 2254 Cases in the United States District Courts; East v. Scott, 55 F.3d 996, 1001 (5th Cir.1995). The petitioners want to study a large sample of Judge Maloney’s cases to see whether a pattern of favoring the prosecution in eases in which he was not bribed emerges, to depose “some of those persons and witnesses who were most intimately associated with Judge Maloney who may be able to provide material information on his behavior in eases where he didn’t get bribes,” and to get hold of any evidence that the federal government might have obtained in its prosecution of Maloney that he really did lean over backwards in favor of the government in cases in which he was not bribed— perhaps in this very case. The first proposal [691]*691would not require formal discovery at all, since Maloney’s cases are a matter of public record. The third too; in the first instance at any rate, all it would require is a perusal of the transcript of Maloney’s trial. It is true that a part of the trial record was sealed, but it was unsealed in August of 1994, so that the petitioners’ lawyers have had a year and a half to look for clues in that record. The second proposal is for a fishing, expedition. Even if the expedition discovered that Maloney did lean over backwards in favor of the prosecution in cases in which he was not bribed, in order to conceal his taking of bribes in other cases, it would not show that he followed the practice in this case. This may be a case in which any judge would have ruled in favor of the government in the instances of which the defendants complain.
A party to an ordinary civil suit need not demonstrate good cause in order to be permitted to conduct discovery. A petitioner for habeas corpus must, because collateral attack on a criminal judgment that has become final is an extraordinary remedy. Without the aid of formal discovery the petitioners’ able counsel could have (and perhaps have) studied the pattern of Judge Maloney’s rulings in cases in which he did and cases in which he did not take bribes, could have (and perhaps have) inventoried his rulings in the present case to see whether they consistently favored the prosecution, and could have (and perhaps have) studied the record of Maloney’s prosecution by the United States for clues to their theory of bias. If none of these public sources of information has yielded any evidence of bias in our case — and none has — the probability is slight that a program of depositions aimed at crooks and their accomplices and likely to be derailed in any event by real and feigned lapses of memory will yield such evidence.
We do not make light of judicial corruption. It has tainted the judicial system of Illinois, caused unjust acquittals, jeopardized convictions, tarnished the legal profession, and raised profound doubts not only about the state’s method of selecting judges but also about the entire political culture of the state. But in the circumstances of this case, corruption is not a constitutional ground for vacating the petitioners’ convictions.
The petitioners raise another issue of bias, this in the context of a claim of ineffective assistance of counsel. One of the jurors was the wife of an Illinois state judge who had once sentenced Bracy to prison for armed robbery. The defendants’ lawyer was aware of this but did not object to her being selected for the jury. Toward the end of the trial, however, the lawyer revealed to the jury that Judge Downing, the juror’s husband (though not identified as such to the jury), had once given Bracy the most severe sentence that he had ever received prior to this case. The petitioners argue that, thus reminded that her husband had dealt harshly with Bracy on a prior occasion, Mrs. Downing was bound to be prejudiced against Bracy and perhaps therefore his codefendant as well. This is too thin a speculation to justify a new trial on the ground of ineffective assistance of counsel and we are not persuaded by the proposal that in lieu of ordering a new trial we order the district court to conduct an evidentiary hearing at which Mrs. Downing would be questioned about what she was thinking when she was a member of the jury fourteen years ago. The defendants were content to have as a juror the wife of the judge who they knew had sentenced Bracy to a long prison term, and the decision to accept her is not and could not plausibly be claimed to be ineffective assistance. Only the lawyer’s slip of the tongue that revealed this fact to her could be thought ineffective assistance. But if so, the likely prejudice was too slight to warrant a new trial, or an evidentiary hearing unlikely to be any more fruitful than, discovery concerning Judge Maloney.
Another alleged irregularity at trial is the judge’s failure to strike remarks made by the prosecutors in closing argument. The worst remark to which an objection has been preserved was: “and if you think I would jeopardize my license, my family, my children, my future to put [a peripheral witness named Dorfman] on in a case and make him lie_” This is claimed to be “vouching” for the truth of the witness’s testimony, which prosecutors are not supposed to do, because [692]*692it crosses the line from advocacy to testimony. Defense counsel had accused the prosecutor of wanting to convict Collins so badly that he “made a guy [Dorfman] come in here and tell you something that he knows is not in that report.” In other words, defense counsel was accusing the prosecutor of having made a witness tell a lie. The prosecutor denied this in the passage that we have quoted. This was- not vouching for the truth of the witness’s testimony. It was denying that the prosecutor had made the witness lie. And anyway vouching is not a violation of any specific constitutional right but at most an irregularity that if shown in a particular case to have been likely to have led to the conviction of an innocent man might be held to be a denial of due process of law in that case. Rodriguez v. Peters, 63 F.3d 546, 558 (7th Cir.1995); Kappos v. Hanks, 54 F.3d 365, 367 (7th Cir.1995). No such inference is possible here.
The next question is the standard for granting an evidentiary hearing when, long after the conviction of a criminal defendant, a prosecution witness steps forward and recants a part of his testimony. Many years after the trial of Bracy and Collins, a private investigator retained by the defendants’ counsel talked to Morris Nellum. Later Nel-lum was interviewed by Bracy’s lawyer and a transcript was made of that interview. Nel-lum did not recant his testimony that Bracy and Collins had committed the murders. He merely tried to exonerate the third murderer, Hooper. But in explaining how he had come to testify against Hooper he made lurid accusations that the police had beaten Bracy and him (Nellum) and threatened him with a sledgehammer and that the prosecutors had told him to lie about such details as when he had first told the police where the pistols used in the murders had been pitched. The jury had been told that Nellum had at first denied knowing the whereabouts of the gun but not that the prosecutors had told him to lie about that denial.
Nellum later repeated a part of his recantation in a deposition that was interrupted when an Illinois prosecutor who had talked to him after the interview with defense counsel, and an Arizona prosecutor, reminded Nellum that he had said he only wanted to testify in front of a judge. In seeking an evidentiary hearing the defendants’ lawyers rely primarily on the transcript of the interview rather than on the interrupted deposition. They argue that it creates enough suspicion that the prosecutors knowingly used perjured testimony at the trial to require an evidentiary hearing to get to the bottom of Nellum’s recantation.
None of the alleged lies concerns a matter vital to the government’s case. Nellum did not deny that he was present when the victims were removed from the apartment, that he drove to the viaduct to pick up Collins, and that he saw Collins toss the pistols into Lake Michigan. Yet if the jury had thought that the police had beaten Nellum and that the prosecutors had coerced him to lie, albeit about details of the offense rather than about the involvement of the defendants, his credibility, already compromised because he was testifying in exchange for the promise of a sentence extraordinarily lenient considering that he had been an accomplice in three murders, might have been so far impaired that the jury would have disbelieved the core as well as the periphery of his testimony.
Given this possibility, we must consider what showing based on newly discovered evidence that a constitutional violation may have been committed at trial (here, the knowing use. of perjured evidence by the prosecution) is necessary before a hearing to determine the truthfulness of the evidence is required. The state argues only weakly that the petitioners should have obtained the recantation sooner, in which event it would not be newly discovered evidence in the relevant sense. Dever v. Kansas State Penitentiary, 36 F.3d 1531, 1536 (10th Cir.1994). So the request for an evidentiary hearing is not barred by a want of diligence, and furthermore it cannot be a condition of the grant of such a hearing that the movant already have in his possession all the evidence that he seeks to develop in the hearing. But equally it cannot be enough that the petitioner has found some new evidence. To reopen a criminal proceeding many years after the defendant was convicted and his conviction affirmed (which in this case occurred more [693]*693than ten years ago) is an extraordinary interference with the finality of the criminal process and requires a demonstration that a hearing would probably yield evidence that would require a new trial at which the petitioner would have a substantial chance of acquittal. In view of the passage of time since, and the disordered social milieu in which, the petitioners committed these murders, it is doubtful whether they could be retried with good prospects for an accurate verdict, even though their guilt of multiple murders committed in cold blood is not in doubt. They cannot be blamed for the fact that it has taken fifteen years for their challenge to their convictions to come to this court; so far as we can tell, they have not abused the postconviction process. But one consequence of the extraordinary delays that are tolerated in capital eases in order to minimize the risk of mistaken execution is that an order for a new trial issued toward the end of the normal postconviction process may have the practical effect of an acquittal. This consideration makes judges hesitate to order new trials on the basis of newly discovered evidence unlikely to be reliable or, even if believed, to undermine confidence in the verdict.
We are mindful that in Townsend v. Sain, 372 U.S. 293, 312, 83 S.Ct. 745, 756, 9 L.Ed.2d 770 (1963), the Supreme Court stated that “where the facts are in dispute, the federal court in habeas corpus must hold an evidentiary hearing” unless the petitioner received a full and fair hearing in state court. This passage from the Townsend opinion continues to be quoted approvingly. See, e.g., Herrera v. Collins, 506 U.S. 390, 399, 113 S.Ct. 853, 860, 122 L.Ed.2d 203 (1993). But it cannot be taken literally, since we know that the district court “may employ a variety of measures in an effort to avoid the need for an evidentiary hearing” on disputed facts, Blackledge v. Allison, 431 U.S. 63, 81, 97 S.Ct. 1621, 1633, 52 L.Ed.2d 136 (1977), including a direction to expand the record to include evidentiary materials that may resolve the factual dispute without a hearing. Id. at 82, 97 S.Ct. at 1633; Rule 7 of the Rules Governing Section 2254 Cases in the United States District Courts.
Likewise, not any old allegation of newly discovered evidence will trigger an evidentiary hearing; the allegation must be “substantial.” Townsend v. Sain, supra, 372 U.S. at 313, 83 S.Ct. at 757. That is common sense. Had Judge Hart in his order of August 24,1994, granted rather than denied the motion for an evidentiary hearing, and the hearing been held later that year, the witnesses, primarily Nellum and the prosecutors and police involved in the investigation and prosecution of the murder eases against Bracy, Collins, and Hooper, would have been testifying about events that had occurred thirteen to fourteen years earlier. So long an interval between the events and the testimony about the events would cast a pall of doubt over the reliability of the testimony. Ours is not the typical situation in which an evidentiary hearing is granted in a posteonviction proceeding. Not only is the interval typically shorter, as in Daniels v. United States, 54 F.3d 290 (7th Cir.1995), and Barkauskas v. Lane, 878 F.2d 1031, 1034 (7th Cir.1989), but when it is as long as it would be here this is usually because the state failed to give the petitioner a full and fair hearing, and the state must not be rewarded for its own denials of due process. When the request for a federal evidentiary hearing is based on evidence gathered long after the event — too late to be presented to the state courts, which cannot be faulted for having failed to provide the petitioner with an evidentiary hearing — the federal court is entitled to insist, as a precondition to granting a hearing, that the petitioner demonstrate that he have a “colorable” claim, meaning by this not that it be nonfrivolous on its face (the usual meaning of “colorable”) but that there be some reason to think it valid. Siripongs v. Calderon, 35 F.3d 1308, 1314 (9th Cir.1994).
Even if Nellum testified under oath to all that he had told Bracy’s lawyer and even if his testimony were believed over that of prosecutors and police almost certain to testify contrary to Nellum, thus establishing that the prosecution made knowing use of perjured testimony, a new trial would not be warranted. The knowing use of perjured testimony by the prosecution, although a [694]*694very serious infringement of the constitutional rights of a criminal defendant, is not an automatic basis for a new trial. There must be a reasonable likelihood that the violation affected the verdict. United States v. Agurs, 427 U.S. 97, 103, 96 S.Ct. 2392, 2397, 49 L.Ed.2d 342 (1976); Giglio v. United States, 405 U.S. 150, 154, 92 S.Ct. 763, 766, 31 L.Ed.2d 104 (1972); United States v. Boyd, 55 F.3d 239, 243 (7th Cir.1995). The evidence of Bracy’s and Collins’s guilt was very powerful, and evidence brought out at the evidentiary hearing which if repeated at a new trial would merely cast doubt on Nel-lum’s credibility would be unlikely to sway a jury. We are imagining a new trial at which all the evidence is the same except that Nellum changes some of the details of his earlier testimony, and, when the changes are thrown in his face, makes lurid accusations against the police and the prosecutors. The core of Nellum’s testimony would be unaffected, and would be richly corroborated by the testimony of the other witnesses whose evidence we summarized at the beginning of this opinion (assuming those witnesses can be found and persuaded to testify or the transcript of their original testimony is admitted at the second trial), and by the guns. Of course we cannot be certain that once Nel-lum started down the recantation path he would go no further than he has done to date. Having completed his bargained for sentence long ago he has no incentive to cooperate with the prosecution. But the more sweeping his recantation, the less credible it is likely to be.
Prosecutors frequently rely heavily on the testimony of members of the criminal class, such as Nellum, who did not “go straight” after completion of his light sentence for his role in the murders. They have no choice. These witnesses from the criminal demimonde are not only unreliable witnesses, as the defendants’ counsel emphasized to the jury in asking them to disbelieve Nellum’s testimony; they are unreliable 'people. We would be imperiling the punishment of dangerous criminals if we established a precedent that would invite the lawyers for convicted defendants (especially in capital cases, not only because of the stakes involved but also because few prisoners other than those under sentence of death are represented in postconviction proceedings) to pester these witnesses to recant many years after they had testified. We do not suggest that the lawyers would act unethically. In the American system of justice the zealous representation of a client is a duty, not an ethical lapse. And in some cases the recantations would be genuine and material and if so they would be a proper basis for ordering a new trial despite the passage of time. But we are given no reason to suppose that Nellum’s recantations are either genuine or material. His original testimony was amply corroborated and his recantation preserved the core of the original testimony intact. And while the interview with Bracy’s lawyer occurred in 1992 and the abortive deposition the following year, in the more than two years that have elapsed since the deposition the defendants’ current lawyers, whose energy and ability cannot be doubted, have failed to obtain any corroboration for Nelhim’s recantation.
We cannot of course determine the credibility of Nellum’s recantation; we can only express our suspicions. These are germane, however, because an evidentiary hearing need not be granted on the basis of newly discovered evidence presented many years after the defendant’s conviction became final unless there is a good reason to expect the hearing to result in an order for a new trial. We cannot find this proposition clearly articulated in any ease, but it seems to us consistent with the results in the cases and sound as a matter of first principles. The circumstances of Nellum’s recantation, the strength of the original evidence, and the fact that the core of his testimony was not recanted persuade us that the request for an evidentiary hearing was properly denied.
We turn to the defendants’ challenge to the sentencing hearing. They complain that their lawyers were not given enough time to obtain evidence of mitigating circumstances for submission to the jury at that hearing. They asked for a continuance immediately after the jury convicted their clients but it was denied. It was properly denied. In Illinois the jury that determines the defendant’s guilt in a capital case also [695]*695determines whether he is to be sentenced to death, unless as in this case the defendant waives his right to be sentenced by the jury. Since it is the same jury, the sentencing hearing perforce follows immediately upon the trial, as it also does when the jury is waived. Defense lawyers know all this and therefore if they wish to gather evidence of mitigating circumstances they must do so before the trial ends, because they will have no time to do so after the trial ends. But in this case the defendants’ lawyers dropped the ball and the defendants argue in the alternative that by doing so the lawyers failed to provide competent assistance of counsel at the sentencing hearing.
Perhaps so, but we need not decide this; for there was no prejudice, and without proof of prejudice a claim of ineffective assistance of counsel cannot succeed. The defendants’ current lawyers have turned up no mitigating circumstances that might have been put before the jury with a fair chance of success. It is true that one of the aggravating circumstances, though limited to Bracy, is that shortly after participating in the triple murder in Chicago he participated in a double murder in Arizona. He had not yet been tried for those murders but the woman who was the wife and daughter of the murder victims and the state’s main witness was brought to Chicago to testify in the sentencing hearing that Bracy was indeed one of the Arizona murderers. The defendants argue that their lawyers failed to present alibi evidence that might have persuaded the jury that Bracy had not committed those murders. We have little patience with this argument. Bracy was later convicted and sentenced to death for the Arizona murders, and the burden of proof that the state bore in convicting him was of course higher than it bore in the sentencing hearing, for the existence of an aggravating circumstance need only be proved by a preponderance of the evidence, provided that at least one such circumstance has been proved beyond a reasonable doubt, thus making the defendant eligible for the death sentence. 720 ILCS 5-9-1 (f); Free v. Peters, 12 F.3d 700, 703 (7th Cir.1993); People v. Ramey, 152 Ill.2d 41, 178 Ill.Dec. 19, 35-36, 604 N.E.2d 275, 291-92 (1992). So if all the evidence that had been before the Arizona jury had been presented to the jury in Chicago, the Chicago jury surely would have found that Bracy had committed the Arizona murders. Not all that evidence was presented, so maybe the jury would have been swayed by alibi evidence that we now know is false. But no principle of justice authorizes throwing out a sentence on the ground that the sentence might have been different had the defendant been allowed to present false testimony. Lockhart v. Fretwell, 506 U.S. 364, 369-71, 113 S.Ct. 838, 842-43, 122 L.Ed.2d 180 (1993); Nix v. Whiteside, 475 U.S. 157, 175-76, 106 S.Ct. 988, 998-99,89 L.Ed.2d 123 (1986).
Bracy objects to some remarks by the prosecutor at the closing argument at the sentencing hearing. The worst was, “Some of us went to Yiet Nam and had to kill for this country, and I will be damned if anybody is going to tell me that what we did in Viet Nam or in any other war was a violation of the Fifth Commandment of the Bible.” This was a response to defense counsel’s argument that to sentence the defendants to death would violate the Ten Commandments, one of which of course is, “Thou shalt not kill.” The prosecutor was pointing out, with unnecessary but not we think fatally prejudicial emphasis, that there is such a thing as justified homicide and that the execution of a duly convicted and sentenced murderer is, plausibly, an illustration of it.
Collins objects to the exclusion of a prospective juror on the basis of his acknowledging in response to a question by the judge that he would “probably” not consider imposing the death penalty. The defendants’ lawyer did not object to excusing this prospective juror for cause. We do not think this was error, and certainly not error of constitutional proportions, given the absence of an objection. Wainwright v. Witt, 469 U.S. 412, 424-26, 429, 105 S.Ct. 844, 852-54, 855 83 L.Ed.2d 841 (1985).
The petitioners present other issues, but they either have too little merit to warrant discussion or they are foreclosed by previous decisions of this court that we are not given any reason to reexamine. We have considered the possibility that the cumulative effect [696]*696of the various irregularities alleged tips the balance in favor of a new trial or a new sentencing hearing but have concluded that it does not. We regret that Judge Maloney presided over the petitioners’ trial but we do not think that the Constitution relieves them from the judgments that the Illinois courts have rendered.
AFFIRMED.